Low and high energy solutions of oscillatory non-autonomous Schrödinger equations with magnetic field

Abstract

We are concerned with the mathematical and asymptotic analysis of solutions to the following nonlinear problem $\begin{array}{l} \{\begin{matrix} - Δ_{A} u = λ β (x) | u |^{q} u + f (| u |) u & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} \end{array}$ where $Δ_{A} u$ is the magnetic Laplace operator, $Ω \subset R^{N}$ is a smooth bounded domain, $A : Ω \mapsto R^{N}$ is the magnetic potential, $u : Ω \mapsto C$ , λ is a real parameter, $β \in L^{\infty} (Ω, R)$ is an indefinite potential, q is nonnegative, and $f : [0, + \infty) \mapsto R$ is a reaction that oscillates either in a neighborhood of the origin or at infinity. We analyze two distinct cases, in close relationship with the oscillatory growth of the reaction. Additionally, we give asymptotic estimates for the norm of the solutions in related function spaces.

Keywords

Magnetic Laplace operator blow-up low- and high-energy solutions oscillation variational methods

1. Introduction

This paper deals with the qualitative and asymptotic analysis of solutions to the following class of nonlinear Dirichlet problems: $\begin{array}{l} (1.1) & \{\begin{matrix} - Δ_{A} u = λ β (x) | u |^{q} u + f (| u |) u & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} \end{array}$ where $Δ_{A} u : = {(\nabla - i A)}^{2} u$ is the magnetic Laplace operator. We suppose that Ω is a smooth bounded domain of $R^{N}$ ( $N ⩾ 2$ ), $A : Ω \subset R^{N} \mapsto R^{N}$ is the magnetic potential, $u : Ω \mapsto C$ , λ is a real parameter, $β \in L^{\infty} (Ω, R)$ is an indefinite potential, q is nonnegative, i is the imaginary unit, and $f : [0, + \infty) \mapsto R$ is a reaction that oscillates either near the origin or at infinity.

There is a lot of literature on the competition phenomena for elliptic equations without magnetic potential in different situations, mainly provided that $A \equiv 0$ . We refer to Ambrosetti–Brezis–Cerami [5] who considered a semilinear elliptic equation involving a concave-convex reaction; see also [6,11,15,18,20,21,29] and the references therein. Many papers have investigated the interactions between distinct types of nonlinearities involving singular terms; see [1,13,16,22] and the references therein.

Problem (1.1) is related with the standing wave solutions $ϕ (x, t) : = e^{- i \frac{E}{ℏ} t} u (x)$ , where E is a real number, of the following nonlinear evolution system $\begin{array}{l} (1.2) & i ℏ \frac{\partial ϕ}{\partial t} = {(\frac{ℏ}{i} \nabla - A (z))}^{2} ϕ + U (z) ϕ - l (| ϕ |^{2}) ϕ, z \in Ω \subset R^{N} . \end{array}$ By straightforward computation wee see that ϕ is a solitary wave for problem (1.2) if and only if u is a solution of the following stationary problem $\begin{array}{l} (1.3) & {(\frac{ℏ}{i} \nabla - A (z))}^{2} u + V (z) u = l (| u |^{2}) u, z \in Ω, \end{array}$ where $V (z) = U (z) - E$ . From a physical point of view, the transition from Quantum Mechanics to Classical Mechanics can be formally performed by sending to zero the Planck constant. Therefore, it is very crucial to investigate the existence of such solutions and their shape in the semiclassical limit, namely as $ℏ \to 0$ .

In recent decades, the perturbations of (1.3) and of inequality value problems without magnetic fields (namely, if $A \equiv 0$ and $h = 1$ ) have been extensively considered. We recall the following basic example $\begin{matrix} (1.4) & \{\begin{matrix} - Δ u = | u |^{q - 2} u & in Ω, \\ u = 0, & on \partial Ω, \end{matrix} \end{matrix}$ where Ω is a smooth bounded domain of $R^{N}$ ( $N ⩾ 2$ ) and $2 < q < 2^{*}$ , $2^{*} = \infty$ if $N = 2$ and $2^{*} = 2 N / (N - 2)$ if $N ⩾ 3$ . Using symmetric version of the Mountain Pass Theorem (see Ambrosetti–Rabinowitz [7]) we see that problem (1.4) possesses infinitely many solutions in $H_{0}^{1} (Ω, R)$ . Finally, a natural idea is to see what happens if the above problem is affected by some kind of perturbation. Consider the following problem $\begin{matrix} (1.5) & \{\begin{matrix} - Δ u = | u |^{q - 2} u + g (x) & in Ω, \\ u = 0, & on \partial Ω . \end{matrix} \end{matrix}$ Bahri–Berestycki [10] and Struwe [30] have proved independently that there is $q_{0} < 2^{*}$ such that for any $g \in L^{2} (Ω, R)$ , problem (1.5) still has infinitely many solutions, if $2 < q < q_{0}$ . Furthermore, Bahri [9] has established that for any $2 < q < q_{0}$ there is a dense open set of $g \in H^{- 1} (Ω, R)$ for which problem (1.5) admits infinitely many solutions.

This paper is motivated by some recent works concerning the following Schrödinger equations with magnetic field: $\begin{array}{l} (1.6) & - Δ_{A} u + V (x) u = | u |^{p - 2} u, x \in Ω \subset R^{N}, N ⩾ 2 . \end{array}$ Here, $u : Ω \mapsto C$ , $- Δ_{A} u : = {(- i \nabla + A)}^{2} u$ , $2 < p ⩽ 2^{*}$ , where $2^{*} = + \infty$ if $N = 2$ and $2^{*} = 2 N / (N - 2)$ if $N ⩾ 3$ , $A : Ω \mapsto R^{N}$ and $V : Ω \mapsto R$ are smooth.

To the best of our knowledge, the first paper in which problem (1.6) has been studied seems to be Esteban–Lions [19]. They have used the concentration-compactness principle and minimization arguments to get the existence of solutions for $N = 2$ and $N = 3$ . More recently, applying constrained minimization and a minimax-type argument, Arioli–Szulkin [8] considered the Schrödinger equation in a magnetic field. They established the existence of non-trivial solutions both in the critical and in the subcritical case, provided that some technical conditions relating to V and A were assumed. We also refer to [4,17] for other results related to the problem (1.1) in the presence of the magnetic field when the nonlinearity has a subcritical growth. We mention the works [3,23] for the critical case and also refer to the recent papers [24,27,31] for the study of various classes of PDEs with magnetic potential.

Molica Bisci–Rădulescu–Servadei [26] considered the following elliptic problem $\begin{array}{l} \{\begin{matrix} - div D (x, \nabla u) = λ β (x) u^{q} + f (u) & in Ω, \\ u ⩾ 0 & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N}$ , $N ⩾ 3$ , is a bounded domain smooth boundary $\partial Ω$ , $u : Ω \mapsto R$ , $q ⩾ 0$ and $λ \in R$ are parameters, $β \in L^{\infty} (Ω, R)$ , while $D : \overline{Ω} \times R^{N} \mapsto R^{N}$ and $f : [0, + \infty) \mapsto R$ are continuous functions. Applying variational and topological methods, they showed that the number of solutions of the problem is influenced by the competition between the power-type nonlinearity $u^{q}$ and the oscillatory reaction f.

In view of the results of Molica Bisci–Rădulescu–Servadei [26], it is natural to ask if the similar kind of result holds for the problem with magnetic field. The aim of this paper is devoted to presenting a positive answer to this question. In such a way, we give a relationship between the number of solutions of problem (1.1) with the reaction $| u |^{q}$ and the oscillatory term f. To some extent, the oscillatory term in our paper f is more extensive than the oscillatory term f considered in the paper of Molica Bisci–Rădulescu–Servadei [26]. Furthermore, as we will see later, thanks to the appearance of the magnetic potential A, problem (1.1) cannot be converted into a pure real-valued problem, hence we must deal directly with a complex-valued problem, which leads to some new technical difficulties and challenges. Our problem is more complicated than the problem without magnetic potential and we need to add technical estimates and employ new idea.

Motivated by the above works, when f oscillates either near the origin or at infinity we consider the number and the behavior of the solutions for problem (1.1) in this paper. In order to achieve our goal, we will employ variational and topological methods together with truncation and comparison techniques. In the subsequent part, we will introduce our main results in two cases, that is, the nonlinearity f oscillates near the origin and at the infinity, respectively. Finally, we point out that the sign of the coefficient $β \in L^{\infty} (Ω, R)$ in problem (1.1) can be uncertain, for example, see [2,12,18,26] and references therein. Since we consider the functions are complex-valued, we emphasize that it is necessary to carefully analyze some estimates and truncation functions used in this paper, see for instance Lemma 2.2, Corollary 2.3 and Theorems 3.1 and 5.1.

This paper is organized as follows. In Section 2 we first outline the variational framework for problem (1.1) and give some important estimates, then we are also ready to show the main results of this paper. In Section 3 we study an auxiliary problem and for it we show the existence of solutions by direct minimization. Finally, in Section 4 we investigate problem (1.1) in the presence of an oscillation term near zero, while Section 5 is devoted to the case of oscillations at infinity. We refer to the monograph by Papageorgiou, Rădulescu and Repovš [28] for some of the abstract methods used in this paper.

2. Variational framework and main results

In this section, we first outline the variational framework for problem (1.1) and give some important estimates.

Let U be an open set in $R^{N}$ . For a function $u : U \mapsto C$ , let us define $\begin{matrix} H^{1} (U, C) : = {u \in L^{2} (U, C) : | \nabla u | \in L^{2} (U, R)} . \end{matrix}$ The space $H^{1} (U, C)$ is a Hilbert space endowed with the scalar product $\begin{matrix} {⟨ u, v ⟩}_{U} : = Re \int_{U} (\nabla u \overline{\nabla v} + u \overline{v}) d x, \end{matrix}$ where Re and the bar denote the real part of a complex number and the complex conjugation, respectively. Moreover, we denote by $‖ u ‖_{H^{1} (U, C)}$ the norm induced by this inner product.

Let the space $H_{0}^{1} (U)$ be the closure of $C_{0}^{\infty} (U, C)$ in $H^{1} (U, C)$ . If $U \neq R^{N}$ , then $C_{0}^{\infty} (U, C)$ is not necessarily dense in $H^{1} (U, C)$ . The subspace $H_{0}^{1} (U)$ of $H^{1} (U, C)$ is frequently applied to study differential equations with “zero boundary conditions” on $\partial U$ , the boundary of U. We refer to Lieb and Loss [25] for more details.

Let Ω be a smooth bounded domain of $R^{N}$ , $N ⩾ 2$ . Note that, for all $u \in H_{0}^{1} (Ω, C)$ we have $\begin{matrix} \int_{Ω} {| \nabla | u | |}^{2} d x ⩽ \int_{Ω} | \nabla u |^{2} d x . \end{matrix}$ By this and Poincaré inequality we get that $\begin{array}{l} (2.1) & \int_{Ω} | u |^{2} d x ⩽ C_{0} \int_{Ω} {| \nabla | u | |}^{2} d x ⩽ C_{0} \int_{Ω} | \nabla u |^{2} d x, \end{array}$ for some $C_{0} > 0$ .

So, we can define the norm of $H_{0}^{1} (Ω, C)$ as follows $\begin{matrix} ‖ u ‖_{H_{0}^{1} (Ω, C)} : = {(\int_{Ω} | \nabla u |^{2} d x)}^{\frac{1}{2}} . \end{matrix}$ Indeed, by (2.1) we see that the norms of $‖ \cdot ‖_{H^{1} (Ω, C)}$ and $‖ \cdot ‖_{H_{0}^{1} (Ω, C)}$ are equivalent.

Let $H_{A} (Ω)$ denote the closure of $C_{0}^{\infty} (Ω, C)$ under the scalar product $\begin{matrix} {⟨ u, v ⟩}_{Ω} : = Re \int_{Ω} \nabla_{A} u \overline{\nabla_{A} v} d x, \end{matrix}$ where $\begin{matrix} \nabla_{A} u : = - i \nabla u - A u . \end{matrix}$ The space $H_{A} (Ω)$ is a Hilbert space equipped with norm $\begin{matrix} ‖ u ‖_{H_{A} (Ω)} : = {(\int_{Ω} | \nabla_{A} u |^{2} d x)}^{\frac{1}{2}} . \end{matrix}$

We give the following well-known diamagnetic inequality which is proved by Esteban–Lions [19] and Lieb–Loss [25, Theorem 7.21]:

Lemma 2.1.
Assume that $u \in H_{A} (Ω)$ , then $| u | \in H_{0}^{1} (Ω, R)$ and $\begin{matrix} (2.2) & | \nabla | u | (x) | = | Re (\nabla u \frac{\overline{u}}{| u |}) | = | Re ((\nabla u - i A u) \frac{\overline{u}}{| u |}) | ⩽ | \nabla_{A} u (x) |, \end{matrix}$ for a.e. $x \in Ω$ , where $| u |$ defined by $| u | (x) = | u (x) |$ .

Furthermore, the embedding $H_{A} (Ω) ↪ L^{μ} (Ω, C)$ is continuous for each $1 ⩽ μ ⩽ 2^{}$ (for any $μ \in [1, + \infty)$ if $N = 2$ ) and it is compact for $1 ⩽ μ < 2^{}$ , where $2^{}$ denotes the critical Sobolev exponent, $2^{} = \infty$ if $N = 2$ and $2^{} = 2 N / (N - 2)$ if $N ⩾ 3$ . In order to conclude our main results, we assume that A satisfies the following hypothesis in this paper:
$(A_{1})$
$A \in L^{υ} (Ω, R^{N})$ , where $υ > N ⩾ 2$ .

We remark that the spaces $H_{A} (Ω)$ and $H_{0}^{1} (Ω, C)$ are incomparable, more exactly, in general $H_{A} (Ω) ⊈ H_{0}^{1} (Ω, C)$ and $H_{0}^{1} (Ω, C) ⊈ H_{A} (Ω)$ . Additionally, they are equivalent if hypothesis $(A_{1})$ is fulfilled, as stated in the following lemma.
Lemma 2.2.
Assume that A satisfy* $(A_{1})$ , then $u \in H_{A} (Ω)$ if and only if $u \in H_{0}^{1} (Ω, C)$ . Moreover, there are $c_{1}, c_{2} > 0$ depending only on Ω such that $\begin{matrix} c_{1} ‖ u ‖_{H_{0}^{1} (Ω, C)} ⩽ ‖ u ‖_{H_{A} (Ω)} ⩽ c_{2} ‖ u ‖_{H_{0}^{1} (Ω, C)} \end{matrix}$ for all $u \in H_{A} (Ω)$ .
Proof.
Firstly, we will prove that $\begin{matrix} u \in H_{0}^{1} (Ω, C) \Rightarrow u \in H_{A} (Ω) and ‖ u ‖_{H_{A} (Ω)} ⩽ c_{2} ‖ u ‖_{H_{0}^{1} (Ω, C)}, \end{matrix}$ for some $c_{2} > 0$ . Since $u \in H_{0}^{1} (Ω, C)$ and then $| \nabla | u | (x) | ⩽ | \nabla u (x) |$ a.e. $x \in Ω$ . Thus we see that $| u | \in H_{0}^{1} (Ω, R)$ , and then we can get that the Sobolev embedding $H_{0}^{1} (Ω, C) ↪ L^{μ} (Ω, C)$ is continuous for each $1 ⩽ μ ⩽ 2^{}$ (for any $μ \in [1, + \infty)$ if $N = 2$ ) and it is compact for $1 ⩽ μ < 2^{}$ . Combining this, $(A_{1})$ and the Hölder inequality we have $\begin{array}{l} (2.3) & \int_{Ω} | A u |^{2} d x ⩽ ‖ A ‖_{L^{υ} (Ω)}^{2} ‖ u ‖_{L^{2 υ / (υ - 2)} (Ω)}^{2} ⩽ c_{3} ‖ A ‖_{L^{υ} (Ω)}^{2} ‖ u ‖_{H_{0}^{1} (Ω, C)}^{2}, \end{array}$ for some $c_{3} > 0$ . Hence we get $\begin{matrix} \int_{Ω} | \nabla_{A} u |^{2} d x ⩽ \int_{Ω} 2 (| \nabla u |^{2} + | A u |^{2}) d x ⩽ c_{2} ‖ u ‖_{H_{0}^{1} (Ω, C)}^{2}, \end{matrix}$ for some $c_{2} > 0$ .

Finally, we will prove that $\begin{matrix} u \in H_{A} (Ω) \Rightarrow u \in H_{0}^{1} (Ω, C) and ‖ u ‖_{H_{A} (Ω)} ⩾ c_{1} ‖ u ‖_{H_{0}^{1} (Ω, C)}, \end{matrix}$ for some $c_{1} > 0$ . Note that $\begin{matrix} \int_{Ω} | \nabla_{A} u |^{2} d x = \int_{Ω} | \nabla u |^{2} d x + 2 Re \int_{Ω} i A \overline{u} \nabla u d x + \int_{Ω} | A u |^{2} d x . \end{matrix}$ It remains to show that $\begin{matrix} \int_{Ω} | \nabla u |^{2} d x + 2 Re \int_{Ω} i A \overline{u} \nabla u d x + \int_{Ω} | A u |^{2} d x ⩾ ε \int_{Ω} | \nabla u |^{2} d x \end{matrix}$ for some positive ε. Arguing indirectly, we find $u_{n}$ with $‖ u_{n} ‖_{H_{0}^{1} (Ω, C)} = 1$ such that $\begin{array}{l} (2.4) & \int_{Ω} | \nabla u_{n} |^{2} d x + 2 Re \int_{Ω} i A \overline{u_{n}} \nabla u_{n} d x + \int_{Ω} | A u_{n} |^{2} d x ⩽ \frac{1}{n} . \end{array}$ Up to a subsequence, still denoted by itself, we may assume that $u_{n} ⇀ u$ in $H_{0}^{1} (Ω, C)$ . Combining $(A_{1})$ , the inequality $‖ u_{n} ‖_{H_{A} (Ω)} ⩽ 1$ and the facts that the embedding $H_{0}^{1} (Ω, C) ↪ L^{μ} (Ω, C)$ and the embedding $H_{A} (Ω) ↪ L^{μ} (Ω, C)$ are compact for $1 ⩽ μ < 2^{}$ we know that $\begin{array}{l} (2.5) & lim_{n \to \infty} Re \int_{Ω} i A \overline{u_{n}} \nabla u_{n} d x = lim_{n \to \infty} Re \int_{Ω} i A \overline{u} \nabla u d x, \\ (2.6) & lim_{n \to \infty} \int_{Ω} | A u_{n} |^{2} d x = lim_{n \to \infty} \int_{Ω} | A u |^{2} d x . \end{array}$

Therefore, we have $\begin{matrix} \int_{Ω} | \nabla_{A} u |^{2} d x = \int_{Ω} | \nabla u |^{2} d x + 2 Re \int_{Ω} i A \overline{u} \nabla u d x + \int_{Ω} | A u |^{2} d x ⩽ 0 . \end{matrix}$ If $u \neq 0$ , this is a contradiction, and if $u = 0$ , from (2.4), (2.5) and (2.6) we deduce that $\begin{matrix} 1 = lim_{n \to \infty} \int_{Ω} | \nabla u_{n} |^{2} d x = 0, \end{matrix}$ a contradiction again.

The proof of Lemma 2.2 is now complete. □

Next, we would like to discuss another useful estimate.
Corollary 2.3.
Assume that hypothesis* $(A_{1})$ is fulfilled, then there is a constant $M_{0} > 1$ such that $\begin{array}{l} {‖ | u | ‖}_{H_{A} (Ω)} ⩽ M_{0} ‖ u ‖_{H_{A} (Ω)}, \end{array}$ for every $u \in H_{A} (Ω)$ . Here, $M_{0}$ is independent of u.
Proof.
Indeed, combining the hypothesis $(A_{1})$ , (2.2), (2.3) and Lemma 2.2 we have $\begin{array}{l} {‖ | u | ‖}_{H_{A} (Ω)}^{2} & = \int_{Ω} {| \nabla | u | |}^{2} d x + \int_{Ω} | A u |^{2} d x \\ ⩽ \int_{Ω} | \nabla_{A} u |^{2} d x + \int_{Ω} | A u |^{2} d x \\ ⩽ ‖ u ‖_{H_{A} (Ω)}^{2} + c_{3} ‖ A ‖_{L^{υ} (Ω)}^{2} ‖ u ‖_{H_{0}^{1} (Ω, C)}^{2} \\ ⩽ (\frac{c_{3}}{c_{1}^{2}} ‖ A ‖_{L^{υ} (Ω)}^{2} + 1) ‖ u ‖_{H_{A} (Ω)}^{2} \\ = M_{0}^{2} ‖ u ‖_{H_{A} (Ω)}^{2}, \end{array}$ for every $u \in H_{A} (Ω)$ and for some constant $M_{0} = {(\frac{c_{3}}{c_{1}^{2}} ‖ A ‖_{L υ (Ω)}^{2} + 1)}^{1 / 2} > 1$ .

The proof of Corollary 2.3 is now complete. □

Now, let us define the space X as follows: $\begin{matrix} X : = {u = e^{i A (0) \cdot x} v (x) : v \in H_{0}^{1} (Ω, R)} \subset H_{0}^{1} (Ω, C) . \end{matrix}$ Using Lemma 2.2 we see that the space X is a Hilbert subspace equipped with the norm $‖ u ‖_{H_{A} (Ω)}$ . By Lemma 2.2 we deduce that $X \subset H_{A} (Ω)$ . In particular, $X = H_{0}^{1} (Ω, R)$ if $A (0) = 0$ .

This section is also devoted to the main results of this paper, we will show the existence of infinitely many solutions for problem (1.1) when f oscillates near the origin and $q ⩾ p - 1$ or f oscillates at infinity and $0 < q ⩽ p - 1$ . We also prove the existence of at least a finite number of solutions when f oscillates near the origin and $0 < q < p - 1$ and f oscillates at infinity and $q > p - 1$ . Where $1 ⩽ p < 2^{*} - 1$ is related to the growth of f either near the origin or at the infinity.

Let $\hat{g}, f : [0, + \infty) \mapsto R$ satisfy $\hat{g} (s) = s f (s)$ . For notational simplicity, in the sequel by $s f (s) \in C ([0, + \infty], R)$ we mean that $\hat{g} \in C ([0, + \infty], R)$ .

In all these cases we suppose that $f : [0, + \infty) \mapsto R$ is a function and $s f (s) \in C ([0, + \infty), R)$ . Also, F is defined by the following function $\begin{array}{l} (2.7) & F (s) : = \int_{0}^{s} f (t) t d t \end{array}$ for every $s ⩾ 0$ .
2.1. Oscillatory behavior near origin

We first assume that the following hypotheses are fulfilled: $\begin{array}{l} (2.8) & \underset{s \to 0^{+}}{lim inf} \frac{f (s)}{s^{p - 1}} : = - ℓ_{0} \in [- \infty, 0), \\ (2.9) & - \infty < \underset{s \to 0^{+}}{lim inf} \frac{F (s)}{s^{2}} ⩽ \underset{s \to 0^{+}}{lim sup} \frac{F (s)}{s^{2}} = + \infty, \end{array}$ where $1 ⩽ p < 2^{*} - 1$ is a parameter and the function F is defined in (2.7).

As an example, let us consider the function $\begin{array}{l} f (s) s = \{\begin{matrix} (α + 1) s^{α} (1 - sin s^{- δ}) + δ s^{α - δ} cos s^{- δ} - (p + 1) γ s^{p} & if s > 0, \\ 0 & if s = 0, \end{matrix} \end{array}$ where α, δ and γ satisfy $0 < δ < α < 1 ⩽ p$ and $γ > 0$ . Note that $s f (s)$ is continuous in $[0, + \infty)$ . We define $\begin{matrix} F (s) = \int_{0}^{s} f (t) t d t = s^{α + 1} (1 - sin s^{- δ}) - γ s^{p + 1}, s > 0 . \end{matrix}$

Another example is given by $\begin{array}{l} f (s) s = \{\begin{matrix} (α + 1) s^{α} {cos}^{2} s^{- δ} - 2 δ^{α - δ} cos s^{- δ} sin s^{- δ} - (p + 1) γ s^{p} & if s > 0, \\ 0 & if s = 0, \end{matrix} \end{array}$ where α, δ and γ satisfy $0 < δ < α < 1 ⩽ p$ and $γ > 0$ . Thanks to these choices of the parameter, the function $s f (s)$ is continuous in $[0, + \infty)$ . We define the function F as follows $\begin{matrix} F (s) = \int_{0}^{s} f (t) t d t = s^{α + 1} {cos}^{2} s^{- δ} - γ s^{p + 1}, s > 0 . \end{matrix}$ In both examples, f and F satisfy assumptions (2.8) and (2.9). We refer to Molica Bisci–Rădulescu–Servadei [26] for more details.

The first main result of this paper establishes the existence of a sequence of low energy solutions, that is, weak solutions with lower and lower energies.

Theorem 2.4.
Assume that A satisfies $(A_{1})$ , $β \in L^{\infty} (Ω, R)$ and $s f (s) \in C ([0, + \infty), R)$ satisfies hypotheses ( 2.8 ) and ( 2.9 ). Assume that one of the following conditions is fulfilled:
$q = p - 1$ , $ℓ_{0} \in (0, + \infty)$ and $λ β (x) < λ_{0}$ a.e. $x \in Ω$ for some $λ_{0} \in (0, ℓ_{0})$ or

$q = p - 1$ , $ℓ_{0} = + \infty$ and $λ \in R$ is arbitrary or

$q > p - 1$ and $λ \in R$ is arbitrary.
Then there is a sequence ${u_{j}}_{j}$ in X of weak solutions of problem ( 1.1 ) such that $\begin{array}{l} (2.10) & lim_{j \to \infty} ‖ u_{j} ‖_{H_{A} (Ω)} = lim_{j \to \infty} ‖ u_{j} ‖_{L^{\infty} (Ω, C)} = 0 . \end{array}$

If $1 < q < p - 1$ , then for every $k \in N$ there is $Λ_{k} > 0$ such that problem ( 1.1 ) has at least k different weak solutions $u_{1}, \dots, u_{k} \in X$ such that $\begin{array}{l} (2.11) & ‖ u_{j} ‖_{H_{A} (Ω)} ⩽ 1 / j and ‖ u_{j} ‖_{L^{\infty} (Ω, C)} ⩽ 1 / j \end{array}$ provided that $| λ | < Λ_{k}$ .

Hypothesis (2.8) implies the existence of solutions for problem (1.1), while hypothesis (2.9) is applied to infer several pieces of information about the number of solutions. We point out that claim $(c)$ includes also the case when the power q is critical or supercritical, that is, the case when $q ⩾ 2^{*}$ .
2.2. Oscillatory behavior at infinity

We also first suppose that the following hypotheses are fulfilled: $\begin{array}{l} (2.12) & \underset{s \to + \infty}{lim inf} \frac{f (s)}{s^{p - 1}} : = - ℓ_{\infty} \in [- \infty, 0), \\ (2.13) & - \infty < \underset{s \to + \infty}{lim inf} \frac{F (s)}{s^{p + 1}} ⩽ \underset{s \to + \infty}{lim sup} \frac{F (s)}{s^{p + 1}} = + \infty, \end{array}$ where $1 ⩽ p < 2^{*} - 1$ is parameter and the function F is defined in (2.7).

As an example, let us consider the following function $\begin{matrix} f (s) = (α + 1) s^{α - 1} (1 - sin s^{δ}) + δ s^{α + δ - 1} cos s^{δ} - (p + 1) γ s^{p - 1}, \end{matrix}$ where α, δ and γ are such that $α > p$ , $δ > 0$ and $γ > 0$ . Owing to these choices of the parameter, the function $s f (s)$ is continuous in $[0, + \infty)$ . The function F is given by $\begin{matrix} F (s) = \int_{0}^{s} f (t) t d t = s^{α + 1} (1 - sin s^{δ}) - γ s^{p + 1}, s > 0 . \end{matrix}$ Also in this case, f and F satisfy assumptions (2.12) and (2.13). We also refer to Molica Bisci–Rădulescu–Servadei [26] for more details.

The following counterpart of Theorem 2.4 yields the existence of high energy solutions, that is, weak solutions with higher and higher energies.

Theorem 2.5.
Assume that A satisfies $(A_{1})$ , $β \in L^{\infty} (Ω, R)$ , and $s f (s) \in C ([0, + \infty), R)$ satisfies hypotheses ( 2.12 ) and ( 2.13 ). Assume that one of the following conditions is fulfilled:
$q = p - 1$ , $ℓ_{\infty} \in (0, + \infty)$ and $λ β (x) < λ_{\infty}$ a.e. $x \in Ω$ for some $λ_{\infty} \in (0, ℓ_{\infty})$ or

$q = p - 1$ , $ℓ_{\infty} = + \infty$ and $λ \in R$ is arbitrary or

$1 < q < p - 1$ and $λ \in R$ is arbitrary.
Then there is a sequence ${u_{j}}_{j}$ in X of weak solutions of problem ( 1.1 ) such that $\begin{array}{l} (2.14) & lim_{j \to \infty} ‖ u_{j} ‖_{L^{\infty} (Ω, C)} = + \infty . \end{array}$

If $q > p - 1$ , then for every $k \in N$ there is $Λ_{k} > 0$ such that problem ( 1.1 ) has at least k different weak solutions $u_{1}, \dots, u_{k} \in X$ such that $\begin{array}{l} (2.15) & ‖ u_{j} ‖_{L^{\infty} (Ω, C)} ⩾ j - 1, j = 1, \dots, k \end{array}$ provided $| λ | < Λ_{k}$ .

Similar to the case of oscillation near the origin, the condition (2.12) is employed here to show the existence of solutions of problem (1.1). Meanwhile, condition (2.13) ensures that the number of these solutions is infinite, when $1 < q ⩽ p - 1$ , and at least a finite, if $q > p - 1$ .
3. An auxiliary problem

In this section, we first define the functions $h : Ω \times [0, + \infty) \mapsto R$ and $g : Ω \times C \mapsto C$ such that $g (x, t) : = h (x, | t |) t$ . Let us consider the following auxiliary problem $\begin{array}{l} (P_{g}^{K}) & \{\begin{matrix} - Δ_{A} u + K (x) | u |^{p - 1} u = g (x, u) & in Ω, \\ u = 0 & on \partial Ω . \end{matrix} \end{array}$ Here we suppose that $K : Ω \mapsto R$ satisfies the following hypotheses $\begin{array}{l} (3.1) & K \in L^{\infty} (Ω, R) with \underset{x \in Ω}{ess inf} K (x) > 0 . \end{array}$ Moreover, we assume that $g : Ω \times [0, + \infty) \mapsto R$ is a Carathéodory function and the following hypotheses are fulfilled: $\begin{array}{l} (3.2) & there is a positive constant M such that | g (x, s) | ⩽ M for a.e. x \in Ω and for every s ⩾ 0; \\ there are two positive constants 0 < σ < η such that g (x, s) ⩽ 0 for a.e. x \in Ω and for any \\ (3.3) & s \in [σ, η] . \end{array}$

In this section, our purpose is devoted to showing that there is a weak solution for the problem ( P g K ). We say that $u \in X$ is a weak solution of problem ( P g K ) if $\begin{array}{l} (3.4) & Re (\int_{Ω} \nabla_{A} u \overline{\nabla_{A} φ} d x + \int_{Ω} K (x) | u |^{p - 1} u \overline{φ} d x - \int_{Ω} g (x, u) \overline{φ} d x) = 0 \end{array}$ for all $φ \in X$ .

Problem ( P g K ) has a variational framework and we define the functional $J_{K, g} : X \mapsto R$ as follows $\begin{array}{l} (3.5) & J_{K, g} (u) = \frac{1}{2} \int_{Ω} | \nabla_{A} u |^{2} d x + \frac{1}{p + 1} \int_{Ω} K (x) | u |^{p + 1} d x - \int_{Ω} G (x, | u |) d x, \end{array}$ where $\begin{array}{l} (3.6) & G (x, s) : = \int_{0}^{s} g (x, t) d t \end{array}$ for $s ⩾ 0$ .

Since the embedding $X ↪ L^{μ} (Ω, C)$ is continuous for each $1 ⩽ μ ⩽ 2^{*}$ (for any $μ \in [1, + \infty)$ if $N = 2$ ), combining these hypotheses (3.1) and (3.2) we can deduce that $J_{K, g}$ is well-defined. Furthermore, using standard arguments we see that $J_{K, g} \in C^{1} (X, R)$ .

Consequently, the critical points of the functional $J_{K, g}$ are weak solutions of problem (3.4). To this end, we introduce the following set $W^{η}$ : $\begin{matrix} W^{η} : = {u = e^{i A (0) \cdot x} v \in X : v \in H_{0}^{1} (Ω, R), ‖ u ‖_{L^{\infty} (Ω, C)} = ‖ v ‖_{L^{\infty} (Ω, R)} ⩽ η}, \end{matrix}$ where the parameter $η > 0$ is given in (3.2).

The next auxiliary property plays an important role in the proof of Theorems 2.4 and 2.5.

Theorem 3.1.
Assume that A satisfies $(A_{1})$ , that $K : Ω \mapsto R$ is a function satisfying ( 3.1 ) and that $g : Ω \times [0, + \infty) \mapsto R$ is a Carathéodory function satisfying ( 3.2 ) and ( 3.3 ). Then the following properties hold:
the functional $J_{K, g} (\cdot)$ is bounded from below on $W^{η}$ and its infimum is achieved at some $u_{η} \in W^{η}$ ;

$| u_{η} (x) | \in [0, σ]$ for a.e. $x \in Ω$ , where $σ > 0$ is the parameter given in ( 3.2 );

$u_{η}$ is a weak solution of problem ( P g K ).

Proof.
First we show claim (i). It is obvious that $W^{η}$ is convex. Furthermore, $W^{η}$ is closed in X. To this end, we define $u_{j} = e^{i A (0) \cdot x} v_{j}$ , $u = e^{i A (0) \cdot x} v \in W^{η}$ and assume that this sequence ${u_{j}}_{j} \subset W^{η}$ satisfies $u_{j} \to u \in X$ as $j \to \infty$ .

We assert that $u \in W^{η}$ . Clearly, $u \in X$ . Moreover, by the hypotheses, we see that ${u_{j}}_{j}$ is bounded in $L^{\infty} (Ω, C)$ . Since $L^{1} (Ω, C)$ is a separable Banach space and its dual space is $L^{\infty} (Ω, C)$ , thus employing [14, Corollary 3.30] we obtain that $u_{j} \to u$ in the weak^* topology of $L^{\infty} (Ω, C)$ as $j \to \infty$ . Therefore, passing to a subsequence, still denoted by itself, applying [14, Proposition 3.13] we have that $\begin{matrix} \underset{j \to \infty}{lim inf} ‖ u_{j} ‖_{L^{\infty} (Ω, C)} ⩾ ‖ u ‖_{L^{\infty} (Ω, C)} . \end{matrix}$ Combining this and $\begin{matrix} ‖ u_{j} ‖_{L^{\infty} (Ω, C)} ⩽ η \end{matrix}$ for any $j \in N$ , we obtain that $\begin{matrix} ‖ v ‖_{L^{\infty} (Ω, R)} = ‖ u ‖_{L^{\infty} (Ω, C)} ⩽ η . \end{matrix}$ By this we get that $u \in W^{η}$ . So, the proof of the claim is now complete.

Hence, by the above several pieces of information we get that $W^{η}$ is convex and closed in X. So, it follows from [14, Theorem 3.7] that $W^{η}$ weakly closed in X.

Next, we are concerned with some properties of $J_{K, g}$ . Firstly, it is obvious that $J_{K, g}$ is sequentially weakly lower semicontinuous. On the other hand, (3.1), (3.2) and the definition of G imply that the functional $J_{K, g}$ is bounded from below on $W^{η}$ . In fact, we have $\begin{array}{l} J_{K, g} (u) & = \frac{1}{2} \int_{Ω} | \nabla_{A} u |^{2} d x + \frac{1}{p + 1} \int_{Ω} K (x) | u |^{p + 1} d x - \int_{Ω} G (x, | u |) d x \\ ⩾ - \int_{Ω} G (x, | u |) d x \\ ⩾ - M \int_{Ω} | u | d x \\ ⩾ - η M L (Ω), \end{array}$ for all $u \in W^{η}$ . Here, $L (Ω)$ is the Lebesgue measure of Ω. By this we can define $α_{η}$ as follows $\begin{array}{l} (3.7) & α_{η} : = inf_{u \in W^{η}} J_{K, g} (u) > - \infty . \end{array}$ By the definition of $α_{η}$ we see that there exist $v_{k} \in H_{0}^{1} (Ω, R)$ and $u_{k} = e^{i A (0) \cdot x} v_{k} \in W^{η}$ such that $\begin{array}{l} (3.8) & α_{η} ⩽ J_{K, g} (u_{k}) < α_{η} + \frac{1}{k}, \end{array}$ for every $k \in N$ .

Additionally, on account of $u_{k} \in W^{η}$ and (3.2), we deduce that $\begin{array}{l} \frac{1}{2} \int_{Ω} | \nabla_{A} u_{k} |^{2} d x + \frac{1}{p + 1} \int_{Ω} K (x) | u_{k} |^{p + 1} d x & = \int_{Ω} G (x, | u_{k} |) d x + J_{K, g} (u_{k}) \\ ⩽ η M L (Ω) + J_{K, g} (u_{k}) \\ ⩽ η M L (Ω) + α_{η} + \frac{1}{k} \\ ⩽ η M L (Ω) + α_{η} + 1, \end{array}$ for every $k \in N$ . By this and (3.1) we get that $\begin{array}{l} (3.9) & ‖ u_{k} ‖_{H_{A} (Ω)}^{2} ⩽ 2 η M L (Ω) + 2 α_{η} + 2, \end{array}$ for every $k \in N$ . Then, passing to a subsequence, still denoted by itself, by the boundeness of ${u_{k}}_{k}$ in X we deduce that there are $v_{η} \in H_{0}^{1} (Ω, R)$ and $u_{η} = e^{i A (0) \cdot x} v_{η} \in X$ such that $\begin{array}{l} (3.10) & \{\begin{matrix} u_{k} \to u_{η} & weakly in X, \\ u_{k} \to u_{η} & strongly in L^{p} (Ω, C), 1 ⩽ p < 2^{}, \\ u_{k} (x) \to u_{η} (x) & a.e. on Ω, \end{matrix} \end{array}$ as $k \to \infty$ .

Now we will prove that $u_{η}$ is the minimum of $J_{K, g} (\cdot)$ . To this end, note that $u_{η} \in X$ , since $W^{η}$ is weakly closed in X. Therefore, from (3.7) we have $\begin{array}{l} (3.11) & J_{K, h} (u_{η}) ⩾ α_{η} . \end{array}$

Furthermore, $J_{K, g}$ is sequentially weakly lower semicontinuous. Combining this and relations (3.8) and (3.10) we get that $\begin{matrix} α_{η} ⩾ \underset{k \to \infty}{lim inf} J_{K, g} (u_{k}) ⩾ J_{K, g} (u_{η}) . \end{matrix}$ By this and (3.11) we deduce that $J_{K, g} (u_{η}) = α_{η}$ . Combining this and (3.7) the proof of claim (i) is now complete.

Next, we will show (ii). To this end, we set σ given in assumption (3.2) and define B as follows $\begin{matrix} B : = {x \in Ω : | u_{η} (x) | = | v_{η} (x) | > σ}, \end{matrix}$ where $u_{η} (x) = e^{i A (0) \cdot x} v_{η} (x)$ , $v_{η} \in H_{0}^{1} (Ω, R)$ .

If we first assume that $L (B) > 0$ , and then we can define the following function $\begin{array}{l} w (x) = \{\begin{matrix} σ & if | u_{η} (x) | = | v_{η} (x) | > σ, \\ v_{η} (x) & if 0 ⩽ | u_{η} (x) | = | v_{η} (x) | ⩽ σ, \end{matrix} \end{array}$ for a.e. $x \in Ω$ .

Clearly, $w \in H_{0}^{1} (Ω, R)$ . By the definition of X we see that $u_{1} (x) : = e^{i A (0) \cdot x} w (x) \in X$ . Furthermore, $0 ⩽ | u_{1} (x) | = | w (x) | ⩽ σ$ for a.e. $x \in Ω$ . Of course, $u_{1} \in W^{η}$ , since $σ < η$ , by hypothesis (3.2).

From the definition of B we see that $u_{1} (x) = e^{i A (0) \cdot x} v_{η} (x) = u_{η}$ for a.e. $x \in Ω ∖ B$ and $u_{1} (x) = e^{i A (0) \cdot x} σ$ for a.e. $x \in B$ . As a result of this, we have that $\begin{array}{l} (3.12) & \int_{Ω} | \nabla_{A} u_{1} |^{2} d x - \int_{Ω} | \nabla_{A} u_{η} |^{2} d x = \int_{B} | \nabla_{A} u_{1} |^{2} d x - \int_{B} | \nabla_{A} u_{η} |^{2} d x . \end{array}$

Case 1: If $meas {x \in B : A (x) \neq 0} > 0$ , by the definition of $w \in H_{0}^{1} (Ω, R)$ and Lemma 2.2 we have $\begin{matrix} 0 < \int_{B} | \nabla_{A} w |^{2} d x ⩽ c_{2}^{2} \int_{B} | \nabla w |^{2} d x = 0, \end{matrix}$ a contradiction.

Case 2*: If $meas {x \in B : A (x) \neq 0} = 0$ , relation (3.12) can be rewritten as follows $\begin{matrix} \int_{Ω} | \nabla_{A} u_{1} |^{2} d x - \int_{Ω} | \nabla_{A} u_{η} |^{2} d x = \int_{B} | \nabla u_{1} |^{2} d x - \int_{B} | \nabla_{A} u_{η} |^{2} d x, \end{matrix}$ from which it follows that $\begin{array}{l} J_{K, g} (u_{1}) - J_{K, g} (u_{η}) & = \frac{1}{2} (\int_{Ω} | \nabla_{A} u_{1} |^{2} d x - \int_{Ω} | \nabla_{A} u_{η} |^{2} d x) \\ + \frac{1}{p + 1} \int_{Ω} K (x) (| u_{1} |^{p + 1} - | u_{η} |^{p + 1}) d x \\ - \int_{Ω} (G (x, | u_{1} |) - G (x, | u_{η} |)) d x \\ = - \frac{1}{2} \int_{B} | \nabla v_{η} |^{2} d x - \frac{1}{2} \int_{B} {| A (0) |}^{2} (| v_{η} |^{2} - σ^{2}) d x \\ + \frac{1}{p + 1} \int_{B} K (x) (σ^{p + 1} - | u_{η} |^{p + 1}) d x \\ (3.13) & - \int_{B} (G (x, σ) - G (x, | u_{η} |)) d x . \end{array}$

Clearly, by the definition of B we obtain that $\begin{array}{l} (3.14) & - \frac{1}{2} \int_{B} | \nabla v_{η} |^{2} d x - \frac{1}{2} \int_{B} {| A (0) |}^{2} (| v_{η} |^{2} - σ^{2}) d x ⩽ 0 . \end{array}$

Using the (3.1) and the definition of the set B, we get that $\begin{array}{l} (3.15) & \frac{1}{p + 1} \int_{B} K (x) (σ^{p + 1} - | u_{η} |^{p + 1}) d x ⩽ 0 . \end{array}$

On the other hand, due to $| u_{η} | ⩽ η$ , combining this, relation (3.2) and the definition of B we have $\begin{array}{l} (3.16) & \int_{B} (G (x, σ) - G (x, | u_{η} |)) d x = \int_{B} \int_{σ}^{| u_{η} (x) |} - g (x, s) d s d x ⩾ 0 . \end{array}$ From (3.13)–(3.16) we obtain that $\begin{array}{l} (3.17) & J_{K, g} (u_{1}) - J_{K, g} (u_{η}) ⩽ 0 . \end{array}$

Additionally, thanks to $u_{1} \in W^{η}$ , we can easily deduce that $J_{K, g} (u_{1}) ⩾ J_{K, g} (u_{η})$ . From this and (3.17) we obtain that $\begin{array}{l} (3.18) & J_{K, g} (u_{1}) = J_{K, g} (u_{η}) . \end{array}$ Since (3.18) is fulfilled and all the integrals in the right-hand side of (3.13) are non-positive, we deduce that every integral term in (3.13) should be zero. In particular, $\begin{matrix} \int_{B} K (x) (| u_{η} |^{p + 1} - σ^{p + 1}) d x = 0 . \end{matrix}$ Using the definition of B and (3.1), we necessarily get that $L (B) = 0$ , contradicting our hypothesis. Hence, claim (ii) is proved.

Finally, we will argue (iii). First we fix $φ \in C_{0}^{\infty} (Ω, R)$ and define $\begin{matrix} ε_{0} : = \frac{η - σ}{‖ φ ‖_{L^{\infty} (Ω, R)} + 1} > 0, \end{matrix}$ where σ and η are given as in (3.2). Furthermore, we define the function $E : [- ε_{0}, ε_{0}] \mapsto R$ as follows $\begin{matrix} E (ε) = J_{K, g} (u_{η} + ε ξ), \end{matrix}$ where $ξ (x) = e^{i A (0) \cdot x} φ (x)$ . Firstly, note that, since (ii) is fulfilled, for every $ε \in [- ε_{0}, ε_{0}]$ we get $\begin{array}{l} | u_{η} (x) + ε ξ (x) | & ⩽ | u_{η} (x) | + | ε | | φ (x) | \\ ⩽ | u_{η} (x) | + \frac{η - σ}{‖ φ ‖_{L^{\infty} (Ω, R)} + 1} | φ (x) | \\ ⩽ σ + η - σ = η, \end{array}$ for a.e. $x \in Ω$ . Hence, $u_{η} + ε ξ \in W^{η}$ .

Owing to (i), we obtain $E (ε) ⩾ E (0)$ for any $ε \in [- ε_{0}, ε_{0}]$ , that is, 0 is an interior minimum point of E. Thus, combining this fact that E is differentiable at 0 we can easily know that $E^{'} (0) = 0$ and so also $⟨ J_{K, h}^{'} (u_{η}), ξ ⟩ = 0$ . Since $ξ \in X$ is arbitrary and by the definition $J_{K, g}$ we see that $u_{η}$ is a weak solution of problem ( P g K ) (hence, a solution of (3.4)).

The proof of Theorem 3.1 is now complete. □

We point out that the function $u \equiv 0$ is a weak solution of problem ( P g K ). Theorem 3.1 cannot ensure that the solution $u_{η}$ of problem ( P g K ) is a non-trivial solution. Nevertheless, as long as the nonlinear term f is properly selected, through Theorem 3.1 we can find the existence of non-trivial solutions of the original problem (1.1).

We state this section by establishing a special function which will be very crucial in the proof of our main theorems. From now on, we assume that $x_{0} \in Ω$ and $r > 0$ satisfy $B (x_{0}, r) \subset Ω$ . For every $s > 0$ we define the following function $z_{s}$ : $\begin{array}{l} (3.19) & z_{s} (x) = \{\begin{matrix} 0 & if x \in Ω ∖ B (x_{0}, r), \\ \frac{2 s}{r} (r - | x - x_{0} |) & if x \in B (x_{0}, r) ∖ B (x_{0}, r / 2), \\ s & if x \in B (x_{0}, r / 2) . \end{matrix} \end{array}$ It is obvious that $z_{s} ⩾ 0$ in Ω and $z_{s} \in H_{0}^{1} (Ω, R)$ . Meanwhile, $‖ z_{s} ‖_{L^{\infty} (Ω, R)} = s$ and $\begin{array}{l} (3.20) & ‖ z_{s} ‖_{H_{0}^{1} (Ω, R)}^{2} = \int_{Ω} | \nabla z_{s} |^{2} d x ⩽ \frac{4 s^{2} w_{N} r^{N}}{r^{2}} \equiv C (r, N) s^{2}, \end{array}$ where $C (r, N) > 0$ is a constant and $w_{N}$ is the volume of the unit ball in $R^{N}$ .

Also, we define the truncation function $τ_{η} : [0, + \infty] \mapsto R$ as follows $\begin{array}{l} (3.21) & τ_{η} (s) : = min {η, s} \end{array}$ for every $s ⩾ 0$ , where the constant $η > 0$ is given in hypothesis (3.3). It is obvious that $τ_{η} \in C ([0, + \infty), R)$ .
4. The case of oscillation at the origin

We first consider the case where the nonlinear term f oscillates near the origin in the section. As a result of Theorem 3.1, we can give an auxiliary result in order to prove Theorem 2.4. More exactly, we will show the existence of infinitely many solutions for problem ( P g K ) when the following hypotheses are fulfilled on the function g: $\begin{array}{l} (4.1) & there is \overline{s} > 0 such that sup_{s \in [0, \overline{s}]} | g (\cdot, s) | \in L^{\infty} (Ω, R); \\ there are two sequences {σ_{j}}_{j} and {η_{j}}_{j} and 0 < η_{j + 1} < σ_{j} < η_{j} \\ and lim_{j \to \infty} η_{j} = 0 such that g (x, s) ⩽ 0 for a.e. x \in Ω and for all \\ (4.2) & s \in [σ_{j}, η_{j}], j \in N; \\ (4.3) & - \infty < \underset{s \to 0^{+}}{lim inf} \frac{G (x, s)}{s^{2}} ⩽ \underset{s \to 0^{+}}{lim sup} \frac{G (x, s)}{s^{2}} = + \infty uniformly for a.e. x \in Ω, \end{array}$ where the function G is given in (3.6).

Under these hypotheses, we have the following theorem for problem ( P g K ).

Theorem 4.1.
Assume that A satisfies $(A_{1})$ , $K : Ω \mapsto R$ satisfies ( 3.1 ) and $g : Ω \times [0, + \infty) \mapsto R$ is a Carathéodory function satisfying ( 4.1 )–( 4.3 ). Then there is a sequence ${u_{j}}_{j} \subset X$ of weak solutions of problem ( P g K ) such that $\begin{array}{l} (4.4) & lim_{j \to \infty} ‖ u_{j} ‖_{H_{A} (Ω)} = lim_{j \to \infty} ‖ u_{j} ‖_{L^{\infty} (Ω, C)} = 0 . \end{array}$
Proof.
Thanks to $η_{j} \to 0$ as $j \to \infty$ , from (4.2), without loss of generality, we may suppose that $\begin{array}{l} (4.5) & σ_{j} < η_{j} < \overline{s} \end{array}$ for j large enough, where $\overline{s} > 0$ given in (4.1).

For every $j \in N$ , let the function $g_{j} : Ω \times [0, + \infty) \mapsto R$ satisfy $\begin{array}{l} (4.6) & g_{j} (x, s) = g (x, τ_{η_{j}} (s)) \end{array}$ for a.e. $x \in Ω$ and $s ⩾ 0$ , where the function $τ_{η_{j}}$ is given in (3.21) with $η = η_{j}$ . Also, we define the following function $\begin{matrix} G_{j} (x, s) : = \int_{0}^{s} g_{j} (x, t) d t \end{matrix}$ for a.e. $x \in Ω$ and $s ⩾ 0$ . In addition, for each $j \in N$ we define the functional $J_{j}$ as follows $\begin{array}{l} (4.7) & J_{j} : = J_{K, g_{j}}, \end{array}$ where $J_{K, g_{j}}$ is given by (3.5), with $g = g_{j}$ . We point out that $J_{j}$ is the energy functional related to problem ( P g K ) when $g = g_{j}$ .

The function $g_{j} (\cdot, s)$ satisfies all the hypotheses of Theorem 3.1 for $j \in N$ sufficiently large. In fact, using (4.1), (4.5) and (4.6), $g_{j} (\cdot, s)$ verifies (3.2). Finally, from (4.2) we see that hypothesis (3.3) holds true. Thus, as a result of Theorem 3.1, for j large enough there exists $u_{j} \in W^{η_{j}}$ such that $\begin{array}{l} (4.8) & min_{u \in W^{η_{j}}} J_{j} (u) = J_{j} (u_{j}), \\ (4.9) & | u_{j} (x) | \in [0, σ_{j}] for a.e. x \in Ω, \\ (4.10) & u_{j} is a weak solution of (P_{g_{j}}^{K}) . \end{array}$

Thanks to the definition of $τ_{η}$ , combining (4.6) and the fact $| u_{j} (x) | ⩽ σ_{j} < η_{j}$ a.e. $x \in Ω$ , one has $\begin{matrix} g_{j} (x, | u_{j} (x) |) = g (x, τ_{η_{j}} (| u_{j} (x) |)) = g (x, | u_{j} (x) |) \end{matrix}$ for a.e. $x \in Ω$ . Thus, from this and (4.10), $u_{j}$ is also a weak solution for problem ( P g K ).

To prove Theorem 4.1, we must show that there are infinitely many distinct elements in the sequence ${u_{j}}_{j}$ . For this purpose, we first prove that $\begin{array}{l} (4.11) & J_{j} (u_{j}) < 0 for j \in N sufficiently large . \end{array}$

Hypothesis (4.3) yields the existence of some $ℓ > 0$ and $ζ \in (0, η_{1})$ such that $\begin{array}{l} (4.12) & \underset{x \in Ω}{ess inf} G (x, s) ⩾ - ℓ s^{2} for all s \in (0, ζ) \end{array}$ and that there is a sequence ${s_{j}}_{j}$ such that $0 < s_{j} \to 0$ as $j \to \infty$ (here we employ the definition of G, to choose $s_{j} > 0$ ) such that $\begin{array}{l} (4.13) & lim_{j \to \infty} \frac{ess {inf}_{x \in Ω} G (x, s_{j})}{s_{j}^{2}} = + \infty . \end{array}$ Consequently, for every $L > 0$ $\begin{array}{l} (4.14) & \underset{x \in Ω}{ess inf} g (x, s_{j}) > L s_{j}^{2} \end{array}$ for $j \in N$ sufficiently large.

Thanks to $σ ↘ 0$ as $j \to \infty$ , we can take a subsequence of ${σ}_{j}$ still denoted by itself such that $\begin{array}{l} (4.15) & s_{j} ⩽ σ_{j} \end{array}$ for any $j \in N$ .

Now we fix $j \in N$ large enough and define $\begin{matrix} z_{j} : = z_{s_{j}} \in H_{0}^{1} (Ω, R) \end{matrix}$ as in (3.21) with $s = s_{j}$ . Thus, $‖ z_{j} ‖_{L^{\infty} (Ω, R)} = s_{j} ⩽ σ_{j} < η_{j}$ from (4.2) and (4.15). Therefore, $0 ⩽ z_{j} (x) ⩽ s_{j} ⩽ σ_{j} < η_{j}$ a.e. $x \in Ω$ and $e^{i A (0) \cdot x} z_{j} \in W^{η_{j}}$ . This implies that for a.e. $x \in Ω$ $\begin{matrix} \int_{0}^{z_{j} (x)} g_{j} (x, t) d t = \int_{0}^{z_{j} (x)} g (x, τ_{η_{j}} (t)) d t = \int_{0}^{z_{j} (x)} g (x, t) d t . \end{matrix}$ From this and on account of Lemma 2.2, (3.1) and (3.20), combining the fact that the Sobolev embedding $H_{0}^{1} (Ω, R) ↪ L^{μ} (Ω, C)$ is continuous for each $1 ⩽ μ ⩽ 2^{*}$ , for j sufficiently large one has: $\begin{array}{l} J_{j} (e^{i A (0) \cdot x} z_{j}) & = \frac{1}{2} \int_{Ω} {| \nabla_{A} (e^{i A (0) \cdot x} z_{j}) |}^{2} d x + \frac{1}{p + 1} \int_{Ω} K (x) | z_{j} |^{p + 1} d x - \int_{Ω} G_{j} (x, z_{j}) d x \\ ⩽ \frac{1}{2} \int_{Ω} {| \nabla_{A} (e^{i A (0) \cdot x} z_{j}) |}^{2} d x + \frac{c_{4}}{p + 1} ‖ K ‖_{L^{\infty} (Ω, R)} \int_{Ω} | \nabla z_{j} |^{2} d x - \int_{Ω} G (x, z_{j}) d x \\ ⩽ C (A, r, N) (1 + \frac{c_{4}}{p + 1} ‖ K ‖_{L^{\infty} (Ω, R)}) s_{j}^{2} - \int_{B (x_{0}, r / 2)} G (x, s_{j}) d x \\ - \int_{B (x_{0}, r) ∖ B (x_{0}, r / 2)} G (x, z_{j}) d x \\ (4.16) & ⩽ (C (A, r, N) (1 + \frac{c_{4}}{p + 1} ‖ K ‖_{L^{\infty} (Ω, R)}) - L {(r / 2)}^{N} w_{N} + ℓ L (Ω)) s_{j}^{2}, \end{array}$ for some $c_{4} > 0$ and $C (A, r, N) > 0$ , due to (4.12), (4.14) and applying the fact $z_{j} (x) < η_{j} < η_{1}$ (being ${η_{j}}_{j}$ decreasing by (4.2)), where $w_{N}$ denotes the volume of the unit ball in $R^{N}$ . Taking $L > 0$ sufficiently large we get that $\begin{matrix} L {(r / 2)}^{N} w_{N} > C (A, r, N) (1 + \frac{c_{4}}{p + 1} ‖ K ‖_{L^{\infty} (Ω, R)}) + ℓ L (Ω), \end{matrix}$ for j sufficiently large we obtain that $\begin{matrix} J_{j} (e^{i A (0) \cdot x} z_{j}) < 0 . \end{matrix}$ Clearly, applying also (4.8), we have that, if j is large enough $\begin{array}{l} (4.17) & J_{j} (u_{j}) = min_{u \in W^{η_{j}}} J_{j} (u) ⩽ J_{j} (e^{i A (0) \cdot x} z_{j}) < 0, \end{array}$ which shows (4.11). Also, this ensures that $u_{j} ≢ 0$ in Ω, since $J_{j} (0) = 0$ .

Now we establish that $\begin{array}{l} (4.18) & lim_{j \to \infty} J_{j} (u_{j}) = 0 . \end{array}$ To this end, note that for $j \in N$ large enough, employing the definition of $G_{j}$ , (4.1), (4.2), (4.5), (4.6) and (4.9), we get that $\begin{array}{l} J_{j} (u_{j}) & ⩾ - \int_{Ω} G_{j} (x, | u_{j} (x) |) d x \\ = - \int_{Ω} \int_{0}^{| u_{j} (x) |} g (x, s) d s d x \\ ⩾ - \int_{Ω} sup_{s \in [0, \overline{s}]} | g (x, s) | | u_{j} (x) | d x \\ (4.19) & ⩾ - L (Ω) {‖ sup_{s \in [0, \overline{s}]} | g (\cdot, s) | ‖}_{L^{\infty} (Ω, R)} σ_{j} . \end{array}$

Thanks to ${lim}_{j \to \infty} σ_{j} = 0$ by (4.2), the above inequality and (4.17) lead to (4.18) and so the claim is now verified.

From (4.11) and (4.18), we get that the sequence ${u_{j}}_{j}$ has infinitely many different elements, thus, there are infinitely many different weak solutions for problem ( P g K ).

Finally, it is necessary to conclude that (4.4) holds true. Note that $‖ u_{j} ‖_{L^{\infty} (Ω, C)} ⩽ σ_{j}$ for $j \in N$ large enough by (4.9), and ${lim}_{j \to \infty} σ_{j} = 0$ (see (4.2)), we can easily see that $‖ u_{j} ‖_{L^{\infty} (Ω, C)} \to 0$ as $j \to \infty$ .

For the latter limit, from (3.1) we get $\begin{array}{l} \frac{1}{2} \int_{Ω} | \nabla_{A} u_{j} |^{2} d x & ⩽ \frac{1}{2} \int_{Ω} | \nabla_{A} u_{j} |^{2} + \frac{1}{p + 1} \int_{Ω} K (x) | u_{j} |^{p + 1} d x \\ < \int_{Ω} G_{j} (x, | u_{j} |) d x = \int_{Ω} G (x, | u_{j} |) d x \\ ⩽ L (Ω) {‖ sup_{s \in [0, \overline{s}]} | g (\cdot, s) | ‖}_{L^{\infty} (Ω, R)} σ_{j}, \end{array}$ on account of (4.1), (4.9) and (4.11).

Hence, from (4.2) we get that $\begin{matrix} lim_{j \to \infty} ‖ u_{j} ‖_{H_{A} (Ω)} = 0 . \end{matrix}$

The proof of Theorem 4.1 is complete. □

Now we are ready to show Theorem 2.4. This strategy will include in applying Theorems 3.1 and 4.1 and choosing appropriately the functions K and g.
4.1. Proof of Theorem 2.4

Proof.
Given $q ⩾ p - 1$ , we first prove that problem (1.1) possesses infinitely many different weak solutions under appropriate hypotheses. Let us start to study the case when $q = p - 1$ and the case when $q > p - 1$ . We will apply Theorem 4.1 to discuss the above two cases.

Firstly, we conclude claim $(a)$ . In this case, we assume that $q = p - 1$ , $ℓ_{0} \in (0, + \infty)$ and $λ \in R$ is fulfilled $λ β (x) < λ_{0}$ a.e. $x \in Ω$ for some $λ_{0} \in (0, ℓ_{0})$ . We select ${\tilde{λ}}_{0} \in (λ_{0}, ℓ_{0})$ and set $\begin{array}{l} (4.20) & K (x) : = {\tilde{λ}}_{0} - λ β (x) and g (x, s) : = {\tilde{λ}}_{0} s^{p} + f (s) s, \end{array}$ a.e. $x \in Ω$ and $s ⩾ 0$ .

Now we state that the functions K and g given in (4.20) verify all the hypotheses of Theorem 4.1. Note that $K \in L^{\infty} (Ω, R)$ and $\begin{matrix} \underset{x \in Ω}{ess inf} K (x) ⩾ {\tilde{λ}}_{0} - λ_{0} > 0, \end{matrix}$ recalling that $β \in L^{\infty} (Ω, R)$ . Thus, (3.1) holds true.

Combining the continuity of $s \to g (\cdot, s)$ and the Weierstrass theorem imply (4.1). Moreover, since for every $x \in Ω$ and $s > 0$ $\begin{matrix} \frac{G (x, s)}{s^{2}} = \frac{{\tilde{λ}}_{0}}{p} s^{p - 1} + \frac{F (s)}{s^{2}}, \end{matrix}$ assumption (2.9) immediately yields (4.3).

It still requires to prove that g satisfies (4.2). To this end, using (2.8) we deduce that there is a sequence ${s_{j}}_{j}$ converging to 0 such that $\begin{array}{l} (4.21) & \frac{f (s_{j})}{s_{j}^{p - 1}} \to - ℓ_{0} \end{array}$ as $j \to \infty$ . Now, since ${\tilde{λ}}_{0} < ℓ_{0}$ by hypothesis, there is $\hat{ε} > 0$ such that ${\tilde{λ}}_{0} + \hat{ε} < ℓ_{0}$ . Combining this and (4.21) we obtain that, for j sufficiently large, say $j ⩾ j^{} \in N$ , $\begin{array}{l} (4.22) & \frac{f (s_{j})}{s_{j}^{p - 1}} < - {\tilde{λ}}_{0} . \end{array}$ Clearly, applying the continuity of $s f (s)$ , there is a neighborhood $(σ_{j}, η_{j})$ of $s_{j}$ such that $\begin{matrix} g (x, s) = {\tilde{λ}}_{0} s^{p} + f (s) s ⩽ 0, \end{matrix}$ for every $x \in Ω$ and any $s \in [σ_{j}, η_{j}]$ and $j ⩾ j^{}$ . Thus (4.2) holds true.

Consequently, we can employ Theorem 4.1 for problem ( P g K ) with K and g given in (4.20). As a result of this, we prove the existence of infinitely many non-trivial solutions ${u_{j}}_{j}$ for problem ( P g K ) and the relation (2.10) holds true. On account of the choice of K and g in (4.20) and $q = p - 1$ , it is easy to get that $u_{j}$ is a weak solution of problem (1.1) and this completes the proof of Theorem 2.4 in the situation $q = p - 1$ .

Next, we show claim $(b)$ . For this goal, we let $q = p - 1$ , $ℓ_{0} = + \infty$ and $λ \in R$ . In this setting we take ${\tilde{λ}}_{0} \in (0, ℓ_{0})$ and $\begin{array}{l} (4.23) & K (x) : = {\tilde{λ}}_{0} and g (x, s) : = (λ β (x) + {\tilde{λ}}_{0}) s^{p} + f (s) s, \end{array}$ a.e. $x \in Ω$ and $s ⩾ 0$ . In this case we can follow the proof of claim $(a)$ and we just replace relation (4.22) with the following one $\begin{array}{l} (4.24) & \frac{f (s_{j})}{s_{j}^{p - 1}} ⩽ - (| λ | ‖ β ‖_{L^{\infty} (Ω, R)} + {\tilde{λ}}_{0}) \end{array}$ for j sufficiently large, and on account of this $\begin{matrix} g (x, s) = (λ β (x) + {\tilde{λ}}_{0}) s^{p} + f (s) s ⩽ (| λ | ‖ β ‖_{L^{\infty} (Ω, R)} + {\tilde{λ}}_{0}) s^{p} + f (s) s . \end{matrix}$

Finally, we show claim $(c)$ . To this end, we set $q > p - 1$ and $λ \in R$ . Let ${\tilde{λ}}_{0} \in (0, ℓ_{0})$ and $\begin{array}{l} (4.25) & K (x) : = {\tilde{λ}}_{0} and g (x, s) : = λ β (x) s^{q + 1} + {\tilde{λ}}_{0} s^{p} + f (s) s, \end{array}$ for a.e. $x \in Ω$ and $s ⩾ 0$ . Also in this case our goal is devoted to showing that K and g given in (4.25) verify the assumptions needed by Theorem 4.1.

Consequently, (3.1) is fulfilled. Furthermore, combining $β \in L^{\infty} (Ω, R)$ , the continuity of $s \to g (\cdot, s)$ and the Weierstrass theorem we deduce that (4.1) holds true. In addition, we obtain that $\begin{matrix} \frac{G (x, s)}{s^{2}} = λ \frac{β (x)}{q + 2} s^{q} + \frac{{\tilde{λ}}_{0}}{p + 1} s^{p - 1} + \frac{F (s)}{s^{2}}, \end{matrix}$ for a.e. $x \in Ω$ and $s > 0$ . Hence, using condition (2.9) and the fact that $q > p - 1 ⩾ 0$ yield (4.3).

Finally, we have that $\begin{array}{l} (4.26) & g (x, s) ⩽ | λ | ‖ β ‖_{L^{\infty} (Ω, R)} s^{q + 1} + {\tilde{λ}}_{0} s^{p} + f (s) s, \end{array}$ for a.e. $x \in Ω$ and any $s ⩾ 0$ . Therefore, from (2.8) we get $\begin{array}{l} (4.27) & \underset{s \to 0^{+}}{lim inf} \frac{h (x, s)}{s^{p - 1}} ⩽ \underset{s \to 0^{+}}{lim inf} (| λ | ‖ β ‖_{L^{\infty} (Ω, R)} s^{q - p + 1} + {\tilde{λ}}_{0} + \frac{f (s)}{s^{p - 1}}) = {\tilde{λ}}_{0} - ℓ_{0} < 0 \end{array}$ uniformly a.e. $x \in Ω$ , due to the choice of q. Hence, there is a sequence ${s_{j}}_{j}$ converging to 0 as $j \to \infty$ such that $g (x, s_{j}) < 0$ for $j \in N$ sufficiently large and uniformly a.e. $x \in Ω$ . Thus, combining the continuity of $s \mapsto g (\cdot, s)$ , there are two sequences ${σ_{j}}_{j}$ , ${η_{j}}_{j}$ such that $0 < η_{j + 1} < σ_{j} < s_{j} < η_{j}$ , ${lim}_{j \to \infty} η_{j} = 0$ , and $\begin{matrix} g (x, s) ⩽ 0, \end{matrix}$ for a.e. $x \in Ω$ and every $s \in [σ_{j}, η_{j}]$ and j sufficiently large. Thus, assumption (4.2) holds true. Similar to the proof of claim $(a)$ , using Theorem 4.1 we can obtain $(c)$ .

Finally, we consider the case when $0 < q < p - 1$ . In this situation our strategy is to use Theorem 3.1 and take the functions K and g appropriately. For this goal, we set ${\tilde{λ}}_{0} \in (0, ℓ_{0})$ , where $ℓ_{0} > 0$ is given in hypothesis (2.8), and define $\begin{array}{l} (4.28) & K (x) : = {\tilde{λ}}_{0} and g (x, s, λ) : = λ β (x) s^{q + 1} + {\tilde{λ}}_{0} s^{p} + f (s) s, \end{array}$ a.e. $x \in Ω$ , $s ⩾ 0$ and $λ \in R$ .

Taking account of the fact that $\begin{matrix} g (x, s, 0) = {\tilde{λ}}_{0} s^{p} + f (s) s, \end{matrix}$ and arguing as in (4.26) and (4.27), we can find that there are the sequences ${σ_{j}}_{j}$ , ${η_{j}}_{j}$ , ${s_{j}}_{j}$ and ${λ_{j}}_{j}$ such that $λ_{j} > 0$ , $\begin{array}{l} (4.29) & 0 < η_{j + 1} < σ_{j} < s_{j} < η_{j} < 1, lim_{j \to \infty} η_{j} = 0, \end{array}$ and $\begin{array}{l} (4.30) & g (x, s, λ) ⩽ 0 \end{array}$ a.e. $x \in Ω$ , for all $s \in [σ_{j}, η_{j}]$ , $λ \in [- λ_{j}, λ_{j}]$ and $j \in N$ sufficiently large.

For any $j \in N$ , let $g_{j} : Ω \times [0, + \infty] \times [- λ_{j}, λ_{j}] \mapsto R$ be the function given by $\begin{array}{l} (4.31) & g_{j} (x, s, λ) = g (x, τ_{η_{j}} (s), λ) \end{array}$ for a.e. $x \in Ω$ and $s ⩾ 0$ . Also, we define the following function and $\begin{array}{l} G_{j} (x, s, λ) : = \int_{0}^{s} g_{j} (x, t, λ) d t \end{array}$ for a.e. $x \in Ω$ , $s ⩾ 0$ , and $λ \in [- λ_{j}, λ_{j}]$ .

Let us start proving that K given in (4.28) and $g_{j}$ satisfy all the hypotheses of Theorem 3.1. Clearly, (3.1) is trivially fulfilled. In addition, the regularity of g and the continuity of $τ_{η}$ state that $g_{j}$ is a Carathéodory function. On the other hand, employing (4.31), (3.21), the continuity of $s \mapsto g (\cdot, s, \cdot)$ and the Weierstrass theorem imply that $g_{j}$ verifies (3.2). Finally, (4.30) and (4.31) imply (3.3) for j sufficiently large. Thus $g_{j}$ verifies all the hypotheses of Theorem 3.1 for j large enough.

Next, we define the following energy functional $J_{j, λ}$ such that $\begin{array}{l} (4.32) & J_{j, λ} : = J_{K, g_{j} (\cdot, \cdot, λ)}, \end{array}$ for any $j \in N$ , where $J_{K, g_{j} (\cdot, \cdot, λ)}$ is the functional given in (3.5), with $g = g_{j} (\cdot, \cdot, λ)$ . Using Theorem 3.1 we obtain that, for j sufficiently large and provided $| λ | ⩽ λ_{j}$ , there is $u_{j, λ} \in W^{η_{j}}$ such that $\begin{array}{l} (4.33) & min_{u \in W^{η_{j}}} J_{j, λ} (u) = J_{j, λ} (u_{j, λ}), \\ (4.34) & | u_{j, λ} (x) | \in [0, σ_{j}] for a.e. x \in Ω \end{array}$ and $\begin{array}{l} (4.35) & u_{j, λ} is a weak solution of (P_{g_{j} (\cdot, \cdot, λ)}^{K}) . \end{array}$ Since for j large enough $\begin{array}{l} (4.36) & | u_{j, λ} (x) | ⩽ σ_{j} < η_{j} \end{array}$ for a.e. $x \in Ω$ , from (4.29) and (4.34), we obtain that $\begin{matrix} g_{j} (x, | u_{j, λ} (x) |, λ) = g (x, | u_{j, λ} (x) |, λ), \end{matrix}$ consequently, from (4.28) it is obviously known that $u_{j, λ}$ is a weak solution of problem (1.1), provided j is large enough and $| λ | ⩽ λ_{j}$ .

It still needs to show that for any $k \in N$ problem (1.1) possesses at least k different solutions, for appropriate values of λ. To this end, first of all note that, taking account of the choices of K and $g_{j} (\cdot, s, λ)$ and (4.36), the functional $J_{j, λ}$ is defined by $\begin{array}{l} J_{j, λ} (u) & = \frac{1}{2} \int_{Ω} | \nabla_{A} u |^{2} d x - \frac{λ}{q + 2} \int_{Ω} β (x) | u |^{q + 2} d x - \int_{Ω} F (x, | u |) d x \\ (4.37) & = J_{j, 0} (u) - \frac{λ}{q + 2} \int_{Ω} β (x) | u |^{q + 2} d x \end{array}$ for all $u \in X$ .

We assert that there is an increasing sequence ${θ_{j}}_{j}$ such that $θ_{j} < 0$ , ${lim}_{j \to \infty} θ_{j} = 0$ and $\begin{array}{l} (4.38) & θ_{j - 1} < J_{j, 0} (u_{j, 0}) < θ_{j} \end{array}$ for $j ⩾ j^{}$ , with $j^{} \in N$ .

Note that the function $\begin{matrix} (x, s) \to g (x, s, 0) = {\tilde{λ}}_{0} s^{p} + f (s) s \end{matrix}$ satisfies hypotheses of Theorem 4.1, in particular due to (2.9). Thus, we can follow as in the proof of Theorem 4.1 (see, in particular, formulas (4.12)–(4.15)) we obtain that there are $ℓ > 0$ and $ζ \in (0, η_{1})$ such that $\begin{array}{l} (4.39) & F (s) ⩾ - ℓ s^{2} for any s \in (0, ξ) \end{array}$ and that there is a sequence ${{\tilde{s}}_{j}}_{j}$ such that $0 < {\tilde{s}}_{j} \to 0$ as $j \to \infty$ such that for any $L > 0$ $\begin{array}{l} (4.40) & F ({\tilde{s}}_{j}) ⩾ L {\tilde{s}}_{j}^{2} \end{array}$ for $j \in N$ sufficiently large. Also, thanks to $σ_{j} ↘ 0$ as $j \to \infty$ , we can take a subsequence of ${σ_{j}}_{j}$ , still denoted itself, such that $\begin{array}{l} (4.41) & {\tilde{s}}_{j} ⩽ σ_{j} \end{array}$ for any $j \in N$ .

Now we fix $j \in N$ large enough and define the following function $\begin{matrix} {\tilde{z}}_{j} : = z_{{\tilde{s}}_{j}} \in H_{0}^{1} (Ω, R) \end{matrix}$ as in (3.19) with $s = {\tilde{s}}_{j}$ . In a similar way to (4.16), combining (4.39) and (4.40) we obtain that for j sufficiently large $\begin{array}{l} (4.42) & J_{j, 0} (u_{j, 0}) ⩽ J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) < - Γ_{1} {\tilde{s}}_{j}^{2} = : γ_{j} < 0 \end{array}$ for a appropriate constant $Γ_{1} > 0$ . Also, similar to the fashion in (4.19) and combining the continuity of $s f (s)$ and (4.36), we get that, for j large enough $\begin{array}{l} J_{j, 0} (u_{j, 0}) & ⩾ - \int_{Ω} F (x, | u_{j, 0} |) d x \\ ⩾ - \int_{Ω} \int_{0}^{| u_{j, 0} (x) |} | f (s) s | d s d x \\ ⩾ - \int_{Ω} \int_{0}^{σ_{j}} | f (s) s | d s d x \\ (4.43) & ⩾ - Γ_{2} σ_{j} = : κ_{j}, \end{array}$ here $Γ_{2}$ is a positive constant.

First of all, note that ${γ_{j}}_{j}$ and ${κ_{j}}_{j}$ are such that $κ_{j} < γ_{j} < 0$ for any $j \in N$ and ${lim}_{j \to \infty} γ_{j} = {lim}_{j \to \infty} κ_{j} = 0$ , since ${σ_{j}}_{j}$ does, owing to (4.29). Hence, we can extract two subsequences, still denoted by ${γ_{j}}_{j}$ and ${κ_{j}}_{j}$ , such that the above properties hold true and the sequences ${γ_{j}}_{j}$ and ${κ_{j}}_{j}$ are non-decreasing. Now, we can define $\begin{array}{l} θ_{j} : = \{\begin{matrix} γ_{j} & if j \in N is even, \\ κ_{j} & if j \in N is odd . \end{matrix} \end{array}$ As a result of this definition and of (4.42) and (4.43) we get that for j sufficiently large $\begin{matrix} θ_{2 m - 1} = κ_{2 m - 1} ⩽ κ_{2 m} ⩽ J_{2 m, 0} (u_{2 m, 0}), J_{2 m, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) < γ_{2 m} = θ_{2 m} . \end{matrix}$ Letting $2 m = j$ , then we concludes (4.38). Note that $θ_{j}$ is independent of λ, since $γ_{j}$ and $κ_{j}$ do. In the same fashion as (4.38), we also obtain that $\begin{array}{l} (4.44) & θ_{j - 1} < J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) < θ_{j} . \end{array}$

Now, for any $j ⩾ j^{}$ , we let $\begin{array}{l} (4.45) & λ_{j}^{'} : = \frac{(q + 2) (J_{j, 0} (u_{j, 0}) - θ_{j - 1})}{(‖ β ‖_{L^{\infty} (Ω, R)} + 1) L (Ω)}, λ_{j}^{″} : = \frac{(q + 2) (θ_{j} - J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}))}{‖ β ‖_{L^{1} (Ω, R)} + 1} . \end{array}$ Note that $λ_{j}^{'}$ and $λ_{j}^{″}$ are strictly positive, owing to (4.38) and (4.44), and they are independent of λ.

Now, for every fixed $k \in N$ , let $\begin{matrix} Λ_{k} : = min {λ_{j^{} + 1}, \dots, λ_{j^{} + k}, λ_{j^{} + 1}^{'}, \dots, λ_{j^{} + k}^{'}, λ_{j^{} + 1}^{″}, \dots, λ_{j^{} + k}^{″}} . \end{matrix}$ Consequently, $Λ_{k} > 0$ is independent of λ. Also, if $| λ | ⩽ λ_{j}$ for every $j = j^{} + 1, \dots, j^{} + k$ . As a result of this, for every $λ \in R$ with $| λ | ⩽ Λ_{k}$ $\begin{matrix} u_{j, λ} is a weak solution of problem (1.1) \end{matrix}$ for every $j = j^{} + 1, \dots, j^{} + k$ .

Let us prove that these solutions are different. For this goal, one has $u_{j, λ} \in W^{η_{j}}$ , from (4.36) and so $\begin{array}{l} (4.46) & J_{j, 0} (u_{j, 0}) = min_{u \in W^{η_{j}}} J_{j, 0} (u) ⩽ J_{j, 0} (u_{j, λ}) . \end{array}$ Combining (4.37) and (4.46), for any λ such that $| λ | ⩽ Λ_{k}$ we get $\begin{array}{l} J_{j, λ} (u_{j, λ}) & = J_{j, 0} (u_{j, λ}) - \frac{λ}{q + 2} \int_{Ω} β (x) | u_{j, λ} |^{q + 2} d x \\ ⩾ J_{j, 0} (u_{j, 0}) - \frac{| λ |}{q + 2} ‖ β ‖_{L^{\infty} (Ω, R)} η_{j}^{q + 2} L (Ω) \\ ⩾ J_{j, 0} (u_{j, 0}) - \frac{Λ_{k}}{q + 2} ‖ β ‖_{L^{\infty} (Ω, R)} L (Ω) \\ ⩾ J_{j, 0} (u_{j, 0}) - \frac{λ_{j}^{'}}{q + 2} ‖ β ‖_{L^{\infty} (Ω, R)} L (Ω) \\ (4.47) & > θ_{j - 1}, \end{array}$ for any $j = j^{} + 1, \dots, j^{} + k$ , owing to (4.29), (4.36), the choice of $Λ_{k}$ and the definition of $λ_{j}^{'}$ .

Additionally, from (4.37) and (4.46) and on account of the fact that $‖ {\tilde{z}}_{j} ‖_{L^{\infty} (Ω, R)} = {\tilde{s}}_{j} ⩽ σ_{j} < η_{j} < 1$ (see (4.29) and (4.41)), for any λ with $| λ | ⩽ Λ_{k}$ we infer that $\begin{array}{l} J_{j, λ} (u_{j, λ}) & = min_{u \in W^{η_{j}}} J_{j, λ} (u) ⩽ J_{j, λ} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) \\ = J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) - \frac{λ}{q + 2} \int_{Ω} β (x) | {\tilde{z}}_{j} |^{q + 2} d x \\ ⩽ J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) - \frac{λ}{q + 2} \int_{{x \in Ω : λ β (x) < 0}} β (x) | {\tilde{z}}_{j} |^{q + 2} d x \\ ⩽ J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) - \frac{λ}{q + 2} \int_{{x \in Ω : λ β (x) < 0}} β (x) d x \\ ⩽ J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) + \frac{| λ |}{q + 2} \int_{{x \in Ω : λ β (x) < 0}} | β (x) | d x \\ ⩽ J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) + \frac{| λ |}{q + 2} \int_{Ω} | β (x) | d x \\ ⩽ J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) + \frac{Λ_{k}}{q + 2} ‖ β ‖_{L^{1} (Ω, R)} \\ ⩽ J_{j, 0} (e^{i A (0) \cdot x} {\tilde{z}}_{j}) + \frac{λ_{j}^{″}}{q + 2} ‖ β ‖_{L^{1} (Ω, R)} \\ (4.48) & < θ_{j}, \end{array}$ for any $j = j^{} + 1, \dots, j^{} + k$ , again owing to the choice of $Λ_{k}$ and the definition of $λ_{j}^{″}$ .

Therefore, by (4.47), (4.48) and the properties of ${θ_{j}}_{j}$ we infer that for any $j = j^{} + 1, \dots, j^{} + k$ $\begin{array}{l} (4.49) & θ_{j - 1} < J_{j, λ} (u_{j, λ}) < θ_{j} < 0, \end{array}$ which implies that $\begin{matrix} J_{1, λ} (u_{1, λ}) < \dots < J_{k, λ} (u_{k, λ}) < 0, \end{matrix}$ that is, the solutions ${u_{1, λ}, \dots, u_{k, λ}}$ are all different and non-trivial, provided $| λ | ⩽ Λ_{k}$ .

Finally, we give an estimate of the $H_{A} (Ω)$ -norm of $u_{j, λ}$ . For this purpose, applying (4.29), (4.36), (4.37) and (4.49) we get that for every $j = j^{} + 1, \dots, j^{} + k$ and $| λ | ⩽ Λ_{k}$ $\begin{array}{l} \frac{1}{2} ‖ u_{j, λ} ‖_{H_{A} (Ω)}^{2} & ⩽ J_{j, λ} (u_{j, λ}) + \frac{λ}{q + 2} \int_{Ω} β (x) | u_{j, λ} |^{q + 2} d x + \int_{Ω} F (| u_{j, λ} |) d x \\ < θ_{j} + \frac{| λ |}{q + 2} ‖ β ‖_{L^{\infty} (Ω, R)} σ_{j}^{q + 2} + \int_{Ω} \int_{0}^{σ_{j}} | f (s) s | d s d x \\ < \frac{| λ |}{q + 2} ‖ β ‖_{L^{\infty} (Ω, R)} σ_{j} + \overline{C} σ_{j}, \end{array}$ for an appropriate positive constant $\overline{C}$ , that is $\begin{matrix} ‖ u_{j, λ} ‖_{H_{A} (Ω)} ⩽ \tilde{C} σ_{j}^{1 / 2}, \end{matrix}$ where $\tilde{C} > 0$ . Since $σ_{j} \to 0$ as $j \to \infty$ , without loss of generality, we may suppose that $\begin{array}{l} (4.50) & σ_{j} ⩽ min {{\tilde{C}}^{- 2}, 1} / j^{2} \end{array}$ and this gives $\begin{matrix} ‖ u_{j, λ} ‖_{H_{A} (Ω)} ⩽ 1 / j^{2} \end{matrix}$ for every $j = j^{} + 1, \dots, j^{} + k$ , provided $| λ | ⩽ Λ_{k}$ , which is the desired claim. Finally, from (4.36) and (4.50) we deduce that $\begin{matrix} ‖ u_{j, λ} ‖_{L^{\infty} (Ω)} ⩽ 1 / j^{2} < 1 / j \end{matrix}$ or any $j = j^{} + 1, \dots, j^{*} + k$ , provided $| λ | ⩽ Λ_{k}$ .

The proof of Theorem 2.4 is now complete. □
5. The case of oscillation at infinity

In this section, we are devoted to considering problem (1.1) in the situation where f oscillates at infinity. To prove Theorem 2.5 we adopt some strategies used before. For the convenience of the reader, we provide several pieces of the information.

Here, we study again problem ( P g K ), under the following hypotheses on the function g: $\begin{array}{l} (5.1) & for every s ⩾ 0 sup_{t \in [0, s]} | g (\cdot, t) | \in L^{\infty} (Ω, R); \\ there are two sequence {σ_{j}}_{j} and {η_{j}}_{j} with 0 < σ_{j} < η_{j} < η_{j + 1} \\ and lim_{j \to \infty} σ_{j} = + \infty such that g (x, s) ⩽ 0 for a.e. x \in Ω and for all \\ (5.2) & s \in [σ_{j}, η_{j}], j \in N; \\ (5.3) & - \infty < \underset{s \to + \infty}{lim inf} \frac{G (x, s)}{s^{p + 1}} ⩽ \underset{s \to + \infty}{lim sup} \frac{G (x, s)}{s^{p + 1}} = + \infty uniformly for a.e. x \in Ω, \end{array}$ where the function G is given in (3.6).

Under these hypotheses, we have the following theorem for problem ( P g K ).

Theorem 5.1.
Assume that A satisfies $(A_{1})$ , $K : Ω \mapsto R$ satisfies ( 3.1 ) and $g : Ω \times [0, + \infty) \mapsto R$ is a Carathéodory function satisfying ( 5.1 )–( 5.3 ). Then there is a sequence ${u_{j}}_{j} \subset X$ of weak solutions of problem ( P g K ) such that $\begin{array}{l} (5.4) & lim_{j \to \infty} ‖ u_{j} ‖_{L^{\infty} (Ω, C)} = + \infty . \end{array}$
Proof.
Firstly, we are concerned with again the function $g_{j}$ and the functional $J_{j}$ given in (4.6) and (4.7), respectively. On account of assumptions (5.1) and (5.2), we can deduce that $g_{j}$ satisfies the hypotheses of Theorem 3.1 for every $j \in N$ . Therefore, for every $j \in N$ , there is $u_{j} \in W^{η_{j}}$ such that $\begin{array}{l} (5.5) & min_{u \in W^{η_{j}}} J_{j} (u) = J_{j} (u_{j}), \\ (5.6) & | u_{j} (x) | \in [0, σ_{j}], \\ (5.7) & u_{j} is a weak solution of (P_{g_{j}}^{K}) . \end{array}$

In a similar fashion arguing as in the proof of Theorem 4.1 and on account of the definition of $g_{j}$ , (5.2) and (5.6), we can conclude that $\begin{matrix} g_{j} (x, | u_{j} (x) |) = g (x, τ_{η_{j}} (| u_{j} (x) |)) = g (x, | u_{j} (x) |) \end{matrix}$ for a.e. $x \in Ω$ . Hence, it follows from (5.7) that $u_{j}$ is also a weak solution of problem ( P g K ).

To conclude the claim of Theorem 5.1, we remain to prove that there are infinitely many different elements in the sequence ${u_{j}}_{j}$ . For this purpose, we first assert that, up to a subsequence, still denoted by itself $\begin{array}{l} (5.8) & lim_{j \to \infty} J_{j} (u_{j}) = - \infty . \end{array}$

For this goal, note that on account of (5.3), there are $ℓ > 0$ and $ξ > 0$ such that $\begin{array}{l} (5.9) & \underset{x \in Ω}{ess inf} G (x, s) ⩾ ℓ s^{p + 1} for any s > ξ \end{array}$ and there is a sequence ${s_{j}}_{j}$ such that $1 < s_{j} \to + \infty$ (here we use the definition of G, to choose $s_{j} > 1$ ) and $\begin{matrix} \underset{j \to \infty}{lim sup} \frac{G (x, s_{j})}{s_{j}^{p + 1}} = + \infty, \end{matrix}$ that is, for every $L > 0$ $\begin{array}{l} (5.10) & \underset{x \in Ω}{ess inf} G (x, s_{j}) > L s_{j}^{p + 1} \end{array}$ for any $j \in N$ large enough.

On account of $σ_{j} ↗ + \infty$ by (5.2), we can extract a subsequence of ${σ_{j}}_{j}$ , still denoted by itself, such that $\begin{array}{l} (5.11) & s_{j} ⩽ σ_{j} \end{array}$ for any $j \in N$ . We fix $j \in N$ and define the following function $\begin{matrix} z_{j} : = z_{s_{j}} \in H_{0}^{1} (Ω, R) \end{matrix}$ as in (3.19) with $s = s_{j}$ . Thus, $‖ z_{j} ‖_{L^{\infty} (Ω, R)} = s_{j}$ , hence, from (5.2) and (5.11), $0 < z_{j} (x) ⩽ σ_{j} < η_{j}$ a.e. $x \in Ω$ and $e^{A (0) \cdot x} z_{j} \in W^{η_{j}}$ . On account of Lemma 2.2, (3.20), (5.9) and (5.10), combining the fact that the Sobolev embedding $H_{0}^{1} (Ω, R) ↪ L^{μ} (Ω, C)$ is continuous for each $1 ⩽ μ ⩽ 2^{}$ we deduce that $\begin{array}{l} J_{j} (e^{A (0) \cdot x} z_{j}) & = \frac{1}{2} \int_{Ω} {| \nabla_{A} (e^{i A (0) \cdot x} z_{j}) |}^{2} d x + \frac{1}{p + 1} \int_{Ω} K (x) | z_{j} |^{p + 1} d x - \int_{Ω} G_{j} (x, z_{j}) d x \\ ⩽ C (A, r, N) s_{j}^{2} + \frac{c_{4}}{p + 1} ‖ K ‖_{L^{\infty} (Ω, R)} \int_{Ω} | \nabla z_{j} |^{2} d x - \int_{B (x_{0}, r / 2)} G (x, s_{j}) d x \\ - \int_{B (x_{0}, r) ∖ B (x_{0}, r / 2) \cap {z_{j} > ξ}} G (x, z_{j}) d x \\ - \int_{B (x_{0}, r) ∖ B (x_{0}, r / 2) \cap {z_{j} ⩽ ξ}} G (x, z_{j}) d x \\ ⩽ (C (A, r, N) (1 + \frac{c_{4}}{p + 1} ‖ K ‖_{L^{\infty} (Ω, R)}) - L {(r / 2)}^{N} w_{N} + ℓ L (Ω)) s_{j}^{p + 1} \\ (5.12) & + {‖ sup_{s \in [0, ξ]} | g (\cdot, s) | ‖}_{L^{\infty} (Ω, R)} L (Ω) ξ . \end{array}$ Taking $L > 0$ large enough, we get $\begin{matrix} L {(r / 2)}^{N} w_{N} > C (A, r, N) (1 + \frac{c_{4}}{p + 1} ‖ K ‖_{L^{\infty} (Ω, R)}) + ℓ L (Ω), \end{matrix}$ and on account of ${lim}_{j \to \infty} s_{j} = + \infty$ , from (5.12) we have $\begin{array}{l} (5.13) & lim_{j \to \infty} J_{j} (e^{i A (0) \cdot x} z_{j}) = - \infty . \end{array}$

Additionally, by (5.5) we obtain $J_{j} (u_{j}) = {min}_{u \in W^{η_{j}}} J_{j} (u) ⩽ J_{j} (e^{i A (0) \cdot x} z_{j})$ , hence, from this and (5.13) claim (5.8) is proved.

As a result of (5.8) we obtain the sequence ${u_{j}}_{j}$ has infinitely many different elements, in particular, $u_{j} ≢ 0$ in Ω, being $J_{j} (0) = 0$ . In fact, we assume that there is only a finite number of elements in the sequence ${u_{j}}_{j}$ , that is, ${u_{1}, \dots, u_{k}}$ for some $k \in N$ . Clearly, the sequence ${J_{j} (u_{j})}_{j}$ reduces to at most the finite set ${J_{1} (u_{1}), \dots, J_{k} (u_{k})}$ and this fact contradicts relation (5.8). Therefore, problem ( P g K ) possesses infinitely many different weak solutions.

Arguing by contradiction we will get (5.4). We suppose that, up to a subsequence, still denoted by itself, there is $L > 0$ such that $\begin{array}{l} (5.14) & ‖ u_{j} ‖_{L^{\infty} (Ω, C)} ⩽ L \end{array}$ for every $j \in N$ .

Thanks to $η_{j} \to + \infty$ as $j \to \infty$ , for j sufficiently large, say $j ⩾ j^{}$ , with $j^{} \in N$ , we get that $η_{j} ⩾ L$ . As a result of this and (5.14), we infer that $\begin{array}{l} (5.15) & u_{j} \in W^{η_{j^{}}} for any j ⩾ j^{} . \end{array}$ Here we applied also the fact that the sequence ${η_{j}}_{j}$ is increasing by (5.2).

Also, on account of the monotonicity of ${η_{j}}_{j}$ , we can know that when $u \in W^{η_{j}}$ and $j < k$ , then $u \in W^{η_{k}}$ , that is $\begin{array}{l} (5.16) & W^{η_{j}} \subseteq W^{η_{k}} \end{array}$ and this yields that a.e. $x \in Ω$ $\begin{array}{l} G_{j} (x, | u |) & = \int_{0}^{| u (x) |} g (x, τ_{η_{j}} (t)) d t \\ = \int_{0}^{| u (x) |} g (x, t) d t \\ (5.17) & = \int_{0}^{| u (x) |} g (x, τ_{η_{k}} (t)) d t = G_{k} (x, | u |), \end{array}$ provided $u \in W^{η_{j}}$ .

Additionally, the sequence ${J_{j} (u_{j})}_{j}$ is non-increasing. In fact, for $j < k$ , combining (5.16) and (5.17) we get $\begin{array}{l} J_{j} (u_{j}) & = min_{u \in W^{η_{j}}} J_{j} (u) \\ = min_{u \in W^{η_{j}}} J_{j} (u) \\ ⩾ min_{u \in W^{η_{k}}} J_{k} (u) \\ (5.18) & = J_{k} (u_{k}) . \end{array}$

Then, from (5.15)–(5.18), for any $j ⩾ j^{}$ we obtain $\begin{array}{l} J_{j^{}} (u_{j^{}}) & ⩾ J_{j} (u_{j}) \\ ⩾ min_{u \in W^{η_{j^{}}}} J_{j} (u) \\ = min_{u \in W^{η_{j^{}}}} J_{j^{}} (u) \\ = J_{j^{}} (u_{j^{}}) . \end{array}$ Thus $\begin{array}{l} J_{j} (u_{j}) = J_{j^{}} (u_{j^{}}) for any j \in j^{} . \end{array}$ This fact contradicts (5.8). So that, $‖ u_{j} ‖_{L^{\infty} (Ω, C)} \to + \infty$ as $j \to \infty$ .

The proof of Theorem 5.1 is now complete. □

For $N ⩾ 3$ , we add the following extra hypothesis on the function h: $\begin{array}{l} (5.19) & sup_{s \in [0, + \infty)} \frac{| h (x, s) |}{1 + s^{2^{} - 2}} < + \infty uniformly a.e. x \in Ω . \end{array}$ Then we can get the following corollary. Corollary 5.2.
All the hypotheses of Theorem* 5.1 are fulfilled. For $N = 2$ , we have that $\begin{matrix} lim_{j \to \infty} ‖ u_{j} ‖_{H_{A} (Ω)} = + \infty, \end{matrix}$ where ${u_{j}}_{j}$ is the sequence of different weak solutions of problem ( P g K ), given by Theorem 5.1 . Furthermore, for $N ⩾ 3$ , we need to assume that the additional assumption ( 5.19 ) holds true. Then, we also have the same result.
Proof.
We consider the case $N ⩾ 3$ and leave the other one to the reader. Arguing by contradiction we suppose that, up to a subsequence, still denoted by itself, there is $L > 0$ such that for every $j \in N$ $\begin{array}{l} (5.20) & ‖ u_{j} ‖_{H_{A} (Ω)} ⩽ L . \end{array}$

Thus, for every $j \in N$ by (5.19) we get $\begin{array}{l} | J_{j} (u_{j}) | & ⩽ ‖ u_{j} ‖_{H_{A} (Ω)}^{2} + \frac{‖ K ‖_{L^{\infty} (Ω, R)}}{p + 1} ‖ u_{j} ‖_{L^{p + 1} (Ω, C)}^{p + 1} + C_{1} \int_{Ω} \int_{0}^{| u_{j} (x) |} (1 + s^{2^{} - 2}) s d s d x \\ (5.21) & ⩽ ‖ u_{j} ‖_{H_{A} (Ω)}^{2} + \frac{‖ K ‖_{L^{\infty} (Ω, R)}}{p + 1} ‖ u_{j} ‖_{L^{p + 1} (Ω, C)}^{p + 1} + C_{2} ‖ u_{j} ‖_{L^{2} (Ω, C)}^{2} + C_{3} ‖ u_{j} ‖_{L^{2^{}} (Ω, C)}^{2^{}} \end{array}$ for appropriate positive constants $C_{1}$ , $C_{2}$ and $C_{3}$ . Thanks to the fact that the Sobolev space X is continuously embedded into $L^{μ} (Ω)$ for any $μ \in [1, 2^{}]$ , from (5.20) and (5.21) we can easily get that the sequence ${J_{j} (u_{j})}_{j}$ is bounded in $R$ . This contradicts (5.8).

The proof of Corollary 5.2 is now complete. □

5.1. Proof of Theorem 2.5

We will apply some techniques used in the proof of Theorem 2.4. The main idea include in employing Theorems 3.1 and 5.1 for problem ( P g K ) and in choosing the functions K and g appropriately.

Proof.

In this case when $q = p - 1$ and $ℓ_{\infty} \in (0, + \infty)$ we fix $λ \in R$ such that $λ β (x) < λ_{\infty}$ a.e. $x \in Ω$ for some $λ_{\infty} \in (0, ℓ_{\infty})$ . In this situation we choose ${\tilde{λ}}_{\infty} \in (λ_{\infty}, ℓ_{\infty})$ and define $\begin{array}{l} (5.22) & K (x) : = {\tilde{λ}}_{\infty} - λ β (x) and g (x, s) : = {\tilde{λ}}_{\infty} s^{p} + f (s) s, \end{array}$ for a.e. $x \in Ω$ and $s ⩾ 0$ , in Theorem 5.1. With the same arguments applied in the proof of Theorem 2.4, it is easily know that K and g verify the hypotheses of Theorem 5.1 and this conclude the claim of Theorem 2.5.

In this situation when $q = p - 1$ and $ℓ_{\infty} = + \infty$ we choose $λ \in R$ and we employ Theorem 5.1 with $\begin{array}{l} (5.23) & K (x) : = {\tilde{λ}}_{\infty} and g (x, s) : = (λ β (x) + {\tilde{λ}}_{\infty}) s^{p} + f (s) s, \end{array}$ for a.e. $x \in Ω$ and $s ⩾ 0$ . The arguments are the same as those applied in the previous case.

In this setting when $0 < q < p - 1$ we select the following functions K and g in Theorem 5.1: $\begin{array}{l} (5.24) & K (x) : = {\tilde{λ}}_{\infty} and g (x, s) : = λ β (x) s^{q + 1} + {\tilde{λ}}_{\infty} s^{p} + f (s) s, \end{array}$ for a.e. $x \in Ω$ and $s ⩾ 0$ , where ${\tilde{λ}}_{\infty} \in (0, ℓ_{\infty})$ , and we argue above.

In this situation when $q > p - 1$ the strategy will include in using Theorem 3.1 for problem ( P g K ) and in choosing the functions K and g appropriately. To this end, set ${\tilde{λ}}_{\infty} \in (λ_{\infty}, ℓ_{\infty})$ , where $ℓ_{\infty} > 0$ is given in hypothesis (2.12), and define $\begin{matrix} K (x) : = {\tilde{λ}}_{\infty} and g (x, s, λ) : = λ β (x) s^{q + 1} + {\tilde{λ}}_{\infty} s^{p} + f (s) s \end{matrix}$ a.e. $x \in Ω$ , $s ⩾ 0$ and $λ \in R$ . In similar fashion arguing as in the proof of Theorem 2.4 we obtain the claim.

Finally, when $q > p - 1$ we can adapt the arguments applied in the proof of Theorem 2.4.

The proof of Theorem 2.5 is now complete. □

Furthermore, we have the following corollary.

Corollary 5.3.

We set $q ⩽ p - 1$ and we assume that all the hypotheses of Theorem 2.5 are fulfilled. For $N = 2$ , we have that $\begin{matrix} lim_{j \to \infty} ‖ u_{j} ‖_{H_{A} (Ω)} = + \infty, \end{matrix}$ where ${u_{j}}_{j}$ is the sequence of different weak solutions of problem ( 1.1 ), given by Theorem 2.5 . Furthermore, for $N ⩾ 3$ we need to suppose that $\begin{array}{l} (5.25) & sup_{s \in [0, + \infty)} \frac{| f (s) |}{1 + s^{2^{*} - 2}} < + \infty . \end{array}$ Then we have the same result.

Proof.

It is sufficient to using Corollary 5.2 with K and g given in (5.22) and (5.23) when $q = p - 1$ , and in (5.24) when $0 < q < p - 1$ . We also consider the case $N ⩾ 3$ and leave the other one to the reader.

We fist study the case when $q = p - 1$ . First of all, Note that $g (x, s) = h (x, s) s$ and $q ⩾ 0$ (being $p ⩾ 1$ ). Also, from (5.25), for $s \in (0, 1]$ $\begin{array}{l} \frac{| h (x, s) |}{1 + s^{2^{*} - 2}} & ⩽ \frac{c_{5} s^{p - 1}}{1 + s^{2^{*} - 2}} + \frac{| f (s) |}{1 + s^{2^{*} - 2}} \\ ⩽ c_{5} s^{p - 1} + \frac{| f (s) |}{1 + s^{2^{*} - 2}} \\ ⩽ c_{5} + sup_{s \in [0, + \infty]} \frac{| f (s) |}{1 + s^{2^{*} - 2}} < + \infty, \end{array}$ for some $c_{5} > 0$ , while $s > 1$ we get $\begin{array}{l} \frac{| h (x, s) |}{1 + s^{2^{*} - 2}} & ⩽ \frac{c_{6} s^{p - 1}}{s^{2^{*} - 2}} + \frac{| f (s) |}{1 + s^{2^{*} - 2}} \\ ⩽ c_{6} s^{p + 1 - 2^{*}} + \frac{| f (s) |}{1 + s^{2^{*} - 2}} \\ ⩽ c_{6} s^{p + 1 - 2^{*}} + sup_{s \in [0, + \infty]} \frac{| f (s) |}{1 + s^{2^{*} - 2}} \\ < + \infty, \end{array}$ for a positive constant $c_{6} > 0$ , and thanks to $s^{p + 1 - 2^{*}} \to 0$ as $s \to + \infty$ . Hence, in any case (5.19) is fulfilled. In the case when $0 < q < p - 1$ we conclude with similar arguments.

The proof of Corollary 5.3 is now complete. □

Footnotes

Acknowledgements

This research is partially supported by the National Natural Science Foundation of China (No. 11971485) and the Fundamental Research Funds for the Central Universities of Central South University (No. 2019zzts211). This paper has been completed while Youpei Zhang was visiting University of Craiova (Romania) with the financial support of China Scholarship Council (No. 201906370079). Youpei Zhang would like to thank China Scholarship Council and Embassy of the People’s Republic of China in Romania.

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