Nodal solutions for double phase Kirchhoff problems with vanishing potentials

Abstract

We consider the following $(p, q)$ -Laplacian Kirchhoff type problem $\begin{array}{l} - (a + b \int_{R^{3}} | \nabla u |^{p} d x) Δ_{p} u - (c + d \int_{R^{3}} | \nabla u |^{q} d x) Δ_{q} u \\ + V (x) (| u |^{p - 2} u + | u |^{q - 2} u) = K (x) f (u) in R^{3}, \end{array}$ where $a, b, c, d > 0$ are constants, $\frac{3}{2} < p < q < 3$ , $V : R^{3} \to R$ and $K : R^{3} \to R$ are positive continuous functions allowed for vanishing behavior at infinity, and f is a continuous function with quasicritical growth. Using a minimization argument and a quantitative deformation lemma we establish the existence of nodal solutions.

Keywords

nodal solutions vanishing potentials Nehari manifold

1. Introduction

This paper deals with the existence of least energy nodal solutions for the following class of quasilinear problems $\begin{array}{l} - (a + b \int_{R^{3}} | \nabla u |^{p} d x) Δ_{p} u - (c + d \int_{R^{3}} | \nabla u |^{q} d x) Δ_{q} u \\ (1.1) & + V (x) (| u |^{p - 2} u + | u |^{q - 2} u) = K (x) f (u) in R^{3}, \end{array}$ where $a, b, c, d > 0$ are constants, $\frac{3}{2} < p < q < 3$ , $V : R^{3} \to R$ and $K : R^{3} \to R$ are positive functions, and f is a continuous function with quasicritical growth.

In recent years, a considerable interest has been devoted to the study of this general class of problems due to the fact that they arise in applications in physics and related sciences.

When $a = c = 1$ and $b = d = 0$ , equation (1.1) becomes a $(p, q)$ -Laplacian problem of the type $\begin{matrix} (1.2) & - Δ_{p} u - Δ_{q} u + V (x) (| u |^{p - 2} u + | u |^{q - 2} u) = K (x) f (u) in R^{3} . \end{matrix}$ As underlined in [27], this equation is related to the more general reaction-diffusion system $\begin{matrix} u_{t} = div (D (u) \nabla u) + c (x, u) and D (u) = | \nabla u |^{p - 2} + | \nabla u |^{q - 2}, \end{matrix}$ which appears in plasma physics, biophysics and chemical reaction design.

In these applications, u represents a concentration, $div (D (u) \nabla u)$ is the diffusion with the diffusion coefficient $D (u)$ , and the reaction term $c (x, u)$ relates to source and loss processes. Usually, the reaction term $c (x, u)$ is a polynomial of u with variable coefficient (see [27]). This kind of problem has been widely investigated by many authors, see for instance [27,31,32,39,42–44,46] and the references therein. In particular, in [17], using a minimization argument and a quantitative deformation lemma, the authors proved the existence of nodal solutions for the following class of $(p, q)$ problems $\begin{array}{l} - div (a (| \nabla u |^{p}) | \nabla u |^{p - 2} \nabla u) + V (x) b (| u |^{p}) | u |^{p - 2} u = K (x) f (u) in R^{N}, \end{array}$ where $N ⩾ 3$ , $2 ⩽ p < N$ , $a, b, f \in C^{1} (R)$ , and $V, K$ are continuous and positive functions (see also [16]).

We stress that in the nonlocal framework, only few recent works deal with the fractional $(p, q)$ -Laplacian. In [25] the authors established the existence, nonexistence and multiplicity for a nonlocal $(p, q)$ -subcritical problem. Ambrosio [7] obtained an existence result for a critical fractional $(p, q)$ -problem via mountain pass theorem. In [21] the authors investigated the existence of infinitely many nontrivial solutions for a class of fractional $(p, q)$ -equations involving concave-critical nonlinearities in bounded domains. Hölder regularity result for nonlocal double phase equations has been established in [29]. Applying suitable variational and topological arguments, in [12] the authors obtained a multiplicity and concentration result for a class of fractional problems with unbalanced growth. We also mention [1,11,36] for other interesting results.

We underline that there is a huge bibliography concerning the nonlinear Schrödinger equation (that is when $p = q = 2$ in (1.2)) $\begin{array}{l} (1.3) & - Δ u + V (x) u = K (x) f (u) in R^{3}, \end{array}$ and we would like to point out that an important class of problems associated with (1.3) is the so called zero mass case, which occurs when the potential V vanishes at infinity. Using several variational methods, many authors attacked this equation; see for instance [2,3,5,6,18,19,22].

When $a = c$ , $b = d (\neq 0)$ and $p = q = 2$ , problem (1.1) becomes the following Kirchhoff equation $\begin{array}{l} (1.4) & - (a + b \int_{R^{3}} | \nabla u |^{2} d x) Δ u + V (x) u = f (x, u) in R^{3} . \end{array}$ This problem is related to the stationary analogue of the Kirchhoff equation [41] $\begin{array}{l} ρ \frac{\partial^{2} u}{\partial t^{2}} - (\frac{p_{0}}{h} + \frac{E}{2 L} \int_{0}^{L} {| \frac{\partial u}{\partial x} |}^{2} d x) \frac{\partial^{2} u}{\partial x^{2}} = 0, \end{array}$ for all $x \in (0, L)$ and $t ⩾ 0$ . This equation is an extension of the classical D’Alembert wave equation taking into account the changes in the length of the strings produced by transverse vibrations. In (1.4), $u (x, t)$ is the lateral displacement of the vibrating string at the coordinate x and the time t, L is the length of the string, h is the cross-section area, E is the Young modulus of the material, ρ is the mass density and $p_{0}$ is the initial axial tension.

The early studies dedicated to the Kirchhoff equation (1.4) were done by Bernstein [20] and Pohozaev [50]. However, the Kirchhoff equation (1.4) began to attract the attention of more researchers only after the work by Lions [45], in which the author introduced a functional analysis approach to study a general Kirchhoff equation in arbitrary dimension with external force term. For more details on classical Kirchhoff problems we refer to [13,15,48,49]. In [34] the authors established the existence of a least energy nodal solution to the following class of nonlocal Schrödinger–Kirchhoff problems $\begin{array}{l} M (\int_{R^{3}} | \nabla u |^{2} d x + \int_{R^{3}} V (x) u^{2} d x) (- Δ u + V (x) u) = K (x) f (u) in R^{3} . \end{array}$ Moreover, when the problem presents symmetry, they proved the existence of infinitely many nontrivial solutions. We also mention [30,33] where the existence of nodal solutions for problems like (1.4) has been obtained.

In the nonlocal framework, Fiscella and Valdinoci [35] proposed the following stationary Kirchhoff model driven by the fractional Laplacian $\begin{array}{l} (1.5) & \{\begin{matrix} - M (\iint_{R^{2 N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y) {(- Δ)}^{s} u = λ f (x, u) + | u |^{2_{s}^{*} - 2} u & in Ω, \\ u = 0 & in R^{N} ∖ Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N}$ is an open bounded set, $2_{s}^{*} = \frac{2 N}{N - 2 s}$ , $N > 2 s$ , $s \in (0, 1)$ , $M : R^{+} \to R^{+}$ is an increasing continuous function which behaves like $M (t) = a + b t$ , with $b ⩾ 0$ , and f is a continuous function. Based on a truncation argument and the mountain pass theorem, the authors established the existence of a non-negative solution to (1.5) for any $λ > λ^{*} > 0$ , where $λ^{*}$ is an appropriate threshold. We also mention [8–10,14,47,51] in which the authors dealt with existence and multiplicity of solutions for (1.5), while concerning the existence and multiplicity of sign-changing solutions for fractional Kirchhoff problems only few results appear in the literature [24,26,40].

Finally, if $a = c$ , $b = d (\neq 0)$ and $p = q \neq 2$ , we have the following p-Laplacian Kirchhoff-type equation $\begin{array}{l} (1.6) & - (a + b \int_{R^{3}} | \nabla u |^{p} d x) Δ_{p} u + V (x) | u |^{p - 2} u = f (x, u) in R^{3} . \end{array}$ Very recently, in [38] using a minimization argument and the Nehari manifold method, the authors investigated the existence of least energy nodal (or sign-changing) solutions to (1.6). We also mention [23,28,37,55] for results regarding Schrö dinger–Kirchhoff equations involving the p-Laplacian.

Motivated by the interest shared by the mathematical community toward $(p, q)$ -Laplacian problems, the goal of the present paper is to study the existence of nodal solutions to (1.1). In order to state precisely our main result, we first introduce the main assumptions on the potentials V and K and on the nonlinearity f.

We assume that $V, K : R^{3} \to R$ are continuous functions and we say that $(V, K) \in K$ if the following conditions are satisfied (see [3]):

$V (x), K (x) > 0$ for all $x \in R^{3}$ and $K \in L^{\infty} (R^{3})$ ;

If $(A_{n}) \subset R^{3}$ is a sequence of Borel sets such that the Lebesgue measure $| A_{n} | ⩽ R$ , for all $n \in N$ and for some $R > 0$ , then $\begin{array}{l} lim_{r \to \infty} \int_{A_{n} \cap B_{ϱ}^{c} (0)} K (x) d x = 0, \end{array}$ uniformly in $n \in N$ , where $B_{ϱ}^{c} (0) : = R^{3} ∖ B_{ϱ} (0)$ .

Furthermore, one of the following conditions is satisfied:

$\frac{K}{V} \in L^{\infty} (R^{3})$ ;

There exists $m \in (q, q^{*})$ such that $\begin{array}{l} \frac{K (x)}{V {(x)}^{\frac{q^{*} - m}{q^{*} - p}}} \to 0 as | x | \to \infty . \end{array}$

Let us point out that the hypotheses on the functions V and K characterize problem (1.1) as a zero mass problem.

Regarding the nonlinearity f, we assume that $f \in C (R, R)$ and f fulfills the following conditions:

${lim}_{| t | \to 0} \frac{f (t)}{| t |^{2 p - 1}} = 0$ if $(V K_{3})$ holds,

${lim}_{| t | \to 0} \frac{f (t)}{| t |^{m - 1}} = 0$ if $(V K_{4})$ holds, with $m \in (q, q^{*})$ defined in $(V K_{4})$ ,

${lim}_{| t | \to \infty} \frac{f (t)}{| t |^{q^{*} - 1}} = 0$ ,

${lim}_{t \to \infty} \frac{F (t)}{t^{2 q}} = \infty$ , where $F (t) : = \int_{0}^{t} f (τ) d τ$ ,

The map $t \mapsto \frac{f (t)}{| t |^{2 q - 1}}$ is strictly increasing for all $| t | > 0$ .

We note that from assumption $(f_{4})$ it follows that $t \mapsto \frac{1}{2 q} f (t) t - F (t)$ is increasing for $t ⩾ 0$ and also that $t \mapsto \frac{1}{2 q} f (t) t - F (t)$ is decreasing for $t ⩽ 0$ (see Remark 2.1 below).

Our main result can be stated as follows:

Theorem 1.1.
Assume that $(V, K) \in K$ and f satisfies conditions $(f_{1})$ (or $({\tilde{f}}_{1})$ ) and $(f_{2})$ – $(f_{4})$ . Then problem ( 1.1 ) admits a least energy sign-changing weak solution. If in addition, f is an odd function, then ( 1.1 ) has infinitely many nontrivial solutions.

A weak solution of problem (1.1) is a function $u \in E$ such that $\begin{array}{l} a \int_{R^{3}} | \nabla u |^{p - 2} \nabla u \cdot \nabla φ d x + b (\int_{R^{3}} | \nabla u |^{p} d x) \int_{R^{3}} | \nabla u |^{p - 2} \nabla u \cdot \nabla φ d x \\ + c \int_{R^{3}} | \nabla u |^{q - 2} \nabla u \cdot \nabla φ d x + d (\int_{R^{3}} | \nabla u |^{q} d x) \int_{R^{3}} | \nabla u |^{q - 2} \nabla u \cdot \nabla φ d x \\ (1.7) & + \int_{R^{3}} V (x) (| u |^{p - 2} u φ + | u |^{q - 2} u φ) d x - \int_{R^{3}} K (x) f (u) φ d x = 0 \end{array}$ for all $φ \in E$ , where $\begin{array}{l} E = {u \in D^{1, p} (R^{3}) \cap D^{1, q} (R^{3}) : \int_{R^{3}} V (x) (| u |^{p} + | u |^{q}) d x < \infty} . \end{array}$ By a sign-changing weak solution to problem (1.1) we mean a function $u \in E$ that satisfies (1.7) with $u^{+} = max {u, 0} \neq 0$ and $u^{-} = min {u, 0} \neq 0$ .

The proof of Theorem 1.1 is achieved by using suitable variational techniques inspired by [4,16,17,34]. In order to study (1.1) we consider the following functional $I : E \to R$ given by $\begin{array}{l} I (u) = & \frac{a}{p} \int_{R^{3}} | \nabla u |^{p} d x + \frac{b}{2 p} {(\int_{R^{3}} | \nabla u |^{p} d x)}^{2} + \frac{c}{q} \int_{R^{3}} | \nabla u |^{q} d x + \frac{d}{2 q} {(\int_{R^{3}} | \nabla u |^{q} d x)}^{2} \\ + \int_{R^{3}} V (x) (\frac{1}{p} | u |^{p} + \frac{1}{q} | u |^{q}) d x - \int_{R^{3}} K (x) F (u) d x . \end{array}$ It is easy to check that $I \in C^{1} (E, R)$ and its differential is given by $\begin{array}{l} ⟨ I^{'} (u), φ ⟩ = & a \int_{R^{3}} | \nabla u |^{p - 2} \nabla u \cdot \nabla φ d x + b (\int_{R^{3}} | \nabla u |^{p} d x) \int_{R^{3}} | \nabla u |^{p - 2} \nabla u \cdot \nabla φ d x \\ + c \int_{R^{3}} | \nabla u |^{q - 2} \nabla u \cdot \nabla φ d x + d (\int_{R^{3}} | \nabla u |^{q} d x) \int_{R^{3}} | \nabla u |^{q - 2} \nabla u \cdot \nabla φ d x \\ + \int_{R^{3}} V (x) (| u |^{p - 2} u φ + | u |^{q - 2} u φ) d x - \int_{R^{3}} K (x) f (u) φ d x . \end{array}$ Then, we define the nodal set $\begin{matrix} M = {w \in N : w^{\pm} \neq 0, ⟨ I^{'} (w), w^{+} ⟩ = ⟨ I^{'} (w), w^{-} ⟩ = 0}, \end{matrix}$ where $\begin{matrix} N = {u \in E ∖ {0} : ⟨ I^{'} (u), u ⟩ = 0} . \end{matrix}$ In order to get least energy nodal (or sign-changing) solutions to (1.1), we minimize the functional $I$ on the nodal set $M$ . Then we prove that the minimum is achieved and, by using a variant of the quantitative deformation lemma, we show that it is a critical point of $I$ . Finally, when the nonlinearity f is odd, we obtain the existence of infinitely many nontrivial weak solutions not necessarily nodals. We point out that our paper extends the results obtained in [34,38].

Problem (1.1) is called nonlocal due to the presence of the Kirchhoff term $(\int_{R^{3}} | \nabla u |^{p} d x) Δ_{p} u$ , this causes some mathematical difficulties which makes the study of such a class of problems particularly interesting. We underline that here we are considering the sum of two Kirchhoff terms: $(\int_{R^{3}} | \nabla u |^{p} d x) Δ_{p} u$ and $(\int_{R^{3}} | \nabla u |^{q} d x) Δ_{q} u$ , with $p < q$ . Moreover, due to the fact that the nonlinearity f is only continuous, one cannot apply standard $C^{1}$ -Nehari manifold arguments due to the lack of differentiability of the associated Nehari manifold $N$ . We were able to overcome this difficulty by borrowing some abstract critical point results obtained in [53]. Furthermore, to produce nodal solutions, instead of using the Miranda Theorem to get critical points of $g_{u} (ξ, λ) = I (ξ u^{+} + λ u^{-})$ we use an iterative process to build a sequence which converges to a critical point of $g_{u} (ξ, λ)$ .

The paper is organized as follows. In Section 2 we introduce the variational structure. In Section 3 we give some preliminary results which overcome the lack of differentiability of the Nehari manifold. Section 4 is devoted to some technical lemmas used in the proof of the main result. In Section 5 we prove Theorem 1.1.
1.1. Notations

We denote by $B_{R} (x)$ the ball of radius R with center x and we set $B_{R}^{c} (x) = R^{3} ∖ B_{R} (x)$ . Let $1 ⩽ r ⩽ \infty$ and $A \subset R^{3}$ . We denote by $| u |_{L^{r} (A)}$ the $L^{r} (A)$ -norm of the function $u : R^{3} \to R$ belonging to $L^{r} (A)$ . When $A = R^{3}$ , we shall simply write $| u |_{r}$ .

2. Variational framework

Let us introduce the space $\begin{array}{l} E = {u \in D^{1, p} (R^{3}) \cap D^{1, q} (R^{3}) : \int_{R^{3}} V (x) (| u |^{p} + | u |^{q}) d x < \infty} \end{array}$ endowed with the norm $\begin{array}{l} ‖ u ‖ = {(\int_{R^{3}} (a | \nabla u |^{p} + V (x) | u |^{p}) d x)}^{\frac{1}{p}} + {(\int_{R^{3}} (c | \nabla u |^{q} + V (x) | u |^{q}) d x)}^{\frac{1}{q}} . \end{array}$

Let us define the Lebesgue space $\begin{array}{l} L_{K}^{r} (R^{3}) = {u : R^{3} \to R : u is measurable and \int_{R^{3}} K (x) | u |^{r} d x < \infty} \end{array}$ equipped with the norm $\begin{matrix} ‖ u ‖_{L_{K}^{r} (R^{3})} = {(\int_{R^{3}} K (x) | u |^{r} d x)}^{\frac{1}{r}} . \end{matrix}$

We recall the following continuous and compactness results whose proofs can be found in [17]:

Lemma 2.1.
Assume that $(V, K) \in K$ .
If $(V K_{3})$ holds, then $E$ is continuously embedded in $L_{K}^{r} (R^{3})$ for every $r \in [q, q^{}]$ .

If $(V K_{4})$ holds, then* $E$ is continuously embedded in $L_{K}^{m} (R^{3})$ .

Lemma 2.2.
Assume that $(V, K) \in K$ .
If $(V K_{3})$ holds, then $E$ is compactly embedded in $L_{K}^{r} (R^{3})$ for every $r \in (q, q^{})$ .

If $(V K_{4})$ holds, then* $E$ is compactly embedded in $L_{K}^{m} (R^{3})$ .

The last lemma of this section is a compactness result related to the nonlinearity (see [17]).
Lemma 2.3.
Assume that $(V, K) \in K$ and f satisfies $(f_{1})$ – $(f_{2})$ or $({\tilde{f}}_{1})$ – $(f_{2})$ . If $(u_{n})$ is a sequence such that $u_{n} ⇀ u$ in $E$ , then $\begin{array}{l} \int_{R^{3}} K (x) F (u_{n}) d x \to \int_{R^{3}} K (x) F (u) d x \end{array}$ and $\begin{array}{l} \int_{R^{3}} K (x) f (u_{n}) u_{n} d x \to \int_{R^{3}} K (x) f (u) u d x . \end{array}$

We conclude this section by giving the following useful remarks. Remark 2.1.
Let us point out that from assumption $(f_{4})$ it follows that $t \mapsto \frac{1}{2 q} f (t) t - F (t)$ is increasing for $t ⩾ 0$ . Indeed, let $0 < t_{2} < t_{1}$ , then using $(f_{4})$ twice we get $\begin{array}{l} \frac{1}{2 q} f (t_{1}) t_{1} - F (t_{1}) & = \frac{1}{2 q} f (t_{1}) t_{1} - F (t_{2}) - \int_{t_{2}}^{t_{1}} f (τ) d τ \\ = \frac{1}{2 q} f (t_{1}) t_{1} - F (t_{2}) - \int_{t_{2}}^{t_{1}} \frac{f (τ)}{τ^{2 q - 1}} τ^{2 q - 1} d τ \\ > \frac{1}{2 q} f (t_{1}) t_{1} - F (t_{2}) - \frac{f (t_{1})}{t_{1}^{2 q - 1}} \int_{t_{2}}^{t_{1}} τ^{2 q - 1} d τ \\ = \frac{1}{2 q} f (t_{1}) t_{1} - F (t_{2}) - \frac{f (t_{1})}{t_{1}^{2 q - 1}} \frac{t_{1}^{2 q} - t_{2}^{2 q}}{2 q} \\ = \frac{1}{2 q} \frac{f (t_{1})}{t_{1}^{2 q - 1}} t_{2}^{2 q} - F (t_{2}) \\ > \frac{1}{2 q} f (t_{2}) t_{2} - F (t_{2}) . \end{array}$ Similarly, it is possible to prove that $t \mapsto \frac{1}{2 q} f (t) t - F (t)$ is decreasing for $t ⩽ 0$ .
Remark 2.2.
Take $u \in E$ with $u^{\pm} \neq 0$ and $ξ, λ ⩾ 0$ , then $\begin{array}{l} {| \nabla (ξ u^{+} + λ u^{-}) |}^{p} = {| \nabla (ξ u^{+}) |}^{p} + {| \nabla (λ u^{-}) |}^{p} \end{array}$ and using the linearity of F and the positivity of K we also have $\begin{array}{l} \int_{R^{3}} K (x) F (ξ u^{+} + λ u^{-}) d x = \int_{R^{3}} K (x) (F (ξ u^{+}) + F (λ u^{-})) d x . \end{array}$ Hence, for any $u \in E$ with $u^{\pm} \neq 0$ and $ξ, λ ⩾ 0$ we have $\begin{array}{l} (2.1) & I (ξ u^{+} + λ u^{-}) = I (ξ u^{+}) + I (λ u^{-}) . \end{array}$ Moreover, $\begin{array}{l} ⟨ I^{'} (ξ u^{+} + λ u^{-}), ξ u^{+} ⟩ = & ξ^{p} \int_{R^{3}} (a {| \nabla u^{+} |}^{p} + V (x) | u |^{p}) d x + b ξ^{2 p} {(\int_{R^{3}} {| \nabla u^{+} |}^{p} d x)}^{2} \\ + ξ^{q} \int_{R^{3}} (c {| \nabla u^{+} |}^{q} + V (x) | u |^{q}) d x + d ξ^{2 q} {(\int_{R^{3}} {| \nabla u^{+} |}^{q} d x)}^{2} \\ - \int_{R^{3}} K (x) f (ξ u^{+}) ξ u^{+} d x \end{array}$ and $\begin{array}{l} ⟨ I^{'} (ξ u^{+} + λ u^{-}), ξ u^{+} ⟩ = & λ^{p} \int_{R^{3}} (a {| \nabla u^{-} |}^{p} + V (x) | u |^{p}) d x + b λ^{2 p} {(\int_{R^{3}} {| \nabla u^{-} |}^{p} d x)}^{2} \\ + λ^{q} \int_{R^{3}} (c {| \nabla u^{-} |}^{q} + V (x) | u |^{q}) d x + d λ^{2 q} {(\int_{R^{3}} {| \nabla u^{-} |}^{q} d x)}^{2} \\ - \int_{R^{3}} K (x) f (λ u^{-}) λ u^{-} d x . \end{array}$

3. Preliminaries

The Nehari manifold associated with $I$ is given by $\begin{matrix} N = {u \in E ∖ {0} : ⟨ I^{'} (u), u ⟩ = 0} . \end{matrix}$ We denote by $\begin{matrix} M = {w \in N : w^{\pm} \neq 0, ⟨ I^{'} (w), w^{+} ⟩ = ⟨ I^{'} (w), w^{-} ⟩ = 0}, \end{matrix}$ and by $S$ the unit sphere on $E$ . We note that $M \subset N$ .

Once f is only continuous, the following results are crucial, since they allow us to overcome the non-differentiability of $N$ .

Lemma 3.1.
Suppose that $(V, K) \in K$ and f satisfies conditions $(f_{1}) - (f_{4})$ . Then the following properties hold:
For each $u \in E ∖ {0}$ , let $φ_{u} : R_{+} \to R$ be defined by $\begin{matrix} φ_{u} (t) = I (t u) . \end{matrix}$ Then there is a unique $t_{u} > 0$ such that $\begin{array}{l} φ_{u}^{'} (t) > 0 for t \in (0, t_{u}) and φ_{u}^{'} (t) < 0 for t \in (t_{u}, \infty); \end{array}$

There is $τ > 0$ , independent of u, such that $t_{u} ⩾ τ$ for every $u \in S$ . Moreover, for each compact set $K \subset S$ , there is $C_{K} > 0$ such that $t_{u} ⩽ C_{K}$ for every $u \in K$ ;

The map $\hat{m} : E ∖ {0} \to N$ given by $\hat{m} (u) : = t_{u} u$ is continuous and $m : = \hat{m} |_{S}$ is a homeomorphism between $S$ and $N$ . Moreover, $m^{- 1} (u) = \frac{u}{‖ u ‖}$ .

Proof.
(a) Let us assume that $(V K_{3})$ holds. Then, using assumptions $(f_{1})$ – $(f_{2})$ given $ε > 0$ there exists $C_{ε} > 0$ such that $\begin{array}{l} | f (t) | ⩽ ε | t |^{p - 1} + C_{ε} | t |^{q^{} - 1} . \end{array}$ Therefore $\begin{array}{l} I (t u) ⩾ & \frac{a}{p} t^{p} \int_{R^{3}} | \nabla u |^{p} d x + \frac{c}{q} t^{q} \int_{R^{3}} | \nabla u |^{q} d x + \int_{R^{3}} V (x) (\frac{t^{p}}{p} | u |^{p} + \frac{t^{q}}{q} | u |^{q}) d x \\ - ε t^{p} \int_{R^{3}} K (x) | u |^{p} d x - C_{ε} t^{q^{}} \int_{R^{3}} K (x) | u |^{q^{}} d x \\ ⩾ & \frac{a}{p} t^{p} \int_{R^{3}} | \nabla u |^{p} d x + \frac{c}{q} t^{q} \int_{R^{3}} | \nabla u |^{q} d x + \int_{R^{3}} V (x) (\frac{t^{p}}{p} | u |^{p} + \frac{t^{q}}{q} | u |^{q}) d x \\ - ε {| \frac{K}{V} |}_{\infty} t^{p} \int_{R^{3}} V (x) | u |^{p} d x - C_{ε}^{'} | K |_{\infty} t^{q^{}} ‖ u ‖^{q^{}} \\ ⩾ & \frac{a}{p} t^{p} \int_{R^{3}} | \nabla u |^{p} d x + (\frac{1}{p} - ε {| \frac{K}{V} |}_{\infty}) t^{p} \int_{R^{3}} V (x) | u |^{p} d x \\ + \frac{t^{q}}{q} \int_{R^{3}} (c | \nabla u |^{q} + V (x) | u |^{q}) d x - C_{ε}^{'} | K |_{\infty} t^{q^{}} ‖ u ‖^{q^{}} . \end{array}$ Choosing $ε \in (0, {(2 p | \frac{K}{V} |_{\infty})}^{- 1})$ , we get $t_{0} > 0$ sufficiently small such that $\begin{array}{l} 0 < φ_{u} (t) = I (t u) for all t \in (0, t_{0}) . \end{array}$ Now, we assume that $(V K_{4})$ is true. Then, there exists a positive constant $C_{m}$ such that, for each $ε \in (0, C_{m})$ we get $R > 0$ such that for any $u \in E$ $\begin{array}{l} \int_{B_{R}^{c} (0)} K (x) | u |^{m} d x ⩽ ε \int_{B_{R}^{c} (0)} (V (x) | u |^{p} + | u |^{q^{}}) d x . \end{array}$ Using assumptions $({\tilde{f}}_{1})$ and $(f_{2})$ , the Hölder and Sobolev inequality we get $\begin{array}{l} I (t u) ⩾ & \frac{a}{p} t^{p} \int_{R^{3}} | \nabla u |^{p} d x + \frac{c}{q} t^{q} \int_{R^{3}} | \nabla u |^{q} d x + \int_{R^{3}} V (x) (\frac{t^{p}}{p} | u |^{p} + \frac{t^{q}}{q} | u |^{q}) d x \\ - C_{1} t^{m} (\int_{B_{R} (0)} K (x) | u |^{m} d x + \int_{B_{R}^{c} (0)} K (x) | u |^{m} d x) - C_{2} t^{q^{}} \int_{R^{3}} K (x) | u |^{q^{}} d x \\ ⩾ & \frac{a}{p} t^{p} \int_{R^{3}} | \nabla u |^{p} d x + \frac{c}{q} t^{q} \int_{R^{3}} | \nabla u |^{q} d x + \int_{R^{3}} V (x) (\frac{t^{p}}{p} | u |^{p} + \frac{t^{q}}{q} | u |^{q}) d x \\ - C_{1} t^{m} | K |_{\frac{q^{}}{q^{} - m}} | u |_{q^{}}^{m} - C_{1} t^{m} ε \int_{B_{R}^{c} (0)} (V (x) | u |^{p} + | u |^{q^{}}) d x - C_{2}^{'} t^{q^{}} | K |_{\infty} ‖ u ‖^{q^{}} \\ ⩾ & \frac{a}{p} t^{p} \int_{R^{3}} | \nabla u |^{p} d x + \frac{c}{q} t^{q} \int_{R^{3}} | \nabla u |^{q} d x + \int_{R^{3}} V (x) (\frac{t^{p}}{p} | u |^{p} + \frac{t^{q}}{q} | u |^{q}) d x \\ - C_{2}^{'} t^{q^{}} | K |_{\infty} ‖ u ‖^{q^{}} - \tilde{C} t^{m} [| K |_{\frac{q^{}}{q^{} - m}} | u |_{q^{}}^{m} + ε ‖ u ‖^{p} + ε ‖ u ‖^{q^{}}] . \end{array}$

Therefore there exists $t_{0} > 0$ sufficiently small such that $\begin{array}{l} 0 < φ_{u} (t) = I (t u) for all t \in (0, t_{0}) . \end{array}$ Let $A \subset supp u$ be a measurable set with finite and positive measure. From $F (t) ⩾ 0$ for any $t \in R$ , $1 < p < q$ , and combining assumptions $(f_{3})$ together with Fatou’s lemma, we obtain $\begin{array}{l} \underset{t \to \infty}{lim sup} \frac{I (t u)}{‖ t u ‖^{2 q}} \\ ⩽ \underset{t \to \infty}{lim sup} {\frac{1}{p} \frac{1}{t^{2 q - p} ‖ u ‖^{2 q - p}} + \frac{b}{2 p} \frac{1}{t^{2 (q - p)} ‖ u ‖^{2 (q - p)}} + \frac{1}{q} \frac{1}{t^{q} ‖ u ‖^{q}} + \frac{d}{2 q} \\ - \int_{A} K (x) \frac{F (t u)}{{(t u)}^{2 q}} {(\frac{u}{‖ u ‖})}^{2 q} d x} \\ ⩽ \frac{d}{2 q} - \underset{t \to \infty}{lim inf} \int_{A} K (x) \frac{F (t u)}{{(t u)}^{2 q}} {(\frac{u}{‖ u ‖})}^{2 q} d x ⩽ - \infty . \end{array}$ Hence there exists $\bar{t} > 0$ large enough for which $φ_{u} (\bar{t}) < 0$ . By virtue of the continuity of $φ_{u}$ and using $(f_{4})$ , there exists $t_{u} > 0$ which is a global maximum of $φ_{u}$ with $t_{u} u \in N$ .

Next, we aim to prove that such $t_{u}$ is the unique critical point of $φ_{u}$ . Assume by contradiction that there exist $0 < t_{1} < t_{2}$ which are critical points of $φ_{u}$ . Then, from the definition of $φ_{u}$ we get $\begin{array}{l} \frac{1}{t_{1}^{2 q - p}} \int_{R^{3}} (a | \nabla u |^{p} + V (x) | u |^{p}) d x + \frac{b}{t_{1}^{2 (q - p)}} {(\int_{R^{3}} | \nabla u |^{p} d x)}^{2} + \frac{1}{t_{1}^{q}} \int_{R^{3}} (c | \nabla u |^{q} + V (x) | u |^{q}) d x \\ + d {(\int_{R^{3}} | \nabla u |^{q} d x)}^{2} - \int_{R^{3}} K (x) \frac{f (t_{1} u)}{{(t_{1} u)}^{2 q - 1}} u^{2 q} d x = 0 \end{array}$ and $\begin{array}{l} \frac{1}{t_{2}^{2 q - p}} \int_{R^{3}} (a | \nabla u |^{p} + V (x) | u |^{p}) d x + \frac{b}{t_{2}^{2 (q - p)}} {(\int_{R^{3}} | \nabla u |^{p} d x)}^{2} + \frac{1}{t_{2}^{q}} \int_{R^{3}} (c | \nabla u |^{q} + V (x) | u |^{q}) d x \\ + d {(\int_{R^{3}} | \nabla u |^{q} d x)}^{2} - \int_{R^{3}} K (x) \frac{f (t_{2} u)}{{(t_{2} u)}^{2 q - 1}} u^{2 q} d x = 0 . \end{array}$ These equalities together with assumption $(f_{4})$ imply that $\begin{array}{l} 0 > & (\frac{1}{t_{2}^{2 q - p}} - \frac{1}{t_{1}^{2 q - p}}) \int_{R^{3}} (a | \nabla u |^{p} + V (x) | u |^{p}) d x + b (\frac{1}{t_{2}^{2 (q - p)}} - \frac{1}{t_{1}^{2 (q - p)}}) {(\int_{R^{3}} | \nabla u |^{p} d x)}^{2} \\ + (\frac{1}{t_{2}^{q}} - \frac{1}{t_{1}^{q}}) \int_{R^{3}} (c | \nabla u |^{q} + V (x) | u |^{q}) d x \\ = & \int_{R^{3}} K (x) (\frac{f (t_{2} u)}{{(t_{2} u)}^{2 q - 1}} - \frac{f (t_{1} u)}{{(t_{1} u)}^{2 q - 1}}) u^{2 q} d x ⩾ 0 \end{array}$ which leads to a contradiction.

(b) By (a) there exists $t_{u} > 0$ such that $φ_{u}^{'} (t_{u}) = 0$ , or equivalently $⟨ I^{'} (t_{u} u), t_{u} u ⟩ = 0$ , and arguing as before, we find a positive τ independent of u such that $t_{u} ⩾ τ$ .

Now, let $K \subset S$ be a compact set and assume by contradiction that there exists $(u_{n}) \subset K$ such that $t_{u_{n}} \to \infty$ . Hence, there exists $u \in K$ such that $u_{n} \to u$ in $E$ . Proceeding as in (a) we can prove that $I (t_{u_{n}} u_{n}) \to - \infty$ in $R$ . Since $t_{u_{n}} u_{n} \in N$ , from Remark 2.1 and recalling that $1 < p < q$ we get $\begin{array}{l} I (t_{u_{n}} u_{n}) = & I (t_{u_{n}} u_{n}) - \frac{1}{2 q} ⟨ I^{'} (t_{u_{n}} u_{n}), t_{u_{n}} u_{n} ⟩ \\ = & (\frac{1}{p} - \frac{1}{2 q}) t_{u_{n}}^{p} \int_{R^{3}} (a | \nabla u_{n} |^{p} + V (x) | u_{n} |^{p}) d x + \frac{b}{2} (\frac{1}{p} - \frac{1}{q}) t_{u_{n}}^{2 p} {(\int_{R^{3}} {| \nabla (u_{n}) |}^{p} d x)}^{2} \\ + \frac{1}{2 q} t_{u_{n}}^{q} \int_{R^{3}} (c | \nabla u_{n} |^{q} + V (x) | u_{n} |^{q}) d x \\ + \int_{R^{3}} K (x) [\frac{1}{2 q} f (t_{u_{n}} u_{n}) t_{u_{n}} u_{n} - F (t_{u_{n}} u_{n})] d x ⩾ 0, \end{array}$ which leads to a contradiction.

(c) Note that $\hat{m}$ , m and $m^{- 1}$ are well defined. In fact, from (a) we deduce that for each $u \in E ∖ {0}$ there exists a unique $\hat{m} (u) \in N$ .

On the other hand, if $u \in N$ then $u \neq 0$ , and we deduce that $m^{- 1} (u) = \frac{u}{‖ u ‖} \in S$ and $m^{- 1}$ is well defined. We point out that $\begin{array}{l} m^{- 1} (m (u)) = m^{- 1} (t_{u} u) = \frac{t_{u} u}{‖ t_{u} u ‖} = u for any u \in S, \\ m (m^{- 1} (u)) = m (\frac{u}{‖ u ‖}) = t_{\frac{u}{‖ u ‖}} \frac{u}{‖ u ‖} = u for any u \in N, \end{array}$ so m is bijective with its inverse $m^{- 1}$ continuous.

Now, let $(u_{n}) \subset E$ and $u \in E ∖ {0}$ such that $u_{n} \to u$ in $E$ . Using (b) we can find $t_{0} > 0$ such that $t_{u_{n}} ‖ u_{n} ‖ = t_{\frac{u}{‖ u ‖}} \to t_{0}$ . Therefore $t_{u_{n}} \to \frac{t_{0}}{‖ u ‖}$ . Using the fact that $t_{u_{n}} u_{n} \in N$ and taking the limit as $n \to \infty$ we deduce that $\frac{t_{0}}{‖ u ‖} u \in N$ and $t_{u} = \frac{t_{0}}{‖ u ‖}$ . This implies that $\hat{m} (u_{n}) \to \hat{m} (u)$ , hence $\hat{m}$ and m are continuous functions. □

Let us define the maps $\begin{matrix} \hat{ψ} : E \to R and ψ : S \to R, \end{matrix}$ by $\hat{ψ} (u) : = I (\hat{m} (u))$ and $ψ : = \hat{ψ} |_{S}$ .

The next result is a consequence of Lemma 2.1 (see [53]).
Proposition 3.1.
Suppose that $(V, K) \in K$ and f fulfills $(f_{1}) - (f_{4})$ . Then the following properties hold:
$\hat{ψ} \in C^{1} (E ∖ {0}, R)$ and $\begin{matrix} ⟨ {\hat{ψ}}^{'} (u), v ⟩ = \frac{‖ \hat{m} (u) ‖}{‖ u ‖} ⟨ I^{'} (\hat{m} (u)), v ⟩ for all u \in E ∖ {0} and v \in E; \end{matrix}$

$ψ \in C^{1} (S, R)$ and $⟨ ψ^{'} (u), v ⟩ = ‖ m (u) ‖ ⟨ I^{'} (m (u)), v ⟩$ , for every $v \in T_{u} S$ ;

If $(u_{n})$ is a ${(PS)}_{d}$ sequence for ψ, then ${m (u_{n})}_{n \in N}$ is a ${(PS)}_{d}$ sequence for $I$ . Moreover, if $(u_{n}) \subset N$ is a bounded ${(PS)}_{d}$ sequence for $I$ , then ${m^{- 1} (u_{n})}_{n \in N}$ is a ${(PS)}_{d}$ sequence for the functional ψ;

u is a critical point of ψ if and only if $m (u)$ is a nontrivial critical point for $I$ . Moreover, the corresponding critical values coincide and $\begin{matrix} inf_{u \in S} ψ (u) = inf_{u \in N} I (u) . \end{matrix}$

We notice that the following equalities hold: $\begin{array}{l} (3.1) & d_{\infty} : = inf_{u \in N} I (u) = inf_{u \in E ∖ {0}} max_{t > 0} I (t u) = inf_{u \in S} max_{t > 0} I (t u) . \end{array}$ In particular, from (a) of Lemma 2.1 and (3.1) it follows that $\begin{matrix} (3.2) & d_{\infty} > 0 . \end{matrix}$
4. Technical lemmas

For each $u \in E$ with $u^{\pm} \neq 0$ , let us introduce the map $g_{u} : [0, \infty) \times [0, \infty) \to R$ defined by $\begin{matrix} g_{u} (ξ, λ) = I (ξ u^{+} + λ u^{-}) . \end{matrix}$

Lemma 4.1.
Suppose that $(V, K) \in K$ and f fulfills $(f_{1})$ – $(f_{4})$ . Then the following properties hold:
The pair $(ξ, λ)$ is a critical point of $g_{u}$ with $ξ, λ > 0$ if and only if $ξ u^{+} + λ u^{-} \in M$ .

The map $g_{u}$ has a unique critical point $(ξ_{+}, λ_{-})$ , with $ξ_{+} = ξ_{+} (u) > 0$ and $λ_{-} = λ_{-} (u) > 0$ which is the unique global maximum point of $g_{u}$ .

The maps $a_{+} (r) : = \frac{\partial g_{u}}{\partial ξ} (r, λ_{-})$ and $a_{-} (r) : = \frac{\partial g_{u}}{\partial λ} (ξ_{+}, r) r$ are such that $a_{+} (r) > 0$ if $r \in (0, ξ_{+})$ , $a_{+} (r) < 0$ if $r \in (ξ_{+}, \infty)$ , $a_{-} (r) > 0$ if $r \in (0, λ_{-})$ and $a_{-} (r) < 0$ if $r \in (λ_{-}, \infty)$ .

Proof.
(i) Let us point out that the gradient of $g_{u}$ is given by $\begin{array}{l} \nabla g_{u} (ξ, λ) & = (\frac{\partial g_{u}}{\partial ξ} (ξ, λ), \frac{\partial g_{u}}{\partial λ} (ξ, λ)) \\ = (⟨ I^{'} (ξ u^{+} + λ u^{-}), u^{+} ⟩, ⟨ I^{'} (ξ u^{+} + λ u^{-}), u^{-} ⟩) \\ = (\frac{1}{ξ} ⟨ I^{'} (ξ u^{+} + λ u^{-}), ξ u^{+} ⟩, \frac{1}{λ} ⟨ I^{'} (ξ u^{+} + λ u^{-}), λ u^{-} ⟩) . \end{array}$ Now, the pair $(ξ, λ)$ , with $ξ, λ > 0$ , is a critical point of $g_{u}$ if and only if $\begin{array}{l} ⟨ I^{'} (ξ u^{+} + λ u^{-}), ξ u^{+} ⟩ = 0 and ⟨ I^{'} (ξ u^{+} + λ u^{-}), λ u^{-} ⟩ = 0 \end{array}$ that is $ξ u^{+} + λ u^{-} \in M$ .

(ii) First we prove that $M \neq \emptyset$ . For each $u \in E$ with $u^{\pm} \neq 0$ and $λ_{0}$ fixed, let us define the function $g_{1} (ξ) : [0, \infty) \to [0, \infty)$ by $g_{1} (ξ) = g_{u} (ξ, λ_{0})$ .

As in Lemma 2.1, the map $g_{1}$ has a maximum positive point and furthermore there exists $ξ_{0} = ξ_{0} (u, λ_{0}) > 0$ such that $g_{1}^{'} (ξ) > 0$ for $ξ \in (0, ξ_{0})$ , $g_{1}^{'} (ξ) < 0$ for $ξ \in (ξ_{0}, \infty)$ and $g_{1}^{'} (ξ_{0}) = 0$ .

Hence, it is well defined the function $η_{1} : [0, \infty) \to [0, \infty)$ defined by $η_{1} (λ) : = ξ (u, λ)$ , where $ξ (u, λ)$ satisfies the properties just mentioned with λ in place of $λ_{0}$ . Exploiting the definition of $g_{1}$ , for all $λ ⩾ 0$ we get $\begin{matrix} (4.1) & g_{1}^{'} (η_{1} (λ)) = \frac{\partial g_{u}}{\partial ξ} (η_{1} (λ), λ) = ⟨ I^{'} (η_{1} (λ) u^{+} + λ u^{-}), η_{1} (λ) u^{+} ⟩ = 0 . \end{matrix}$ Note that, when $u^{\pm} \neq 0$ and the support of $u^{+}$ and $u^{-}$ are disjoint in $R^{3}$ , it follows that (4.1) is equivalent to $\begin{array}{l} η_{1} {(λ)}^{p} \int_{R^{3}} (a {| \nabla u^{+} |}^{p} + V (x) {| u^{+} |}^{p}) d x + b η_{1} {(λ)}^{2 p} {(\int_{R^{3}} {| \nabla u^{+} |}^{p} d x)}^{2} \\ + η_{1} {(λ)}^{q} \int_{R^{3}} (c {| \nabla u^{+} |}^{q} + V (x) {| u^{+} |}^{q}) d x + d η_{1} {(λ)}^{2 q} {(\int_{R^{3}} {| \nabla u^{+} |}^{q} d x)}^{2} \\ (4.2) & = \int_{R^{3}} K (x) f (η_{1} (λ) u^{+}) η_{1} (λ) u^{+} d x . \end{array}$ First we note that $η_{1}$ is a continuous map. Indeed, let $(λ_{n})$ be a sequence such that $λ_{n} \to λ_{0}$ as $n \to \infty$ in $R$ , and assume that $η_{1} (λ_{n}) \to \infty$ as $n \to \infty$ . We aim to prove that $(η_{1} (λ_{n}))$ is bounded. By contradiction, let us suppose that there us a subsequence, still denoted by $(λ_{n})$ , such that $η_{1} (λ_{n}) \to \infty$ as $n \to \infty$ . In particular, for n sufficiently large we have that $η_{1} (λ_{n}) ⩾ λ_{n}$ . From (4.2) we get $\begin{array}{l} \frac{1}{η_{1} {(λ_{n})}^{2 q - p}} \int_{R^{3}} (a {| \nabla u^{+} |}^{p} + V (x) {| u^{+} |}^{p}) d x + \frac{b}{η_{1} {(λ_{n})}^{2 (q - p)}} {(\int_{R^{3}} {| \nabla u^{+} |}^{p} d x)}^{2} \\ + \frac{1}{η_{1} {(λ_{n})}^{q}} \int_{R^{3}} (c {| \nabla u^{+} |}^{q} + V (x) {| u^{+} |}^{q}) d x + d {(\int_{R^{3}} {| \nabla u^{+} |}^{q} d x)}^{2} \\ = \int_{R^{3}} K (x) \frac{f (η_{1} (λ_{n}) u^{+})}{{(η_{1} (λ_{n}) u^{+})}^{2 q - 1}} {(u^{+})}^{2 q} d x, \end{array}$ recalling that $η_{1} (λ_{n}) \to \infty$ as $n \to \infty$ , $λ_{n} \to λ_{0}$ as $n \to \infty$ and exploiting $(f_{3})$ , $(f_{4})$ and Fatou’s lemma, we get a contradiction. This shows that $(η_{1} (λ_{n}))$ is bounded. So there exists $ξ_{0} ⩾ 0$ such that $η_{1} (λ_{n}) \to ξ_{0}$ as $n \to \infty$ . Now, using (4.2) with $λ = λ_{n}$ and taking $n \to \infty$ we deduce $\begin{array}{l} ξ_{0}^{p} \int_{R^{3}} (a {| \nabla u^{+} |}^{p} + V (x) {| u^{+} |}^{p}) d x + b ξ_{0}^{2 p} {(\int_{R^{3}} {| \nabla u^{+} |}^{p} d x)}^{2} \\ + ξ_{0}^{q} \int_{R^{3}} (c {| \nabla u^{+} |}^{q} + V (x) {| u^{+} |}^{q}) d x + d ξ_{0}^{2 q} {(\int_{R^{3}} {| \nabla u^{+} |}^{q} d x)}^{2} \\ = \int_{R^{3}} K (x) f (ξ_{0} u^{+}) ξ_{0} u^{+} d x \end{array}$ that is $\begin{matrix} g_{1}^{'} (ξ_{0}) = \frac{\partial g_{u}}{\partial ξ} (ξ_{0}, λ_{0}) = 0 . \end{matrix}$ Hence, $ξ_{0} = η_{1} (λ_{0})$ which implies that $η_{1}$ is a continuous map.

Moreover, $η_{1} (0) > 0$ . Indeed, if we suppose by contradiction that there exists a sequence $(λ_{n})$ such that $η_{1} (λ_{n}) \to 0^{+}$ and $λ_{n} \to 0$ as $n \to \infty$ , then gathering (4.2) with $(f_{1})$ we get $\begin{array}{l} d {(\int_{R^{3}} {| \nabla u^{+} |}^{q} d x)}^{2} ⩽ \int_{R^{3}} K (x) \frac{f (η_{1} (λ_{n}) u^{+})}{{(η_{1} (λ_{n}) u^{+})}^{2 q - 1}} {(u^{+})}^{2 q} d x \to 0, \end{array}$ which gives a contradiction. Finally, we can also see that $η_{1} (λ) ⩽ s$ for s sufficiently large.

In a similar fashion, for each $ξ_{0} ⩾ 0$ we define $g_{2} (λ) = g_{u} (ξ_{0}, λ)$ , and we can introduce a map $η_{2}$ that satisfies the same properties as $η_{1}$ . In particular, there exists a positive constant $A_{1}$ such that for each $ξ, λ ⩾ A_{1}$ it holds that $η_{1} (λ) ⩽ λ$ and $η_{2} (ξ) ⩽ ξ$ .

Let $A_{2} : = max {{max}_{λ \in [0, A_{1}]} η_{1} (λ), {max}_{ξ \in [0, A_{1}]} η_{2} (ξ)}$ and set $A : = max {C_{1}, C_{2}}$ .

Next, we introduce the map $Φ : [0, A] \times [0, A] \to R^{2}$ defined by $Φ (ξ, λ) = (η_{1} (λ), η_{2} (ξ))$ . First we note that Φ is a continuous map due to the continuity of $η_{1}$ and $η_{2}$ , moreover for every $s \in [0, A]$ we can see that $\begin{array}{l} if λ ⩾ A_{1} then η_{1} (λ) ⩽ λ ⩽ A \\ if λ ⩽ A_{1} then η_{1} (λ) ⩽ max_{λ \in [0, A_{1}]} η_{1} (λ) ⩽ A_{2} ⩽ A \end{array}$ and similarly $\begin{array}{l} if ξ ⩾ A_{1} then η_{2} (ξ) ⩽ t ⩽ A, \\ if ξ ⩽ A_{1} then η_{2} (ξ) ⩽ max_{ξ \in [0, A_{1}]} η_{1} (ξ) ⩽ A_{2} ⩽ A, \end{array}$ hence $Φ ([0, A] \times [0, A]) \subset [0, A] \times [0, A]$ . Applying Brouwer’s fixed point theorem there exists $(ξ_{+}, λ_{-}) \in [0, A] \times [0, A]$ such that $\begin{array}{l} Φ (ξ_{+}, λ_{-}) = (η_{1} (s_{-}), η_{2} (t_{+})) = (ξ_{+}, λ_{-}) . \end{array}$ Since $η_{1}, η_{2}$ are positive functions, $ξ_{+}, λ_{-} > 0$ . In addition $\nabla g_{u} (ξ_{+}, λ_{-}) = 0$ , hence $(ξ_{+}, λ_{-})$ is a critical point of $g_{u}$ .

Next, we show the uniqueness of $(ξ_{+}, λ_{-})$ . First, take $w \in M$ . By $w = w^{+} + w^{-}$ and the definition of $g_{w}$ it follows that $\nabla g_{w} (1, 1) = (0, 0)$ , hence $(1, 1)$ is a critical point of $g_{w}$ . Our aim is to show that $(1, 1)$ is the unique critical point of $g_{w}$ with positive coordinates. With this goal, let $(ξ_{0}, λ_{0})$ be a critical point go $g_{w}$ with $0 < ξ_{0} ⩽ λ_{0}$ . Using $\frac{\partial g_{w}}{\partial ξ} (ξ_{0}, λ_{0}) = 0$ , which is equivalent to $⟨ I^{'} (ξ_{0} w^{+} + λ_{0} w^{-}), ξ_{0} w^{+} ⟩ = 0$ , we can see that $\begin{array}{l} \frac{1}{ξ_{0}^{2 q - p}} \int_{R^{3}} (a {| \nabla w^{+} |}^{p} + V (x) {| w^{+} |}^{p}) d x + \frac{b}{ξ_{0}^{2 (q - p)}} {(\int_{R^{3}} {| \nabla w^{+} |}^{p} d x)}^{2} \\ + \frac{1}{ξ_{0}^{q}} \int_{R^{3}} (c {| \nabla w^{+} |}^{q} + V (x) {| w^{+} |}^{q}) d x + d {(\int_{R^{3}} {| \nabla w^{+} |}^{q} d x)}^{2} \\ = \int_{R^{3}} K (x) \frac{f (ξ_{0} w^{+})}{{(ξ_{0} w^{+})}^{2 q - 1}} {(w^{+})}^{2 q} d x . \end{array}$ Exploiting the fact that $w \in M$ we have $\begin{array}{l} \int_{R^{3}} (a {| \nabla w^{+} |}^{p} + V (x) {| w^{+} |}^{p}) d x + b {(\int_{R^{3}} {| \nabla w^{+} |}^{p} d x)}^{2} \\ + \int_{R^{3}} (c {| \nabla w^{+} |}^{q} + V (x) {| w^{+} |}^{q}) d x + d {(\int_{R^{3}} {| \nabla w^{+} |}^{q} d x)}^{2} \\ = \int_{R^{3}} K (x) f (w^{+}) w^{+} d x \end{array}$ and subtracting we have $\begin{array}{l} (\frac{1}{ξ_{0}^{2 q - p}} - 1) \int_{R^{3}} (a {| \nabla w^{+} |}^{p} + V (x) {| w^{+} |}^{p}) d x + b (\frac{1}{ξ_{0}^{2 q - 2 p}} - 1) {(\int_{R^{3}} {| \nabla w^{+} |}^{p} d x)}^{2} \\ + (\frac{1}{ξ_{0}^{q}} - 1) \int_{R^{3}} (c {| \nabla w^{+} |}^{q} + V (x) {| w^{+} |}^{q}) d x \\ (4.3) & = \int_{R^{3}} K (x) (\frac{f (ξ_{0} w^{+})}{{(ξ_{0} w^{+})}^{2 q - 1}} - \frac{f (w^{+})}{{(w^{+})}^{2 q}}) {(w^{+})}^{2 q} d x . \end{array}$ Using (4.3) and $(f_{4})$ we get $ξ_{0} ⩾ 1$ .

Similarly, from $\frac{\partial g_{w}}{\partial λ} (ξ_{0}, λ_{0}) = 0$ we obtain $\begin{array}{l} \frac{1}{λ_{0}^{2 q - p}} \int_{R^{3}} (a {| \nabla w^{-} |}^{p} + V (x) {| w^{-} |}^{p}) d x + \frac{b}{λ_{0}^{2 (q - p)}} {(\int_{R^{3}} {| \nabla w^{-} |}^{p} d x)}^{2} \\ + \frac{1}{λ_{0}^{q}} \int_{R^{3}} (c {| \nabla w^{-} |}^{q} + V (x) {| w^{-} |}^{q}) d x + d {(\int_{R^{3}} {| \nabla w^{-} |}^{q} d x)}^{2} \\ = \int_{R^{3}} K (x) \frac{f (λ_{0} w^{-})}{{(λ_{0} w^{-})}^{2 q - 1}} {(w^{-})}^{2 q} d x . \end{array}$ Note that from $w \in M$ we also deduce that $\begin{array}{l} \int_{R^{3}} (a {| \nabla w^{-} |}^{p} + V (x) {| w^{-} |}^{p}) d x + b {(\int_{R^{3}} {| \nabla w^{-} |}^{p} d x)}^{2} \\ + \int_{R^{3}} (c {| \nabla w^{-} |}^{q} + V (x) {| w^{-} |}^{q}) d x + d {(\int_{R^{3}} {| \nabla w^{-} |}^{q} d x)}^{2} \\ = \int_{R^{3}} K (x) f (w^{-}) w^{-} d x . \end{array}$ Subtracting these last two equality and using assumption $(f_{4})$ we get $0 < ξ_{0} ⩽ λ_{0} ⩽ 1$ . Hence $ξ_{0} = λ_{0} = 1$ , this shows that $(1, 1)$ is the unique critical point of $g_{w}$ with positive coordinates.

Next, take $u \in E$ such that $u^{\pm} \neq 0$ . Let $(ξ_{1}, λ_{1})$ and $(ξ_{2}, λ_{2})$ be two critical points of $g_{u}$ such that $ξ_{i}, λ_{i} > 0$ for $i = 1, 2$ . Define $\begin{array}{l} U_{1} = ξ_{1} u^{+} + λ_{1} u^{-} and U_{2} = ξ_{2} u^{+} + λ_{2} u^{-} . \end{array}$ Then we have that $U_{1}, U_{2} \in M$ and $U_{1}^{\pm} \neq 0$ . Furthermore, recalling that $ξ_{1}, λ_{1} > 0$ we have $\begin{array}{l} \frac{ξ_{2}}{ξ_{1}} U_{1}^{+} + \frac{λ_{2}}{λ_{1}} U_{1}^{-} = \frac{ξ_{2}}{ξ_{1}} ξ_{1} u^{+} + \frac{λ_{2}}{λ_{1}} λ_{1} u^{-} = ξ_{2} u^{+} + λ_{2} u^{-} = U_{2} \in M \end{array}$ hence from (i) we deduce that $(\frac{ξ_{2}}{ξ_{1}}, \frac{λ_{2}}{λ_{1}})$ is a critical point of $g_{U_{1}}$ . Due to the fact that $U_{1} \in M$ we infer that $\frac{ξ_{2}}{ξ_{1}} = \frac{λ_{2}}{λ_{1}} = 1$ , that is $ξ_{1} = ξ_{2}$ and $λ_{1} = λ_{2}$ , from which follows the uniqueness.

Now we prove that $g_{u}$ has a maximum global point. Let $Ω^{+} \subset supp u^{+}$ and $Ω^{-} \subset supp u^{-}$ be positive with finite measure. Gathering (2.1) with $(f_{3})$ and Fatou’s lemma we get $\begin{array}{l} g_{u} (ξ, λ) = & I (ξ u^{+}) + I (λ u^{-}) \\ = & {ξ^{p} \int_{R^{3}} (a {| \nabla u^{+} |}^{p} + V (x) {| u^{+} |}^{p}) d x + \frac{b}{2 p} ξ^{2 p} {(\int_{R^{3}} {| \nabla u^{+} |}^{p} d x)}^{2} \\ + ξ^{q} \int_{R^{3}} (c {| \nabla u^{+} |}^{q} + V (x) {| u^{+} |}^{q}) d x + \frac{d}{2 q} ξ^{2 q} {(\int_{R^{3}} {| \nabla u^{+} |}^{q} d x)}^{2} \\ - \int_{Ω^{+}} K (x) F (ξ u^{+}) d x} \\ + {λ^{p} \int_{R^{3}} (a {| \nabla u^{-} |}^{p} + V (x) {| u^{-} |}^{p}) d x + \frac{b}{2 p} λ^{2 p} {(\int_{R^{3}} {| \nabla u^{-} |}^{p} d x)}^{2} \\ + λ^{q} \int_{R^{3}} (c {| \nabla u^{-} |}^{q} + V (x) {| u^{-} |}^{q}) d x + \frac{d}{2 q} λ^{2 q} {(\int_{R^{3}} {| \nabla u^{-} |}^{q} d x)}^{2} \\ - \int_{Ω^{-}} K (x) F (λ u^{-}) d x} \\ ⩽ & {ξ^{p} \int_{R^{3}} (a {| \nabla u^{+} |}^{p} + V (x) {| u^{+} |}^{p}) d x + \frac{b}{2 p} ξ^{2 p} {(\int_{R^{3}} {| \nabla u^{+} |}^{p} d x)}^{2} \\ + ξ^{q} \int_{R^{3}} (c {| \nabla u^{+} |}^{q} + V (x) {| u^{+} |}^{q}) d x + \frac{d}{2 q} ξ^{2 q} {(\int_{R^{3}} {| \nabla u^{+} |}^{q} d x)}^{2} \\ - C_{1} ξ^{2 q} \int_{Ω^{+}} K (x) {(u^{+})}^{2 q} d x} \\ + {λ^{p} \int_{R^{3}} (a {| \nabla u^{-} |}^{p} + V (x) {| u^{-} |}^{p}) d x + \frac{b}{2 p} λ^{2 p} {(\int_{R^{3}} {| \nabla u^{-} |}^{p} d x)}^{2} \\ + λ^{q} \int_{R^{3}} (c {| \nabla u^{-} |}^{q} + V (x) {| u^{-} |}^{q}) d x + \frac{d}{2 q} λ^{2 q} {(\int_{R^{3}} {| \nabla u^{-} |}^{q} d x)}^{2} \\ - C_{1} λ^{2 q} \int_{Ω^{-}} K (x) {(u^{-})}^{2 q} d x} \\ + C_{2} | K |_{\infty} (| Ω^{+} | + | Ω^{-} |) \to - \infty as | (ξ, λ) | \to \infty . \end{array}$ Combining the fact that $g_{u}$ is a continuous function with $g_{u} (ξ, λ) \to - \infty$ as $| (ξ, λ) | \to \infty$ , we conclude that $g_{u}$ assumes a global maximum in $(\bar{ξ}, \bar{λ}) \in (0, \infty) \times (0, \infty)$ . Using (2.1), for any $ξ, λ ⩾ 0$ we get $\begin{array}{l} g_{u} (ξ, 0) + g_{u} (0, λ) = I (ξ u^{+}) + I (λ u^{-}) = I (ξ u^{+} + λ u^{-}) = g_{u} (ξ, λ), \end{array}$ therefore $\begin{array}{l} 0 < max_{ξ ⩾ 0} g_{u} (ξ, 0) < max_{ξ, λ ⩾ 0} g_{u} (ξ, λ) = g_{u} (\bar{ξ}, \bar{λ}) \end{array}$ and $\begin{array}{l} 0 < max_{λ ⩾ 0} g_{u} (0, λ) < max_{ξ, λ ⩾ 0} g_{u} (ξ, λ) = g_{u} (\bar{ξ}, \bar{λ}) \end{array}$ showing that $(\bar{ξ}, \bar{λ}) \in (0, \infty) \times (0, \infty)$ . By virtue of the uniqueness of the critical point of $g_{u}$ we have that $(ξ_{+}, λ_{-}) = (\bar{ξ}, \bar{λ})$ , hence $(ξ_{+}, λ_{-})$ is the unique global maximum of $g_{u}$ .

(iii) From Lemma 3.1(a) we get $\frac{\partial g_{u}}{\partial ξ} (r, λ_{-}) > 0$ if $r \in (0, ξ_{+})$ , $\frac{\partial g_{u}}{\partial ξ} (ξ_{+}, λ_{-}) = 0$ and $\frac{\partial g_{u}}{\partial ξ} (r, λ_{-}) > 0$ if $r \in (ξ_{+}, \infty)$ . Similarly for $a_{-} (r)$ . □

Proceeding as in [38] we can prove the following lemma.
Lemma 4.2.
If $(u_{n}) \subset M$ and $u_{n} ⇀ u$ in $E$ , then $u \in E$ and $u^{\pm} \neq 0$ .

Now, we denote by $\begin{array}{l} (4.4) & c_{\infty} = inf_{u \in M} I (u) . \end{array}$ From $M \subset N$ it follows that $c_{\infty} ⩾ d_{\infty} > 0$ .
5. Proof of Theorem 1.1

Let $(u_{n}) \subset M$ be such that $\begin{array}{l} (5.1) & I (u_{n}) \to c_{\infty} in R . \end{array}$ First we show that $(u_{n})$ is bounded in $E$ . Suppose that there exists a subsequence still denoted by $(u_{n})$ such that $\begin{matrix} ‖ u_{n} ‖ \to \infty as n \to \infty . \end{matrix}$ Set $v_{n} = \frac{u_{n}}{‖ u_{n} ‖}$ for all $n \in N$ . Hence $(v_{n})$ is bounded in $E$ so by Lemma 2.2 we may assume that $\begin{array}{l} (5.2) & \begin{aligned} v_{n} ⇀ v in E, \\ v_{n} \to v a.e. in R^{3}, \\ v_{n} \to v in L^{r} (R^{3}) for r \in (q, q^{*}) . \end{aligned} \end{array}$ Now, from $u_{n} = ‖ u_{n} ‖ v_{n}$ it follows that $\begin{array}{l} ‖ u_{n} ‖ v_{n}^{+} + ‖ u_{n} ‖ v_{n}^{-} = ‖ u_{n} ‖ v_{n} = u_{n} \in M \end{array}$ and by Lemma 4.1 we have $ξ_{+} (v_{n}) = λ_{-} (v_{n}) = ‖ u_{n} ‖$ . Recalling that $(ξ_{+}, λ_{-})$ is the unique global maximum point of $g_{v_{n}}$ with positive coordinates, for any $ξ > 0$ we infer $\begin{array}{l} I (u_{n}) = & I (‖ u_{n} ‖ v_{n}) \\ = & I (ξ_{+} (v_{n}) v_{n}^{+} + λ_{-} (v_{n}) v_{n}^{-}) \\ = & g_{v_{n}} (ξ_{+} (v_{n}), λ_{-} (v_{n})) ⩾ g_{v_{n}} (ξ, ξ) = I (ξ v_{n}) \\ = & \frac{ξ^{p}}{p} \int_{R^{3}} (a | \nabla v_{n} |^{p} + V (x) | v_{n} |^{p}) d x + \frac{b}{2 p} ξ^{2 p} {(\int_{R^{3}} | \nabla v_{n} |^{p} d x)}^{2} \\ + \frac{ξ^{q}}{q} \int_{R^{3}} (c | \nabla v_{n} |^{q} + V (x) | v_{n} |^{q}) d x + \frac{d}{2 q} ξ^{2 q} {(\int_{R^{3}} | \nabla v_{n} |^{q} d x)}^{2} \\ (5.3) & - \int_{R^{3}} K (x) F (ξ v_{n}) d x . \end{array}$ Note that $‖ v_{n} ‖ = 1$ , hence $\begin{array}{l} \int_{R^{3}} (a | \nabla v_{n} |^{p} + V (x) | v_{n} |^{p}) d x ⩽ 1 and \int_{R^{3}} (c | \nabla v_{n} |^{q} + V (x) | v_{n} |^{q}) d x ⩽ 1 . \end{array}$ Using $1 < p < q$ , and assuming without loss of generality that $ξ > 1$ so that $ξ^{q} > ξ^{p}$ , and exploiting the following inequality $\begin{matrix} a^{q} + b^{q} ⩾ C_{q} {(a + b)}^{q} for all a, b ⩾ 0 and q > 1 \end{matrix}$ from (5.3) we deduce $\begin{array}{l} I (u_{n}) & ⩾ \frac{ξ^{p}}{p} \int_{R^{3}} (a | \nabla v_{n} |^{p} + V (x) | v_{n} |^{p}) d x + \frac{ξ^{q}}{q} \int_{R^{3}} (c | \nabla v_{n} |^{q} + V (x) | v_{n} |^{q}) d x \\ - \int_{R^{3}} K (x) F (ξ v_{n}) d x \\ ⩾ \frac{ξ^{p}}{q} C_{q} ‖ v_{n} ‖^{q} - \int_{R^{3}} K (x) F (ξ v_{n}) d x \\ (5.4) & = \frac{ξ^{p}}{q} C_{q} - \int_{R^{3}} K (x) F (ξ v_{n}) d x . \end{array}$ Assume by contradiction that $v = 0$ . From (5.2) and Lemma 2.3 we deduce that for any $ξ > 0$ $\begin{array}{l} (5.5) & lim_{n \to \infty} \int_{R^{3}} K (x) F (ξ v_{n}) d x = 0 . \end{array}$ Taking the limit in (5.4), and using (5.1) and (5.5) we have $\begin{array}{l} c_{\infty} ⩾ \frac{ξ^{p}}{q} C_{q} for any ξ > 1, \end{array}$ which gives a contradiction. Therefore $v \neq 0$ .

On the other hand $\begin{array}{l} \frac{I (u_{n})}{‖ u_{n} ‖^{2 q}} = & \frac{1}{p ‖ u_{n} ‖^{2 q}} \int_{R^{3}} (a | \nabla u_{n} |^{p} + V (x) | u_{n} |^{p}) d x + \frac{b}{2 p ‖ u_{n} ‖^{2 q}} {(\int_{R^{3}} | \nabla u_{n} |^{p} d x)}^{2} \\ + \frac{1}{q ‖ u_{n} ‖^{2 q}} \int_{R^{3}} (c | \nabla u_{n} |^{q} + V (x) | u_{n} |^{q}) d x + \frac{d}{2 q ‖ u_{n} ‖^{2 q}} {(\int_{R^{3}} | \nabla u_{n} |^{q} d x)}^{2} \\ - \int_{R^{3}} K (x) \frac{F (u_{n})}{‖ u_{n} ‖^{2 q}} d x \\ ⩽ & \frac{1}{p ‖ u_{n} ‖^{2 q - p}} + \frac{b}{2 p ‖ u_{n} ‖^{2 (q - p)}} + \frac{1}{q ‖ u_{n} ‖^{q}} + \frac{d}{2 q} \\ (5.6) & - \int_{R^{3}} K (x) \frac{F (‖ u_{n} ‖ v_{n})}{{(‖ u_{n} ‖ v_{n})}^{2 q}} {(\frac{u_{n}}{‖ u_{n} ‖})}^{2 q} d x . \end{array}$ Combining assumption $(f_{3})$ together with Fatou’s lemma we get $\begin{array}{l} (5.7) & lim_{n \to \infty} \int_{R^{3}} K (x) \frac{F (‖ u_{n} ‖ v_{n})}{{(‖ u_{n} ‖ v_{n})}^{2 q}} {(\frac{u_{n}}{‖ u_{n} ‖})}^{2 q} d x = + \infty \end{array}$ so taking the limit in (5.6) we get a contradiction in view of (5.1), $‖ u_{n} ‖ \to \infty$ as $n \to \infty$ and (5.7). So $(u_{n})$ is a bounded sequence in $E$ and there exists $u \in E$ such that $u_{n} ⇀ u$ in $E$ . From Lemma 4.2 we have $u^{\pm} \neq 0$ and by Lemma 4.1 there are $ξ_{+}, λ_{-} > 0$ such that $ξ_{+} u^{+} + λ_{-} u^{-} \in M$ , from which $\begin{array}{l} \frac{1}{λ_{-}^{2 q - p}} \int_{R^{3}} (a {| \nabla u^{-} |}^{p} + V (x) {| u^{-} |}^{p}) d x + \frac{b}{λ_{-}^{2 (q - p)}} {(\int_{R^{3}} {| \nabla u^{-} |}^{p} d x)}^{2} \\ + \frac{1}{λ_{-}^{q}} \int_{R^{3}} (c {| \nabla u^{-} |}^{q} + V (x) {| u^{-} |}^{q}) d x + d {(\int_{R^{3}} {| \nabla u^{-} |}^{q} d x)}^{2} \\ (5.8) & = \int_{supp u^{-}} K (x) \frac{f (λ_{-} u^{-})}{{(λ_{-} u^{-})}^{2 q - 1}} {(u^{-})}^{2 q} d x . \end{array}$

Our aim is to prove that $ξ_{+} = λ_{-} = 1$ . Without loss of generality, let us suppose that $0 < ξ_{+} ⩽ λ_{-}$ . First we prove that $0 < ξ_{+} ⩽ λ_{-} ⩽ 1$ . Note that from $u_{n} ⇀ u$ in $E$ and exploiting Lemma 2.3 we have $\begin{array}{l} (5.9) & lim_{n \to \infty} \int_{R^{3}} K (x) f (u_{n}^{\pm}) u_{n}^{\pm} d x = \int_{R^{3}} K (x) f (u^{\pm}) u^{\pm} d x \end{array}$ and also $\begin{array}{l} (5.10) & lim_{n \to \infty} \int_{R^{3}} K (x) F (u_{n}^{\pm}) d x = \int_{R^{3}} K (x) F (u^{\pm}) d x, \end{array}$ and combining $(u_{n}) \subset M$ with Fatou’s lemma we get $\begin{array}{l} ⟨ I^{'} (u), u^{\pm} ⟩ ⩽ \underset{n \to \infty}{lim inf} ⟨ I^{'} (u_{n}), u_{n}^{\pm} ⟩ = 0, \end{array}$ which yields $\begin{array}{l} \int_{R^{3}} (a {| \nabla u^{-} |}^{p} + V (x) {| u^{-} |}^{p}) d x + b {(\int_{R^{3}} {| \nabla u^{-} |}^{p} d x)}^{2} \\ + \int_{R^{3}} (c {| \nabla u^{-} |}^{q} + V (x) {| u^{-} |}^{q}) d x + d {(\int_{R^{3}} {| \nabla u^{-} |}^{q} d x)}^{2} \\ (5.11) & ⩽ \int_{supp u^{-}} K (x) f (u^{-}) u^{-} d x . \end{array}$ Subtracting (5.8) and (5.11) we obtain $\begin{array}{l} (\frac{1}{λ_{-}^{2 q - p}} - 1) \int_{R^{3}} (a {| \nabla u^{-} |}^{p} + V (x) {| u^{-} |}^{p}) d x + b (\frac{1}{λ_{-}^{2 (q - p)}} - 1) {(\int_{R^{3}} {| \nabla u^{-} |}^{p} d x)}^{2} \\ + (\frac{1}{λ_{-}^{q}} - 1) \int_{R^{3}} (c {| \nabla u^{-} |}^{q} + V (x) {| u^{-} |}^{q}) d x \\ ⩾ \int_{supp u^{-}} K (x) (\frac{f (λ_{-} u^{-})}{{(λ_{-} u^{-})}^{2 q - 1}} - \frac{u^{-}}{{(u^{-})}^{2 q - 1}}) {(u^{-})}^{2 q} d x \end{array}$ and using assumption $(f_{3})$ we deduce $0 < λ_{-} ⩽ 1$ . Hence, $0 < ξ_{+} ⩽ λ_{-} ⩽ 1$ .

Next we show that $\begin{array}{l} (5.12) & I (ξ_{+} u^{+} + λ_{-} u^{-}) = c_{\infty} . \end{array}$

Now, from (4.4), $0 < ξ_{+} ⩽ λ_{-} ⩽ 1$ , assumption $(f_{4})$ , (5.9) and (5.10) we obtain $\begin{array}{l} c_{\infty} ⩽ & I (ξ_{+} u^{+} + λ_{-} u^{-}) \\ = & I (ξ_{+} u^{+} + λ_{-} u^{-}) - \frac{1}{2 q} ⟨ I^{'} (ξ_{+} u^{+} + λ_{-} u^{-}), ξ_{+} u^{+} + λ_{-} u^{-} ⟩ \\ = & (\frac{1}{p} - \frac{1}{2 q}) \int_{R^{3}} (a {| \nabla (ξ_{+} u^{+} + λ_{-} u^{-}) |}^{p} + V (x) {| ξ_{+} u^{+} + λ_{-} u^{-} |}^{p}) d x \\ + \frac{b}{2} (\frac{1}{p} - \frac{1}{q}) {(\int_{R^{3}} {| \nabla (ξ_{+} u^{+} + λ_{-} u^{-}) |}^{p} d x)}^{2} \\ + \frac{1}{2 q} \int_{R^{3}} (c {| \nabla (ξ_{+} u^{+} + λ_{-} u^{-}) |}^{q} + V (x) {| ξ_{+} u^{+} + λ_{-} u^{-} |}^{q}) d x \\ + \int_{R^{3}} K (x) (\frac{1}{2 q} f (ξ_{+} u^{+}) (ξ_{+} u^{+}) - F (ξ_{+} u^{+})) d x \\ + \int_{R^{3}} K (x) (\frac{1}{2 q} f (λ_{-} u^{-}) (λ_{-} u^{-}) - F (λ_{-} u^{-})) d x \\ = & (\frac{1}{p} - \frac{1}{2 q}) \int_{R^{3}} (a | \nabla u |^{p} + V (x) | u |^{p}) d x + \frac{b}{2} (\frac{1}{p} - \frac{1}{q}) {(\int_{R^{3}} | \nabla u |^{p} d x)}^{2} \\ + \frac{1}{2 q} \int_{R^{3}} (c | \nabla u |^{q} + V (x) | u |^{q}) d x \\ + \int_{R^{3}} K (x) (\frac{1}{2 q} f (u^{+}) u^{+} - F (u^{+})) d x + \int_{R^{3}} K (x) (\frac{1}{2 q} f (u^{-}) u^{-} - F (u^{-})) d x \\ = & (\frac{1}{p} - \frac{1}{2 q}) \int_{R^{3}} (a | \nabla u |^{p} + V (x) | u |^{p}) d x + \frac{b}{2} (\frac{1}{p} - \frac{1}{q}) {(\int_{R^{3}} | \nabla u |^{p} d x)}^{2} \\ + \frac{1}{2 q} \int_{R^{3}} (c | \nabla u |^{q} + V (x) | u |^{q}) d x + \int_{R^{3}} K (x) (\frac{1}{2 q} f (u) u - F (u)) d x \\ ⩽ & \underset{n \to \infty}{lim inf} {(\frac{1}{p} - \frac{1}{2 q}) \int_{R^{3}} (a | \nabla u_{n} |^{p} + V (x) | u_{n} |^{p}) d x + \frac{b}{2} (\frac{1}{p} - \frac{1}{q}) {(\int_{R^{3}} | \nabla u_{n} |^{p} d x)}^{2} \\ + \frac{1}{2 q} \int_{R^{3}} (c | \nabla u_{n} |^{q} + V (x) | u_{n} |^{q}) d x + \int_{R^{3}} K (x) (\frac{1}{2 q} f (u_{n}) u - F (u_{n})) d x} \\ = & \underset{n \to \infty}{lim inf} {I (u_{n}) - \frac{1}{2 q} ⟨ I^{'} (u_{n}), u_{n} ⟩} = c_{\infty} \end{array}$ which implies that (5.12) holds true. In particular it follows that $ξ_{+} = λ_{-} = 1$ .

Next, we prove that the minimum point $u = u^{+} + u^{-}$ is a critical point of $I$ . Assume by contradiction that $I^{'} (u) \neq 0$ . Then, due to the continuity of $I^{'}$ we can find $α, β > 0$ such that $‖ I^{'} (v) ‖ ⩾ β$ for all $v \in E$ with $‖ v - u ‖ ⩽ 3 α$ .

Define $D = [\frac{1}{2}, \frac{3}{2}] \times [\frac{1}{2}, \frac{3}{2}]$ and $E^{\pm} = {u \in E : u^{\pm} \neq 0}$ , and let us consider the function $G_{u} : D \to E^{\pm}$ defined by setting $\begin{matrix} G_{u} (ξ, λ) = ξ u^{+} + λ u^{-} . \end{matrix}$ Using Lemma 4.1 we can see that $I (G_{u} (1, 1)) = c_{\infty}$ and $I (G_{u} (ξ, λ)) < c_{\infty}$ in $D ∖ {(1, 1)}$ .

Define $\begin{matrix} τ = max_{(ξ, λ) \in \partial D} I (G_{u} (ξ, λ)), \end{matrix}$ then $τ < c_{\infty}$ .

Set $\tilde{S} = {v \in E : ‖ v - u ‖ ⩽ α}$ and choose $ε = min {\frac{1}{4} (c_{\infty} - γ), \frac{α β}{8}}$ . By Theorem 2.3 in [54] there exists a deformation $η \in C ([0, 1] \times E, E)$ such that the following assertions hold:

$η (ξ, v) = v$ if $v \notin E^{- 1} ([c_{\infty} - 2 ε, c_{\infty} + 2 ε])$ ;

$I (η (1, v)) ⩽ c_{\infty} - ε$ for each $v \in E$ with $‖ v - u ‖ ⩽ α$ and $I (v) ⩽ c_{\infty} + ε$ ;

$I (η (1, v)) ⩽ I (η (0, v)) = I (v)$ for all $v \in E$ .

From (b) and (c) we get $\begin{matrix} (5.13) & max_{(ξ, λ) \in \partial D} I (η (1, G_{u} (ξ, λ))) < c_{\infty} . \end{matrix}$ Now we prove that $\begin{matrix} (5.14) & η (1, G_{u} (D)) \cap M \neq \emptyset \end{matrix}$ because the definition of $c_{\infty}$ and (5.14) contradict (5.13).

Let us define $\begin{array}{l} Φ_{u} (ξ, λ) = η (1, G_{u} (ξ, λ)), \\ ψ_{0} (ξ, λ) = (⟨ I^{'} (G_{u} (ξ, 1)), ξ u^{+} ⟩, ⟨ I^{'} (G_{u} (1, λ)), λ u^{-} ⟩), \\ ψ_{1} (ξ, λ) = (\frac{1}{ξ} ⟨ I^{'} (Φ_{u} (ξ, 1)), Φ_{u}^{+} (ξ, 1) ⟩, \frac{1}{λ} ⟨ I^{'} (Φ_{u} (1, λ)), Φ_{u}^{-} (1, λ) ⟩) . \end{array}$

Exploiting Lemma 4.1(iii), the $C^{1}$ function $γ_{+} (ξ) = g_{u} (ξ, 1)$ has a unique global maximum point $ξ = 1$ . By density, given $ε > 0$ small enough, there is $γ_{+, ε} \in C^{\infty} ([\frac{1}{2}, \frac{3}{2}])$ such that $‖ γ_{+} - γ_{+, ε} ‖_{C^{1} ([\frac{1}{2}, \frac{3}{2}])} < ε$ with $ξ_{+}$ being the unique maximum global point of $γ_{+, ε}$ in $[\frac{1}{2}, \frac{3}{2}]$ . Hence, $‖ γ_{+}^{'} - γ_{+, ε}^{'} ‖_{C ([\frac{1}{2}, \frac{3}{2}])} < ε$ , $γ_{+, ε}^{'} (1) = 0$ and $γ_{+, ε}^{″} (1) < 0$ . Analogously, set $γ_{-} (λ) = g_{u} (1, λ)$ , then there exists $γ_{-, ε} \in C^{\infty} ([\frac{1}{2}, \frac{3}{2}])$ such that $‖ γ_{-}^{'} - γ_{-, ε}^{'} ‖_{C ([\frac{1}{2}, \frac{3}{2}])} < ε$ , $γ_{+, ε}^{'} (1) = 0$ and $γ_{+, ε}^{″} (1) < 0$ .

Let us define $ψ_{ε} \in C^{\infty} (D)$ by setting $\begin{matrix} ψ_{ε} (ξ, λ) = (ξ γ_{+, ε}^{'} (ξ), λ γ_{-, ε}^{'} (λ)) . \end{matrix}$ We note that $‖ ψ_{ε} - ψ_{0} ‖_{C (D)} < \frac{3 \sqrt{2}}{2} ε$ , $(0, 0) \notin ψ_{ε} (\partial D)$ , and $(0, 0)$ is a regular value of $ψ_{ε}$ in D.

Since $(1, 1)$ is the unique solution of $ψ_{ε} (ξ, λ) = (0, 0)$ in D, by the definition of Brouwer’s degree, we can infer that, for ε small enough, it holds $\begin{matrix} (5.15) & deg (ψ_{0}, D, (0, 0)) = deg (ψ_{ε}, D, (0, 0)) = sgn Jac (ψ_{ε}) (1, 1), \end{matrix}$ where $Jac (ψ_{ε})$ is the Jacobian determinant of $ψ_{ε}$ and sgn denotes the sign function.

We note that $\begin{array}{l} (5.16) & Jac (ψ_{ε}) (1, 1) = [γ_{+, ε}^{'} (1) + γ_{+, ε}^{″} (1)] \times [γ_{-, ε}^{'} (1) + γ_{-, ε}^{″} (1)] = γ_{+, ε}^{″} (1) \times γ_{-, ε}^{″} (1) > 0, \end{array}$ so combining (5.15) with (5.16) we find $\begin{array}{l} deg (ψ_{0}, D, (0, 0)) = sgn [γ_{+, ε}^{″} (1) \times γ_{-, ε}^{″} (1)] = 1 . \end{array}$

By the definition of τ and the fact that $ε = min {\frac{1}{4} (c_{\infty} - γ), \frac{α β}{8}}$ we have that for any $(ξ, λ) \in \partial D$ $\begin{array}{l} I (G_{u} (ξ, λ)) & ⩽ max_{(ξ, λ) \in \partial D} I (G_{u} (ξ, λ)) < \frac{1}{2} (τ + c_{\infty}) = c_{\infty} - 2 (\frac{c_{\infty} - τ}{4}) ⩽ c_{\infty} - 2 ε . \end{array}$ This and (a )yields that $G_{u} = Φ_{u}$ on $\partial D$ . Therefore, $ψ_{1} = ψ_{0}$ on $\partial D$ and consequently $\begin{matrix} deg (ψ_{1}, D, (0, 0)) = deg (ψ_{0}, D, (0, 0)) = 1, \end{matrix}$ which shows that $ψ_{1} (ξ, λ) = (0, 0)$ for some $(ξ, λ) \in D$ .

Now, in order to verify that (5.14) holds, we prove that $\begin{matrix} (5.17) & ψ_{1} (1, 1) = (⟨ I^{'} (Φ_{u} (ξ, 1)), Φ_{u} {(1, 1)}^{+} ⟩, ⟨ I^{'} (Φ_{u} (1, 1)), Φ_{u} {(1, 1)}^{-} ⟩) = 0 . \end{matrix}$ As a matter of fact, (5.17) and the fact that $(1, 1) \in D$ , yield $Φ_{u} (1, 1) = η (1, G_{u} (1, 1)) \in M$ .

We argue as follows. If the zero $(ξ, λ)$ of $ψ_{1}$ obtained above is equal to $(1, 1)$ there is nothing to prove. Otherwise, we take $0 < δ_{1} < min {| ξ - 1 |, | λ - 1 |}$ and consider $\begin{matrix} D_{1} = [1 - \frac{δ_{1}}{2}, 1 + \frac{δ_{1}}{2}] \times [1 - \frac{δ_{1}}{2}, 1 + \frac{δ_{1}}{2}] . \end{matrix}$

Therefore $(ξ, λ) \in D ∖ D_{1}$ . Hence, we can repeat for $D_{1}$ the same argument used for D, so that we can find a couple $(ξ_{1}, λ_{1}) \in D_{1}$ such that $ψ_{1} (ξ_{1}, λ_{1}) = 0$ . If $(ξ_{1}, λ_{1}) = (1, 1)$ , there is nothing to prove. Otherwise, we can continue with this procedure and find in the n-th step that (5.17) holds, or produce a sequence $(ξ_{n}, λ_{n}) \in D_{n - 1} ∖ D_{n}$ which converges to $(1, 1)$ and such that $\begin{matrix} (5.18) & ψ_{1} (ξ_{n}, λ_{n}) = 0, for every n \in N . \end{matrix}$ Thus, taking the limit as $n \to \infty$ in (5.18) and using the continuity of $ψ_{1}$ we get (5.17). Therefore, $u = u^{+} + u^{-}$ is a critical point of $I$ .

Finally, we consider the case when f is odd. Clearly, the functional ψ is even. In the light of (3.2) and $c_{\infty} ⩾ d_{\infty} > 0$ we can see that ψ is bounded from below in $S$ . Moreover, using Lemma 2.2 and Lemma 2.3, we deduce that ψ satisfies the Palais–Smale condition on $S$ . Hence, applying Proposition 3.1 and [52], we conclude that $I$ has infinitely many critical points.

Footnotes

Acknowledgements

The first author was partly supported by the GNAMPA Project 2020 entitled: Studio Di Problemi Frazionari Nonlocali Tramite Tecniche Variazionali. The second author was partly supported by Slovenian research agency grants P1-0292, N1-0114, N1-0083, N1-0064 and J1-8131.

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