Nonlinear Schrödinger equations involving exponential critical growth with unbounded or vanishing potentials

Abstract

In this paper we deal with the following class of nonlinear Schrödinger equations $\begin{matrix} - Δ u + V (| x |) u = λ Q (| x |) f (u), x \in R^{2}, \end{matrix}$ where $λ > 0$ is a real parameter, the potential V and the weight Q are radial, which can be singular at the origin, unbounded or decaying at infinity and the nonlinearity $f (s)$ behaves like $e^{α s^{2}}$ at infinity. By performing a variational approach based on a weighted Trudinger–Moser type inequality proved here, we obtain some existence and multiplicity results.

Keywords

Nonlinear Schrödinger equation unbounded or decaying radial potentials exponential critical growth weighted Trudinger–Moser type inequality

1. Introduction and main results

In this paper, we are concerned with semilinear elliptic equations of the form $\begin{matrix} (P) & - Δ u + V (| x |) u = λ Q (| x |) f (u), x \in R^{2}, \end{matrix}$ where $λ > 0$ is a parameter, $V, Q : (0, \infty) \to R$ are radial weights, which can by singular at the origin, unbounded or decaying at infinity and the nonlinearity $f : R \to R$ is continuous and has exponential critical growth at infinity, i.e., there exists $α_{0} > 0$ such that $\begin{matrix} (1.1) & lim_{| s | \to + \infty} \frac{| f (s) |}{e^{α s^{2}}} = \{\begin{matrix} 0 & if α > α_{0}, \\ + \infty & if α < α_{0} . \end{matrix} \end{matrix}$

The study of the stationary equation ( P ) is motivated by the study of standing wave solutions of the nonlinear Schrödinger equation, see e.g., [10, 15, 24, 28] and references therein. Our starting point here are the works [29, 30], where the authors proved some weighted Sobolev embedding theorems, and there is a growing recent interest in applications of these results in the study of partial differential equations, see for example [3, 9, 11, 26, 27].

We will assume the following assumptions on the radial functions V and Q:

$V : (0, \infty) \to R$ is continuous, $V > 0$ and there are $a_{0}, a > - 2$ such that $\begin{matrix} \underset{r \to 0^{+}}{lim sup} V (r) r^{- a_{0}} < + \infty and \underset{r \to + \infty}{lim inf} V (r) r^{- a} > 0 . \end{matrix}$

$Q : (0, \infty) \to R$ is continuous, $Q > 0$ and there are $b_{0}, b > - 2$ such that $\begin{matrix} 0 < D_{0} : = \underset{r \to 0^{+}}{lim inf} Q (r) r^{- b_{0}} ⩽ \underset{r \to 0^{+}}{lim sup} Q (r) r^{- b_{0}} < + \infty, \underset{r \to + \infty}{lim sup} Q (r) r^{- b} < + \infty . \end{matrix}$

We highlight here that Ambrosetti–Felli–Malchiodi [5] and do Ó–Sani–Zhang [16] studied equation ( P ) by assuming that V and Q satisfy the following assumption: there are $A_{1}, A_{2}, A_{3} > 0$ such that $\begin{matrix} A_{1} {(1 + | x |^{α})}^{- 1} ⩽ V (x) ⩽ A_{2} and 0 < Q (x) ⩽ A_{3} {(1 + | x |^{β})}^{- 1}, \end{matrix}$ where $0 < α < 2$ and $β > 0$ . Thus, when V and Q are radial we are considering a more general class of potentials than the one in [5, 16].

In the papers [29, 30], the authors studied the existence of solutions for the equation ( P ) when $f (u) = | u |^{p - 2} u$ with $2 < p < 2 N / (N - 2)$ , if $N ⩾ 3$ , and $2 < p < + \infty$ , if $N = 2$ . Our main purpose is to obtain solutions when the nonlinearity f has exponential critical growth. Precisely, besides the critical growth condition (1.1), we also assume the following conditions:

$f (s) = o (| s |^{γ - 1})$ as $s \to 0$ , where $γ : = max {2, 2 (2 + 2 b - a) / (a + 2)}$ ;

there exists $θ > γ$ such that $0 < θ F (s) ⩽ f (s) s$ for all $s \neq 0$ , where $F (s) : = \int_{0}^{s} f (t) d t$ ;

the following limit holds: ${lim}_{| s | \to + \infty} f (s) s / F (s) = \infty$ .

In view of hypothesis

(Q)

, there exists

r_{0} > 0

such that

Q (r) ⩾ D_{0} r^{b_{0}} / 2

, for all

0 < r ⩽ r_{0}

. For this fixed

r_{0}

, we will assume the following assumption:

there exists $β_{0} > 2 {(b_{0} + 2)}^{2} / D_{0} α_{0} r_{0}^{b_{0} + 2}$ such that ${lim inf}_{| s | \to + \infty} f (s) s e^{- α_{0} s^{2}} ⩾ β_{0}$ .

We quote that conditions

(f_{3})

and

(f_{4})

have already been considered in others works, see for instance [17, 21]. It is important to mention that similar issues have been addressed in the paper [3], where the authors proved the existence of positive solutions for equation ( P ) by assuming similar hypotheses

(V)

and

(Q)

with a, b in the range

- 2 < b < (a - 2) / 2

and f with exponential critical growth. To improve this condition, inspired by the paper [20], we used a change of variables to obtain a sharp weighted Trudinger–Moser type inequality. We also mention that our hypotheses on f include the ones in [3].

In order to present our main results, we need some notations. As usual, we denote by $C_{0}^{\infty} (R^{2})$ the space of infinitely differentiable functions with compact support and for $1 ⩽ p < + \infty$ we define the weighted Lebesgue space $\begin{matrix} L^{p} (R^{2}; Q) : = {u : R^{2} \to R measurable : \int_{R^{2}} Q (| x |) | u |^{p} d x < + \infty} . \end{matrix}$ As in the paper [16], we consider the Sobolev space $\begin{matrix} E : = {u \in L_{loc}^{2} (R^{2}) : | \nabla u | \in L^{2} (R^{2}) and \int_{R^{2}} V (| x |) u^{2} d x < + \infty}, \end{matrix}$ which is a Hilbert space when endowed with inner product $\begin{matrix} {⟨ u, v ⟩}_{E} : = \int_{R^{2}} [\nabla u \nabla v + V (| x |) u v] d x, \end{matrix}$ and its correspondent norm $‖ u ‖ : = {⟨ u, u ⟩}_{E}^{1 / 2}$ . Using standard arguments one can prove that $C_{0}^{\infty} (R^{2})$ is dense in E. Furthermore, the subspace $\begin{matrix} E_{r} : = {u \in E : u is radial} \end{matrix}$ is closed in E and thus it is a Hilbert space itself. In this context, by a weak solution of ( P ) we understand a function $u \in E$ such that $\begin{matrix} \int_{R^{2}} [\nabla u \nabla φ + V (| x |) u φ] d x = λ \int_{R^{2}} Q (| x |) f (u) φ d x, for all φ \in C_{0}^{\infty} (R^{2}) . \end{matrix}$

Remark 1.1.
We observe that condition $a, b > - 2$ seems to be necessary to the existence of weak solutions to equation ( P ). To illustrate, let $u \in E$ be a solution of the model equation $\begin{matrix} (1.2) & - Δ u + | x |^{a} u = | x |^{b} f (u), x \in R^{2}, \end{matrix}$ and for $μ > 0$ , consider $u_{μ} (x) = u (μ x)$ the continuous path in E converging to u, as $μ \to 1$ . Since u is a critical point of the energy functional $\begin{matrix} J (u) = \frac{1}{2} \int_{R^{2}} | \nabla u |^{2} d x + \frac{1}{2} \int_{R^{2}} | x |^{a} u^{2} d x - \int_{R^{2}} | x |^{b} F (u) d x, \end{matrix}$ it should satisfy $\frac{d J}{d μ} (u_{μ}) |_{μ = 1} = 0$ and taking into account that $‖ \nabla u_{μ} ‖_{L^{2} (R^{2})}$ is constant, by performing a change of variables we get $\begin{matrix} (1.3) & \frac{(a + 2)}{2} \int_{R^{2}} | x |^{a} u^{2} d x = (b + 2) \int_{R^{2}} | x |^{b} F (u) d x . \end{matrix}$ Therefore, equation (1.2) has no nonzero weak solution if $(a, b)$ belongs to the region $R = ((- \infty, - 2] \times (- 2, \infty)) \cup ((- 2, \infty) \times (- \infty, - 2])$ . Furthermore, if u is a nonzero weak solution of equation (1.2) we see that $\begin{matrix} \int_{R^{2}} | \nabla u |^{2} d x + \int_{R^{2}} | x |^{a} u^{2} d x = \int_{R^{2}} | x |^{b} f (u) u d x, \end{matrix}$ which combined with (1.3), yields $\begin{matrix} \int_{R^{2}} | \nabla u |^{2} d x = \int_{R^{2}} | x |^{b} (f (u) u - \frac{2 (b + 2)}{(a + 2)} F (u)) d x . \end{matrix}$ Consequently, equation (1.2) has no nonzero weak solution if $f (u) u ⩽ 2 (b + 2) / (a + 2) F (u)$ , and hence $θ > γ ⩾ γ_{0} : = 2 (b + 2) / (a + 2)$ in the hypothesis $(f_{2})$ is a necessary condition for the existence when $a = b$ . For related nonexistence result, see for instance, [2, 7] and references therein.

In this setting our first result can be stated as follows.
Theorem 1.2.
Assume that (1.1), $(f_{1})$ – $(f_{4})$ , $(V)$ and $(Q)$ hold. Then, for each $λ > 0$ , the problem ( P ) possesses a nonzero weak solution $u_{λ} \in E_{r}$ satisfying, $\begin{matrix} (1.4) & 0 ⩽ u_{λ} (x) ⩽ c_{0} exp (- c_{1} | x |^{(a + 2) / 4}), x \in R^{2}, \end{matrix}$ for some constants $c_{0}, c_{1} > 0$ depending only on λ.

The crucial ingredient to prove Theorem 1.2, and also in the proof of our next Theorem 1.3, is a weighted Trudinger–Moser type inequality. This inequality, combined with a suitable estimate of the minimax level, yields compactness of the Palais–Smale sequence.

Our second main result concerns the multiplicity of solutions to equation ( P ) for large $λ > 0$ . To this purpose, we shall assume in addition the following local hypothesis:

there exists $ν > γ$ such that ${lim inf}_{s \to 0} F (s) | s |^{- ν} > 0$ .

Our multiplicity result is stated as follows.
Theorem 1.3.
Assume that (1.1), $(f_{1})$ – $(f_{5})$ , $(V)$ and $(Q)$ hold. If in addition, f is odd, there exists a sequence $(λ_{k}) \subset R_{+}$ with $λ_{k} \to \infty$ such that for all $λ > λ_{k}$ , ( P ) has at least k pairs of weak solutions in $E_{r}$ .

For instance, one can check that Theorem 1.2 and Theorem 1.3 apply for the model equation $\begin{matrix} - Δ u + | x |^{a} u = λ | x |^{b} (| u |^{ν - 2} u + | u |^{q - 2} u (e^{u^{2}} - 1)), x \in R^{2}, \end{matrix}$ with $a, b > - 2$ , $λ > 0$ , $ν > γ$ and $q ⩾ γ$ , where γ is defined in $(f_{1})$ . Thus, this class of equations includes the Henon and singular equations ones, which correspond to $a, b > 0$ and $a, b < 0$ , respectively.
Remark 1.4.
It is worth to mention that related existence and nonexistence results in the quasilinear case have been obtained recently in the paper [7], where the authors consider similar hypotheses on the weight functions V and Q. However, here we do not assume $b < a$ , which includes a wide class of weight function. To this, we need to improve the Trudinger–Moser inequality, proving that the range of α is sharp (see Theorem 2.5). Moreover, in our Trudinger–Moser inequality the exponent is critical. Similar results also were obtained in the paper [4].

The remainder of the paper is organized as follows. In Section 2, we introduce our variational setting and prove a weighted Trudinger–Moser type inequality. Finally, in Section 3, we present the proofs of Theorem 1.2 and Theorem 1.3.
2. A sharp Trudinger–Moser inequality

In this section we introduce the variational framework and prove a weighted Trudinger–Moser type inequality, which is a key ingredient in the proof of Theorem 1.2 and Theorem 1.3. For the proof, we borrow some ideas of [13, 20, 25]. We start off by collecting some well-known results that we shall use throughout.

Lemma 2.1 ([29, Lemma 4]).

Assume $(V)$ . If $u \in E$ and $R > 0$ , then $u \in H^{1} (B_{R})$ and $‖ u ‖_{H^{1} (B_{R})} ⩽ C_{R} ‖ u ‖$ , with $C_{R} > 0$ .

Lemma 2.2 ([29, Theorem 2]).

Assume that $(V)$ and $(Q)$ hold. Then the embedding $E_{r} ↪ L^{p} (R^{2}; Q)$ is continuous for all $γ ⩽ p < \infty$ . Furthermore, if $- 2 < b < a$ the embedding is compact for $γ ⩽ p < \infty$ and if $- 2 < a ⩽ b$ the embedding is compact for all $γ < p < \infty$ .

For easy reference, it follows from assumptions $(V)$ and $(Q)$ that for every $0 < R_{0} < R_{1}$ there are positive constants $C_{0}$ , $C_{1}$ , $C_{3}$ , $C_{4}$ such that $\begin{matrix} (2.1) & \{\begin{matrix} V (| x |) ⩽ C_{0} | x |^{a_{0}}, C_{1} | x |^{b_{0}} ⩽ Q (| x |) ⩽ C_{2} | x |^{b_{0}} & if 0 < | x | < R_{0}, \\ C_{3} | x |^{a} ⩽ V (| x |), Q (| x |) ⩽ C_{4} | x |^{b} & if | x | > R_{1} . \end{matrix} \end{matrix}$

In view of Lemma 2.2, it is natural to look for a weighted Trudinger–Moser inequality on the space $E_{r}$ determined by the Young function $\begin{matrix} Φ_{α, j_{0}} (s) : = e^{α s^{2}} - \sum_{j = 0}^{j_{0} - 1} \frac{α^{j}}{j!} s^{2 j}, s \in R, \end{matrix}$ where $α > 0$ and $j_{0} : = [γ / 2] = inf {j \in N : j ⩾ γ / 2}$ . For this purpose, the next lemma plays an important role in the proof of the optimal exponent of our weighted Trudinger–Moser type inequality.

Lemma 2.3.
Assume $(V)$ and consider the so-called Moser’s sequence (see e.g., [19]) given by $\begin{matrix} M_{n} (x, r) = \frac{1}{{(2 π)}^{1 / 2}} \{\begin{matrix} {(log n)}^{1 / 2} & if | x | ⩽ r / n, \\ \frac{log (r / | x |)}{{(log n)}^{1 / 2}} & if r / n ⩽ | x | ⩽ r, \\ 0 & if | x | ⩾ r . \end{matrix} \end{matrix}$ Then $‖ M_{n} ‖^{2} ⩽ 1 + o_{n} (1)$ , where $o_{n} (1)$ denotes a quantity which goes to zero as $n \to \infty$ .
Proof.
First, one can easily check that $‖ \nabla M_{n} ‖_{L^{2} (R^{2})} = 1$ . On the other hand, we can write $\begin{matrix} \int_{R^{2}} V (| x |) M_{n}^{2} d x = \int_{B_{r / n}} V (| x |) M_{n}^{2} d x + \int_{B_{r} ∖ B_{r / n}} V (| x |) M_{n}^{2} d x . \end{matrix}$ By (2.1), with $R_{0} = r$ we get $\begin{matrix} \int_{B_{r / n}} V (| x |) M_{n}^{2} d x ⩽ C_{0} log n \int_{0}^{r / n} s^{a_{0} + 1} d s = \frac{C_{0} log n}{(a_{0} + 2)} {(\frac{r}{n})}^{a_{0} + 2} = o_{n} (1) . \end{matrix}$ Considering the change of variables $t = log (r / s)$ we get $\begin{matrix} \int_{B_{r} ∖ B_{r / n}} V (| x |) M_{n}^{2} d x ⩽ \frac{C_{0}}{log n} \int_{r / n}^{r} {log}^{2} (r / s) s^{a_{0} + 1} d s = \frac{C_{0} r^{a_{0} + 2}}{log n} \int_{0}^{log n} t^{2} e^{- (a_{0} + 2) t} d t, \end{matrix}$ and integrating by parts twice, we obtain $\begin{matrix} \int_{0}^{log n} t^{2} e^{- (a_{0} + 2) t} d t = (- \frac{{log}^{2} n}{(a_{0} + 2) n^{a_{0} + 2}} - \frac{2 log n}{{(a_{0} + 2)}^{2} n^{a_{0} + 2}} - \frac{2}{{(a_{0} + 2)}^{3} n^{a_{0} + 2}} + \frac{2}{{(a_{0} + 2)}^{3}}), \end{matrix}$ which implies the desired result. □

Now we prove a weighted Trudinger–Moser type inequality in balls.
Lemma 2.4.
Let $R > 0$ be fixed and assume that $(V)$ and $(Q)$ hold. Then for all $α > 0$ and $u \in E_{r}$ , it holds that $Q (| x |) e^{α u^{2}} \in L^{1} (B_{R})$ . Moreover, $\begin{matrix} L (α, V, Q, R) : = sup_{‖ u ‖ ⩽ 1} \int_{B_{R}} Q (| x |) e^{α u^{2}} d x < \infty, \end{matrix}$ if and only if $0 < α ⩽ α_{2} : = 4 π (1 + b_{0} / 2)$ .
Proof.
Let $α > 0$ and $R_{0} > 0$ to be chosen later. We shall split the proof into two cases.

Case 1: Assume $b_{0} ⩽ a_{0}$ . For each $u \in E_{r}$ and $a_{0} > - 2$ , inspired by the paper [20], we consider the function $\begin{matrix} w (r) : = {(\frac{a_{0} + 2}{2})}^{1 / 2} u (H (r)), for all r ⩾ 0, \end{matrix}$ where $H (r) = {(\frac{a_{0} + 2}{2})}^{2 / (a_{0} + 2)} r^{2 / (a_{0} + 2)}$ . Carrying out a straightforward computation one has $\begin{matrix} (2.2) & \int_{B_{R_{0}}} | \nabla w |^{2} d x = \int_{B_{H (R_{0})}} | \nabla u |^{2} d x and \int_{B_{R_{0}}} w^{2} d x = \int_{B_{H (R_{0})}} | x |^{a_{0}} u^{2} d x . \end{matrix}$ Moreover, there exists $C_{1} = C_{1} (a_{0}, b_{0}) > 0$ such that $\begin{matrix} (2.3) & \int_{B_{R_{0}}} | x |^{δ} e^{(\frac{2 α}{a_{0} + 2}) w^{2}} d x = C_{1} \int_{B_{H (R_{0})}} | x |^{b_{0}} e^{α u^{2}} d x, \end{matrix}$ where $δ = - 2 (a_{0} - b_{0}) / (a_{0} + 2) \in (- 2, 0]$ , since $b_{0} ⩽ a_{0}$ . By (2.1), there exists $C_{2} > 0$ such that $C_{2} | x |^{a_{0}} ⩽ V (| x |)$ for all $0 < | x | ⩽ R_{0}$ . Thus, by (2.2) we get $\begin{matrix} \int_{B_{R_{0}}} | \nabla w |^{2} d x + C_{2} \int_{B_{R_{0}}} w^{2} d x ⩽ \int_{B_{H (R_{0})}} | \nabla u |^{2} d x + \int_{B_{H (R_{0})}} V (| x |) u^{2} d x, \end{matrix}$ and consequently $w \in H^{1} (B_{R_{0}})$ . Now following a scheme similar to the one in [25] we define $\overline{w} \in H_{0}^{1} (B_{R_{0}})$ by $\begin{matrix} \overline{w} (x) : = \{\begin{matrix} w (| x |) - w (R_{0}) & if 0 ⩽ | x | ⩽ R_{0}, \\ 0 & if | x | ⩾ R_{0}, \end{matrix} \end{matrix}$ and using Young’s inequality we see that $\begin{matrix} w^{2} (x) ⩽ {\overline{w}}^{2} (x) + {\overline{w}}^{2} (x) w^{2} (R_{0}) + 1 + w^{2} (R_{0}) = v^{2} (x) + c^{2}, \end{matrix}$ with $\begin{matrix} v (x) : = \overline{w} (x) {(1 + w^{2} (R_{0}))}^{1 / 2} \in H_{0}^{1} (B_{R_{0}}) and c : = {(1 + w^{2} (R_{0}))}^{1 / 2} . \end{matrix}$ According to (2.3) and [1, Theorem 2.1] we have $\begin{matrix} (2.4) & C_{1} \int_{B_{H (R_{0})}} | x |^{b_{0}} e^{α u^{2}} d x = \int_{B_{R_{0}}} | x |^{δ} e^{(\frac{2 α}{a_{0} + 2}) w^{2}} d x ⩽ e^{c^{2}} \int_{B_{R_{0}}} | x |^{δ} e^{(\frac{2 α}{a_{0} + 2}) v^{2}} d x < \infty . \end{matrix}$ By (2.1) with $R_{1} = H (R_{0})$ , $Q (| x |) ⩽ C_{3} | x |^{b_{0}}$ if $0 < | x | ⩽ H (R_{0})$ . Consequently, $Q (| x |) e^{α u^{2}} \in L^{1} (B_{R})$ for $R_{0}$ sufficiently large such that $H (R_{0}) > R$ . On the other hand, observe that, by (2.2), if $‖ u ‖ ⩽ 1$ we have $\begin{matrix} \int_{B_{R_{0}}} | \nabla w |^{2} d x + \int_{R^{2}} V (| x |) u^{2} d x = \int_{B_{H (R_{0})}} | \nabla u |^{2} d x + \int_{R^{2}} V (| x |) u^{2} d x ⩽ ‖ u ‖^{2} ⩽ 1, \end{matrix}$ that is, $\begin{matrix} \int_{B_{R_{0}}} | \nabla \overline{w} |^{2} d x = \int_{B_{R_{0}}} | \nabla w |^{2} d x ⩽ 1 - \int_{R^{2}} V (| x |) u^{2} d x . \end{matrix}$ Therefore, this inequality and the definition of v gives us the estimate $\begin{matrix} \int_{B_{R_{0}}} | \nabla v |^{2} d x ⩽ (1 + w^{2} (R_{0})) (1 - \int_{R^{2}} V (| x |) u^{2} d x) ⩽ 1 - \int_{R^{2}} V (| x |) u^{2} d x + w^{2} (R_{0}) . \end{matrix}$ Thus, by the definition of w we get $\begin{matrix} \int_{B_{R_{0}}} | \nabla v |^{2} d x ⩽ 1 - \int_{R^{2}} V (| x |) u^{2} d x + (\frac{a_{0} + 2}{2}) u^{2} (H (R_{0})) . \end{matrix}$ Then, using the following version of the so-called Radial Lemma (see e.g., [3, Lemma 2.1]) due to Strauss [28] $\begin{matrix} (2.5) & | u (x) | ⩽ C_{0} ‖ u ‖ | x |^{- (a + 2) / 4}, if | x | ≫ 1, \end{matrix}$ we have that $‖ \nabla v ‖_{L^{2} (B_{R_{0}})} ⩽ 1$ for $R_{0}$ sufficiently large. Now, we observe that $α ⩽ 4 π (1 + b_{0} / 2)$ if and only if $(2 / (a_{0} + 2)) α ⩽ 4 π (1 + δ / 2)$ and hence by [1, Theorem 2.1] we have $\begin{matrix} sup_{‖ v ‖_{H_{0}^{1} (B_{R_{0}})} ⩽ 1} \int_{B_{R_{0}}} | x |^{δ} e^{(\frac{2 α}{a_{0} + 2}) v^{2}} d x < \infty . \end{matrix}$ Once again using that $Q (| x |) ⩽ C_{2} | x |^{b_{0}}$ for all $0 < | x | ⩽ H (R_{0})$ , from the last estimate and (2.4) we infer that $\begin{matrix} L (α, V, Q, R) ⩽ L (α, V, Q, H (R_{0})) = sup_{‖ u ‖ ⩽ 1} \int_{B_{H (R_{0})}} Q (| x |) e^{α u^{2}} d x < \infty, \end{matrix}$ for $R_{0}$ sufficiently large such that $H (R_{0}) > R$ .

Case 2: Assume $a_{0} < b_{0}$ . In this case, we consider the function $\begin{matrix} w (r) : = {(\frac{b_{0} + 2}{2})}^{1 / 2} u (H (r)), for all r ⩾ 0, \end{matrix}$ where $H (r) = {(\frac{b_{0} + 2}{2})}^{2 / (b_{0} + 2)} r^{2 / (b_{0} + 2)}$ . Once again, a straightforward computation shows that $\begin{matrix} \int_{B_{R_{0}}} | \nabla w |^{2} d x = \int_{B_{H (R_{0})}} | \nabla u |^{2} d x and \int_{B_{R_{0}}} w^{2} d x = \int_{B_{H (R_{0})}} | x |^{b_{0}} u^{2} d x . \end{matrix}$ Moreover, there exists $C_{3} = C_{3} (a_{0}, b_{0}) > 0$ such that $\begin{matrix} \int_{B_{R_{0}}} e^{(\frac{2 α}{b_{0} + 2}) w^{2}} d x = C_{1} \int_{B_{H (R_{0})}} | x |^{b_{0}} e^{α u^{2}} d x . \end{matrix}$ Then, using that $a_{0} < b_{0}$ and $(V)$ we find $C_{4} > 0$ such that $\begin{matrix} \int_{B_{R_{0}}} | \nabla w |^{2} d x + C_{4} \int_{B_{R_{0}}} w^{2} d x ⩽ \int_{B_{H (R_{0})}} | \nabla u |^{2} d x + \int_{B_{H (R_{0})}} V (| x |) u^{2} d x . \end{matrix}$ Now, repeating the same argument as in the proof of Case 1 and applying the classical Trudinger–Moser inequality we conclude that $L (α, V, Q, R) < \infty$ if $2 α / (b_{0} + 2) ⩽ 4 π$ , that is, $α ⩽ 4 π (1 + b_{0} / 2)$ . Next, we will prove that $L (α, V, Q, R) = \infty$ whenever $α > α_{2}$ . In fact, setting ${\tilde{M}}_{n} = M_{n} / ‖ M_{n} ‖$ we see that ${\tilde{M}}_{n} \in E_{r}$ and $‖ {\tilde{M}}_{n} ‖ = 1$ . By Lemma 2.3 if $| x | ⩽ r / n$ we have $\begin{matrix} (2.6) & {\tilde{M}}_{n}^{2} (x, r) = \frac{M_{n}^{2}}{‖ M_{n} ‖^{2}} = \frac{log n}{2 π ‖ M_{n} ‖^{2}} ⩾ \frac{log n}{2 π (1 + o_{n} (1))} . \end{matrix}$ Once again, by assumption $(Q)$ , there exists $C_{5} > 0$ such that $Q (| x |) ⩾ C_{5} | x |^{b_{0}}$ for all $0 < | x | ⩽ r$ . Thus, for large n $\begin{matrix} \int_{B_{R}} Q (| x |) e^{α {\tilde{M}}_{n}^{2}} d x ⩾ C_{6} r^{b_{0} + 2} n^{α {(2 π)}^{- 1} {(1 + o_{n} (1))}^{- 1} - (b_{0} + 2)}, \end{matrix}$ which goes to infinity, since $α {(2 π)}^{- 1} - (b_{0} + 2) > 0$ implies that $α {(2 π)}^{- 1} {(1 + o_{n} (1))}^{- 1} - (b_{0} + 2) > 0$ for large n and this completes the proof. □

We are now ready to prove our sharp weighted Trudinger–Moser type inequality in the whole space $R^{2}$ .
Theorem 2.5.
Let $j_{0} = [γ / 2]$ and assume that $(V)$ and $(Q)$ hold. Then, for all $α > 0$ and $u \in E_{r}$ , it holds that $Q (| x |) Φ_{α, j_{0}} (u) \in L^{1} (R^{2})$ . Moreover, $\begin{matrix} L (α, V, Q, \infty) : = sup_{‖ u ‖ ⩽ 1} \int_{R^{2}} Q (| x |) Φ_{α, j_{0}} (u) d x < \infty, \end{matrix}$ if and only if $0 < α ⩽ α_{2} : = 4 π (1 + b_{0} / 2)$ .
Proof.
For $R > 0$ and $u \in E_{r}$ we split the integral as $\begin{matrix} (2.7) & \int_{R^{2}} Q (| x |) Φ_{α, j_{0}} (u) d x = \int_{B_{R}} Q (| x |) Φ_{α, j_{0}} (u) d x + \int_{B_{R}^{c}} Q (| x |) Φ_{α, j_{0}} (u) d x . \end{matrix}$ Since $Q (| x |) Φ_{α, j_{0}} (u) ⩽ Q (| x |) e^{α u^{2}}$ , by Lemma 2.4 it is enough to estimate the second integral on the right-hand side of (2.7). For that, by (2.1) there exists $C_{1} > 0$ such that $Q (| x |) ⩽ C_{1} | x |^{b}$ for all $| x | ⩾ R$ and according to (2.5) we have $u^{2 j} (x) ⩽ {(C ‖ u ‖)}^{2 j} | x |^{- j (a + 2) / 2}$ for $| x | ⩾ R$ . If $2 j_{0} ⩾ γ$ we can choose $j_{1} \in N$ , $j_{1} ⩾ 1$ such that $(b + 2) - \frac{j (a + 2)}{2} < - 1$ , for $j ⩾ j_{1} > j_{0}$ . Consequently, we can estimate $\begin{matrix} (2.8) & \int_{B_{R}^{c}} Q (| x |) Φ_{α, j_{0}} (u) d x ⩽ \sum_{j = j_{0}}^{j_{1} - 1} \frac{α^{j}}{j!} \int_{B_{R}^{c}} Q (| x |) u^{2 j} d x + C_{2} R^{- 1} e^{α C^{2} ‖ u ‖^{2}} . \end{matrix}$ Since $2 j_{0} ⩾ γ$ , by the continuous embedding $E_{r} ↪ L^{p} (R^{2}; Q)$ with $γ ⩽ p < \infty$ , we get $\begin{matrix} \int_{B_{R}^{c}} Q (| x |) Φ_{α, j_{0}} (u) d x ⩽ C_{3} \sum_{j = j_{0}}^{j_{1} - 1} ‖ u ‖^{2 j} + C_{2} R^{- 1} e^{α C ‖ u ‖^{2}} < \infty, \end{matrix}$ and taking the supremum over $u \in E_{r}$ with $‖ u ‖ ⩽ 1$ we conclude the proof. □

We quote here that Theorem 2.5 improves the Trudinger–Moser inequality proved in [3] in the case that $γ = 2$ , i.e., $- 2 < b ⩽ a$ where the authors obtain a similar result for $0 < α < α_{2} : = min {4 π, 4 π (1 + b_{0} / 2)}$ .

As a consequence of Theorem 2.5 we have the following version of a convergence result due to Lions [18].
Corollary 2.6.
Let $j_{0} = [γ / 2]$ and assume that $(V)$ and $(Q)$ hold. Let $(v_{n}) \subset E_{r}$ with $‖ v_{n} ‖ = 1$ and suppose that $v_{n} ⇀ v$ in $E_{r}$ with $‖ v ‖ < 1$ . Then, for each $0 < β < α_{2} {(1 - ‖ v ‖^{2})}^{- 1}$ , up to a subsequence, it holds that $\begin{matrix} sup_{n \in N} \int_{R^{2}} Q (| x |) Φ_{β, j_{0}} (v_{n}) d x < \infty . \end{matrix}$
Proof.
Since $v_{n} ⇀ v$ in $E_{r}$ and $‖ v_{n} ‖ = 1$ we see that $\begin{matrix} ‖ v_{n} - v ‖^{2} = 1 - 2 ⟨ v_{n}, v ⟩ + ‖ v ‖^{2} \to 1 - ‖ v ‖^{2} < \frac{α_{2}}{β}, as n \to \infty . \end{matrix}$ Thus, for large $n \in N$ we have $β ‖ v_{n} - v ‖^{2} < β^{'} < α_{2}$ for some $β^{'} > 0$ . Moreover, observing that $β v_{n}^{2} ⩽ β (1 + ε^{2}) {(v_{n} - v)}^{2} + β (1 + \frac{1}{ε^{2}}) v^{2}$ and applying Young’s inequality with $1 / r_{1} + 1 / r_{2} = 1$ and $r_{1} > 1$ such that $r_{1} β (1 + ε^{2}) ‖ v_{n} - v ‖^{2} ⩽ α_{2}$ one has $\begin{matrix} e^{β v_{n}^{2}} d x ⩽ \frac{1}{r_{1}} e^{r_{1} β (1 + ε^{2}) ‖ v_{n} - v ‖^{2} {(\frac{v_{n} - v}{‖ v_{n} - v ‖})}^{2}} + \frac{1}{r_{2}} e^{r_{2} β (1 + \frac{1}{ε^{2}}) v^{2}} . \end{matrix}$ For every $R > 0$ , multiplying the above inequality by $Q (x)$ and invoking Lemma 2.4 we obtain $\begin{matrix} sup_{n \in N} \int_{B_{R}} Q (| x |) Φ_{β, j_{0}} (v_{n}) d x ⩽ sup_{n \in N} \int_{B_{R}} Q (| x |) e^{β v_{n}^{2}} d x < \infty . \end{matrix}$ On the other hand, as $2 j_{0} ⩾ γ$ , we can use inequality (2.8) with $v_{n}$ to conclude that $\begin{matrix} sup_{n \in N} \int_{B_{R}^{c}} Q (| x |) Φ_{β, j_{0}} (v_{n}) d x < \infty, \end{matrix}$ and hence the proof is complete. □

The proof Theorem 1.2 will be reached by using variational approach. For this purpose, we start off by considering $α > α_{0}$ , as in the hypothesis (1.1) and $q ⩾ 1$ . Thus, from $(f_{1})$ , for any given $ε > 0$ , there exist constants $C_{1}, C_{2} > 0$ such that $\begin{matrix} (2.9) & | f (s) | ⩽ ε | s |^{γ - 1} + C_{1} | s |^{q - 1} Φ_{α, j_{0}} (s), for all s \in R, \end{matrix}$ and $\begin{matrix} (2.10) & | F (s) | ⩽ \frac{ε}{2} | s |^{γ} + C_{2} | s |^{q} Φ_{α, j_{0}} (s), for all s \in R . \end{matrix}$ Consider the energy functional associated with equation ( P ) given by $\begin{matrix} J_{λ} (u) = \frac{1}{2} ‖ u ‖^{2} - λ \int_{R^{2}} Q (| x |) F (u) d x, for all u \in E_{r} . \end{matrix}$ By using that, for all $r ⩾ 1$ the elementary inequality (see e.g., [32, Lemma 2.1]) $\begin{matrix} (2.11) & {(Φ_{α, j_{0}} (s))}^{r} ⩽ Φ_{r α, j_{0}} (s), for all s \in R, \end{matrix}$ holds, it follows from (2.10), Lemma 2.2 and Theorem 2.5 that $J_{λ}$ is well defined and standard arguments show that $J_{λ} \in C^{1} (E_{r}, R)$ with derivative given by: $\begin{matrix} J_{λ}^{'} (u) v = \int_{R^{2}} (\nabla u \nabla v + V (| x |) u v) d x - λ \int_{R^{2}} Q (| x |) f (u) v d x, for all v \in E_{r} . \end{matrix}$
Remark 2.7.
Since the value of $λ > 0$ is not relevant in the proof of Theorem 1.2, we restrict our analysis to $λ = 1$ and to simplify notation we denote $J_{1}$ by J.

Inspired by [8, Lemma 5.1], we have the following version of Palais’ Principle of Symmetric Criticality due to Palais [22]. Proposition 2.8.
Every critical point of J in $E_{r}$ is a weak solution of ( P ).
Proof.
Let $u \in E_{r}$ and consider the linear functional $T_{u} : E \to R$ defined by $\begin{matrix} T_{u} (w) : = \int_{R^{2}} \nabla u \nabla w d x + \int_{R^{2}} V (| x |) u w d x - \int_{R^{2}} Q (| x |) f (u) w d x . \end{matrix}$ First, we are going to check that $T_{u}$ is well defined and continuous on E, which is enough to estimate the last term. For each $w \in E$ , by (2.9) with $q = γ + 1$ and $ε = 1$ we get a constant $C_{1} > 0$ such that $\begin{matrix} (2.12) & \int_{R^{2}} Q (| x |) | f (u) w | d x ⩽ \int_{R^{2}} Q (| x |) | u |^{γ - 1} | w | d x + C_{1} \int_{R^{2}} Q (| x |) | u |^{γ} Φ_{α, j_{0}} (u) | w | d x . \end{matrix}$ Let us to analyze the last two integrals above. For any $R > 0$ , we can write $\begin{matrix} \int_{R^{2}} Q (| x |) | u |^{γ - 1} | w | d x = \int_{B_{R}} Q (| x |) | u |^{γ - 1} | w | d x + \int_{B_{R}^{c}} Q (| x |) | u |^{γ - 1} | w | d x . \end{matrix}$ Now using Hölder’s inequality and Lemma 2.2 we get $C_{2} > 0$ such that $\begin{matrix} (2.13) & \int_{B_{R}} Q (| x |) | u |^{γ - 1} | w | d x ⩽ C_{2} ‖ u ‖^{γ - 1} ‖ w ‖_{L^{γ} (B_{R}, Q)} . \end{matrix}$ Choosing $p_{1} > 1$ such that $p_{1} b_{0} > - 2$ , we see that $| x |^{p_{1} b_{0}} \in L^{1} (B_{R})$ , and hence we can use $(Q)$ together with Hölder’s inequality with $1 / p_{1} + 1 / p_{2} = 1$ to get $\begin{matrix} ‖ w ‖_{L^{γ} (B_{R}, Q)}^{γ} ⩽ C_{3} {(\int_{B_{R}} | x |^{p_{1} b_{0}} d x)}^{\frac{1}{p_{1}}} {(\int_{B_{R}} | w |^{p_{2} γ} d x)}^{\frac{1}{p_{2}}} ⩽ C_{4} {(\int_{B_{R}} | w |^{p_{2} γ} d x)}^{\frac{1}{p_{2}}} . \end{matrix}$ From this and the continuous embedding $E ↪ L^{p_{2} γ} (B_{R})$ (see Lemma 2.1), we deduce that $\begin{matrix} (2.14) & ‖ w ‖_{L^{γ} (B_{R}, Q)} ⩽ C_{5} ‖ w ‖, \end{matrix}$ which combined with (2.13) implies that $\begin{matrix} (2.15) & \int_{B_{R}} Q (| x |) | u |^{γ - 1} | w | d x ⩽ C_{6} ‖ w ‖ . \end{matrix}$ On the other hand, by (2.1), inequality (2.5) and the fact that $b - (γ - 2) (a + 2) / 4 ⩽ a$ , for $| x | > R$ one has $\begin{matrix} Q (| x |) | u |^{γ - 1} ⩽ C_{7} ‖ u ‖^{γ - 2} | x |^{b - (γ - 2) (a + 2) / 4} | u | ⩽ C_{8} | x |^{a} | u | ⩽ C_{9} V (| x |) | u | . \end{matrix}$ Consequently, we get $\begin{matrix} \int_{B_{R}^{c}} Q (| x |) | u |^{γ - 1} | w | d x ⩽ C_{9} ‖ u ‖ ‖ w ‖ = C_{10} ‖ w ‖ \end{matrix}$ This, combined with (2.15), implies $\begin{matrix} (2.16) & \int_{R^{2}} Q (| x |) | u |^{γ - 1} | w | d x ⩽ C_{11} ‖ w ‖ . \end{matrix}$ Next, we estimate the second integral on the hight-hand side of (2.12). For $R > 0$ we also split $\begin{matrix} \int_{R^{2}} Q (| x |) | u |^{γ} Φ_{α, j_{0}} (u) | w | d x = T_{1} (w) + T_{2} (w), \end{matrix}$ where $\begin{matrix} T_{1} (w) : = \int_{B_{R}} Q (| x |) | u |^{γ} Φ_{α, j_{0}} (u) | w | d x and T_{2} (w) : = \int_{B_{R}^{c}} Q (| x |) | u |^{γ} Φ_{α, j_{0}} (u) | w | d x . \end{matrix}$ Invoking Hölder’s inequality, (2.11), Lemma 2.2, Lemma 2.4, and (2.14), we see that $\begin{matrix} | T_{1} (w) | ⩽ ‖ u ‖_{L^{q_{1} γ} (B_{R}, Q)}^{γ} {(\int_{B_{R}} Q (| x |) Φ_{q_{2} α, j_{0}} (u) d x)}^{\frac{1}{q_{2}}} ‖ w ‖_{L^{q_{3}} (B_{R}, Q)} ⩽ C_{12} ‖ w ‖, \end{matrix}$ for $q_{1}, q_{2}, q_{3} > 1$ satisfying $1 / q_{1} + 1 / q_{2} + 1 / q_{3} = 1$ and $q_{3} = γ$ . On the other hand, using that $b - γ (a + 2) / 2 ⩽ a$ , from Hölder’s inequality, $(V)$ , $(Q)$ , (2.5), (2.11) and Theorem 2.5 we get $\begin{matrix} | T_{2} (w) | & ⩽ C_{16} {(\int_{B_{R}^{c}} | x |^{b} | u |^{2 γ} w^{2} d x)}^{\frac{1}{2}} {(\int_{B_{R}^{c}} Q (| x |) Φ_{2 α, j_{0}} (u) d x)}^{\frac{1}{2}} \\ ⩽ C_{17} ‖ u ‖^{γ} {(\int_{B_{R}^{c}} | x |^{b - γ (a + 2) / 2} w^{2} d x)}^{1 / 2} \\ ⩽ C_{18} {(\int_{B_{R}^{c}} | x |^{a} w^{2} d x)}^{1 / 2} ⩽ C_{19} ‖ w ‖ . \end{matrix}$ Therefore, $\begin{matrix} \int_{R^{2}} Q (| x |) | u |^{γ} Φ_{α, j_{0}} (u) | w | d x ⩽ C_{20} ‖ w ‖ . \end{matrix}$ This, together with (2.12) and (2.16), implies that $T_{u}$ is continuous. Now, suppose that $u \in E_{r}$ is a critical point of J, i.e., $T_{u} (w) = 0$ for all $w \in E_{r}$ . By the Riesz Representation Theorem in the space E, there exists a unique $\overline{u} \in E$ such that $T_{u} (\overline{u}) = ‖ \overline{u} ‖^{2} = ‖ T_{u} ‖_{E^{'}}^{2}$ , where $E^{'}$ denotes the dual space of E. Let $O (2)$ denotes the group of orthogonal transformations in $R^{2}$ . Since V, Q and $u_{r}$ are radial, by using change of variables, one has for each $w \in E$ $\begin{matrix} T_{u} (g w) = T_{u} (w) and ‖ g w ‖ = ‖ w ‖, for all g \in O (2) . \end{matrix}$ Applying this with $w = \overline{u}$ , by uniqueness, $g \overline{u} = \overline{u}$ , for all $g \in O (2)$ , which means that $\overline{u} \in E_{r}$ and consequently $T_{u} (\overline{u}) = 0$ , that is, $‖ T_{u} ‖_{E^{'}} = 0$ which implies that $T_{u} (w) = 0$ , for all $w \in E$ and this concludes the proof. □

3. Proofs of Theorem 1.2 and Theorem 1.3

In view of Proposition 2.8 we are going to get solutions of ( P ) looking for critical points of J. We first prove that J satisfies the Mountain Pass geometry.

Lemma 3.1.

Assume that (1.1), $(f_{1})$ – $(f_{2})$ , $(V)$ and $(Q)$ hold. Then

there exist $τ, ρ > 0$ such that $J (u) ⩾ τ$ if $‖ u ‖ = ρ$ ;

there exists $e \in E_{r}$ , with $‖ e ‖ > ρ$ , such that $J (e) < 0$ .

Proof.

For every $q > γ ⩾ 2$ , by Hölder’s inequality with exponents $1 / r_{1} + 1 / r_{2} = 1$ together with (2.11) we get $\begin{matrix} \int_{R^{2}} Q (| x |) | s |^{q} Φ_{α, j_{0}} (u) d x ⩽ ‖ u ‖_{L^{q r_{1}} (R^{2}, Q)}^{q} {(\int_{R^{2}} Q (| x |) Φ_{r_{2} α ‖ u ‖, j_{0}} (\frac{u}{‖ u ‖}) d x)}^{1 / r_{2}} . \end{matrix}$ Choosing $‖ u ‖ = ρ < {(α_{2} / r_{2} α)}^{1 / 2}$ , we can apply Theorem 2.5 and use inequality (2.10) to get $C_{3} > 0$ such that $\begin{matrix} \int_{R^{2}} Q (| x |) F (u) d x ⩽ \frac{ε}{2} ‖ u ‖_{L^{γ} (R^{2}, Q)}^{γ} + C_{3} ‖ u ‖_{L^{q r_{1}} (R^{2}, Q)}^{q}, \end{matrix}$ for every $ε > 0$ . Hence according to Lemma 2.2, there exist constants $C_{4}, C_{5} > 0$ such that $\begin{matrix} J (u) ⩾ \frac{1}{2} ‖ u ‖^{2} - \frac{C_{4} ε}{2} ‖ u ‖^{γ} - C_{5} ‖ u ‖^{q}, \end{matrix}$ which gives us (i), if $γ > 2$ . In case that $γ = 2$ , we obtain the result by choosing $ε > 0$ sufficiently small. To prove (ii), we consider a function $φ \in C_{0, rad}^{\infty} (R^{2}) ∖ {0}$ and denote its support by $supp φ$ . From $(f_{2})$ there exist $θ > γ ⩾ 2$ and constants $A, B > 0$ such that $F (s) ⩾ A | s |^{θ} - B$ , for all $s \in R$ . Thus, for all $t > 0$ , it holds that $\begin{matrix} J (t φ) ⩽ \frac{t^{2}}{2} ‖ φ ‖^{2} - A t^{θ} \int_{supp φ} Q (| x |) | φ |^{θ} d x + B \int_{supp φ} Q (| x |) d x . \end{matrix}$ Since $θ > 2$ , we obtain (ii) by taking $e = t φ$ with $t > 0$ sufficiently large and this concludes the proof. □

In view of Lemma 3.1 the minimax level $\begin{matrix} c = inf_{g \in Γ} max_{t \in [0, 1]} J (g (t)), \end{matrix}$ with $Γ = {g \in C ([0, 1], E_{r}) : g (0) = 0 and J (g (1)) < 0}$ is positive. According to the Mountain Pass Theorem without the Palais–Smale condition (see e.g., [12]) we obtain a Palais–Smale sequence ( ${(PS)}_{c}$ for short) $(u_{n}) \subset E_{r}$ at the level c, that is, $\begin{matrix} J (u_{n}) \to c and J^{'} (u_{n}) \to 0 . \end{matrix}$ Whenever $(u_{n}) \subset E_{r}$ is a ${(PS)}_{c}$ sequence, we will show next that, up to a subsequence, $u_{n} ⇀ u$ in $E_{r}$ . In order to prove that u is a weak solution of ( P ) we will need the following compactnesses result:

Lemma 3.2.

Assume (1.1), $(f_{1})$ – $(f_{3})$ , $(V)$ and $(Q)$ . If $(u_{n}) \subset E_{r}$ is a ${(PS)}_{c}$ sequence to J then $(u_{n})$ is bounded in $E_{r}$ and, up to a subsequence, it holds that:

$Q (| x |) f (u_{n}) \to Q (| x |) f (u)$ in $L_{loc}^{1} (R^{2})$ ;

$Q (| x |) F (u_{n}) \to Q (| x |) F (u)$ in $L^{1} (R^{2})$ .

Proof.

By hypothesis, we have $\begin{matrix} \frac{1}{2} ‖ u_{n} ‖^{2} - \int_{R^{2}} Q (| x |) F (u_{n}) d x = c + o_{n} (1) \end{matrix}$ and $\begin{matrix} ‖ u_{n} ‖^{2} - \int_{R^{2}} Q (| x |) f (u_{n}) u_{n} d x = o_{n} (‖ u_{n} ‖) . \end{matrix}$ Thus, for every $θ > γ ⩾ 2$ we get constants $C_{1}, C_{2} > 0$ such that $\begin{matrix} (3.1) & 1 + C_{1} + C_{2} ‖ u_{n} ‖ ⩾ (\frac{1}{2} - \frac{1}{θ}) ‖ u_{n} ‖^{2} + \int_{R^{2}} Q (| x |) (\frac{1}{θ} f (u_{n}) u_{n} - F (u_{n})) d x, \end{matrix}$ which yields that $(u_{n})$ is bounded by $(f_{2})$ . As a consequence we have the estimate $\begin{matrix} \int_{R^{2}} Q (| x |) f (u_{n}) u_{n} d x ⩽ C_{3}, for all n \in N . \end{matrix}$ Then, up to a subsequence, $u_{n} ⇀ u$ in $E_{r}$ and $Q (| x |) u_{n} \to Q (| x |) u$ in $L_{loc}^{1} (R^{2})$ by Lemma 2.1. From (2.9), Lemma 2.1 and Theorem 2.5 we see that $Q (| x |) f (u) \in L_{loc}^{1} (R^{2})$ . Therefore, thanks to [14, Lemma 2.1] we conclude that (i) holds true. In order to prove (ii), for every $R > 0$ we write $\begin{matrix} \int_{R^{2}} Q (| x |) (F (u_{n}) - F (u)) d x = I_{n} (B_{R}) + I_{n} (B_{R}^{c}), \end{matrix}$ where for $Ω = B_{R}$ or $Ω = B_{R}^{c}$ , $\begin{matrix} I_{n} (Ω) : = \int_{Ω} Q (| x |) (F (u_{n}) - F (u)) d x . \end{matrix}$ First we check that, for all $R > 0$ fixed we have $\begin{matrix} (3.2) & lim_{n \to \infty} I_{n} (B_{R}) = 0 . \end{matrix}$ In fact, for any $ε > 0$ , according to Egoroff’s Theorem there exists a measurable set $Ω \subset B_{R}$ with $| Ω | < ε$ such that $u_{n} (x) \to u (x)$ uniformly in $B_{R} ∖ Ω$ , and consequently $\begin{matrix} (3.3) & | I_{n} (B_{R}) | ⩽ \int_{Ω} Q (| x |) F (u) d x + \int_{Ω} Q (| x |) F (u_{n}) d x + o_{n} (1) . \end{matrix}$ From (2.10), for $q ⩾ γ$ we see that $\begin{matrix} \int_{Ω} Q (| x |) F (u) d x ⩽ \frac{ε}{2} \int_{Ω} Q (| x |) | u |^{γ} d x + C_{2} \int_{Ω} Q (| x |) | u |^{q} Φ_{α, j_{0}} (u) d x . \end{matrix}$ On the other hand, by Lemma 2.2, (2.11) and Theorem 2.5 we have $\begin{matrix} \int_{Ω} Q (| x |) | u |^{q} Φ_{α, j_{0}} (u) d x & ⩽ ‖ Q ‖_{L^{q_{3}} (Ω)} ‖ u ‖_{L^{q q_{1}} (R^{2}, Q)}^{q} {(\int_{Ω} Q (| x |) Φ_{q_{2} α, j_{0}} (u) d x)}^{1 / q_{2}} \\ ⩽ C_{3} {(\int_{Ω} Q (| x |) d x)}^{1 / q_{3}}, \end{matrix}$ whenever $q_{1}, q_{2}, q_{3} > 1$ satisfy $1 / q_{1} + 1 / q_{2} + 1 / q_{3} = 1$ . From hypothesis $(Q)$ , there exists $C_{4} > 0$ such that $Q (| x |) ⩽ C_{4} | x |^{b_{0}}$ , for all $0 < | x | ⩽ R$ . Since $b_{0} > - 2$ we can choose $r_{1} > 1$ with $1 / r_{1} + 1 / r_{2} = 1$ such that $r_{1} b_{0} > - 2$ and hence $\begin{matrix} \int_{Ω} Q (| x |) d x ⩽ C_{5} {(\int_{Ω} | x |^{r_{1} b_{0}} d x)}^{1 / r_{1}} | Ω |^{1 / r_{2}} ⩽ C_{6} ε^{1 / r_{2}} . \end{matrix}$ Thus, we conclude that $\begin{matrix} (3.4) & \int_{Ω} Q (| x |) F (u) d x ⩽ C_{7} ε^{1 / r_{2} q_{3}} . \end{matrix}$ Next, we estimate the second integral on the right-hand side of (3.3). To do this, from (3.1), it follows that $\begin{matrix} (3.5) & \int_{R^{2}} Q (| x |) (\frac{1}{θ} f (u_{n}) u_{n} - F (u_{n})) d x ⩽ C_{8} . \end{matrix}$ For any $ε > 0$ , we can choose $θ_{0} > θ$ such that $\begin{matrix} (3.6) & 0 < \frac{θ C_{8}}{(θ_{0} - θ)} < ε . \end{matrix}$ By hypothesis $(f_{3})$ there exists $s_{0} > 0$ such that $θ_{0} F (s) ⩽ f (s) s$ for any $| s | ⩾ s_{0}$ . Furthermore, by (3.5) we infer that $\begin{matrix} \int_{{| u_{n} | ⩾ s_{0}}} Q (| x |) (f (u_{n}) u_{n} - θ F (u_{n})) d x ⩽ θ C_{8}, \end{matrix}$ and consequently by $(f_{2})$ we get $\begin{matrix} (θ_{0} - θ) \int_{{| u_{n} | ⩾ s_{0}}} Q (| x |) F (u_{n}) d x \\ = \int_{{| u_{n} | ⩾ s_{0}}} Q (| x |) (θ_{0} F (u_{n}) - f (u_{n}) u_{n} + f (u_{n}) u_{n} - θ F (u_{n})) d x \\ ⩽ θ C_{8}, \end{matrix}$ which combined with (3.6) implies that $\begin{matrix} (3.7) & \int_{{| u_{n} | ⩾ s_{0}}} Q (| x |) F (u_{n}) d x < ε . \end{matrix}$ On the other hand, since $Q (| x |) f (u_{n}) \to Q (| x |) f (u)$ in $L^{1} (B_{R})$ there exists $g \in L^{1} (B_{R})$ such that $Q (| x |) | f (u_{n}) | ⩽ g$ a.e in $B_{R}$ . From assumption $(f_{2})$ , we have $\begin{matrix} Q (| x |) F (u_{n}) ⩽ \frac{1}{θ} Q (| x |) f (u_{n}) (u_{n}) ⩽ \frac{1}{θ} g s_{0} a.e. in Ω \cap {| u_{n} | < s_{0}} . \end{matrix}$ Then, by applying the Lebesgue Dominated Convergence Theorem and using (3.4) we obtain $\begin{matrix} lim_{n \to \infty} \int_{Ω \cap {| u_{n} | < s_{0}}} Q (| x |) F (u_{n}) d x = \int_{Ω \cap {| u | < s_{0}}} Q (| x |) F (u) d x < C_{7} ε^{1 / r_{2} q_{3}} . \end{matrix}$ Since $\begin{matrix} \int_{Ω} Q (| x |) F (u_{n}) d x = \int_{Ω \cap {| u_{n} | ⩾ s_{0}}} Q (| x |) F (u_{n}) d x + \int_{Ω \cap {| u_{n} | < s_{0}}} Q (| x |) F (u_{n}) d x, \end{matrix}$ from (3.7) we find $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{Ω} Q (| x |) F (u_{n}) d x ⩽ ε + C_{7} ε^{1 / r_{2} q_{3}} . \end{matrix}$ Since $ε > 0$ is arbitrary, the last estimate above together with (3.3), and (3.4) imply that (3.2) holds true.

Next, we will prove that for $R > 0$ sufficiently large ${lim}_{n \to \infty} I_{n} (B_{R}^{c}) = 0$ . For this, using that $Q (| x |) F (u) \in L^{1} (R^{2})$ for any $ε > 0$ we can choose $R > 0$ sufficiently large such that $\begin{matrix} (3.8) & | I_{n} (B_{R}^{c}) | ⩽ \int_{B_{R}^{c}} Q (| x |) F (u) d x + \int_{B_{R}^{c}} Q (| x |) F (u_{n}) d x < ε + \int_{B_{R}^{c}} Q (| x |) F (u_{n}) d x . \end{matrix}$ To estimate the last integral above, since $(u_{n})$ is bounded, using Hölder’s inequality with $1 / r_{1} + 1 / r_{2} = 1$ we get $\begin{matrix} (3.9) & \int_{B_{R}^{c}} Q (| x |) | u_{n} |^{q} Φ_{α, j_{0}} (u_{n}) d x ⩽ ‖ u_{n} ‖_{L^{r_{1} q} (B_{R}^{c}, Q)}^{q} {(\int_{B_{R}^{c}} Q (| x |) Φ_{r_{2} α, j_{0}} (u_{n}) d x)}^{1 / r_{2}} . \end{matrix}$ This together with inequality (2.10) implies that $\begin{matrix} (3.10) & \begin{matrix} \int_{B_{R}^{c}} Q (| x |) F (u_{n}) d x & ⩽ \frac{ε}{2} \int_{B_{R}^{c}} Q (| x |) | u_{n} |^{γ} d x + C_{9} \int_{B_{R}^{c}} Q (| x |) | u_{n} |^{q} Φ_{α, j_{0}} (u_{n}) d x \\ ⩽ C_{10} ε + C_{11} {(\int_{B_{R}^{c}} Q (| x |) Φ_{r_{2} α, j_{0}} (u_{n}) d x)}^{1 / r_{2}}, \end{matrix} \end{matrix}$ for every $ε > 0$ and $n \in N$ . Now using inequality (2.8) with u replaced to $u_{n}$ we get $\begin{matrix} \int_{B_{R}^{c}} Q (| x |) Φ_{r_{2} α, j_{0}} (u_{n}) d x ⩽ \sum_{j = j_{0}}^{j_{1} - 1} \frac{α^{j}}{j!} \int_{B_{R}^{c}} Q (| x |) u_{n}^{2 j} d x + C_{12} R^{- 1} e^{r_{2} α C^{2} ‖ u_{n} ‖^{2}} . \end{matrix}$ Since $(u_{n})$ is bounded by Lemma 2.2 we deduce that $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{B_{R}^{c}} Q (| x |) Φ_{r_{2} α, j_{0}} (u_{n}) d x ⩽ \sum_{j = j_{0}}^{j_{1} - 1} \frac{α^{j}}{j!} \int_{B_{R}^{c}} Q (| x |) u^{2 j} d x + C_{12} R^{- 1} e^{r_{2} α C^{2} C_{13}}, \end{matrix}$ that goes to zero as $R \to \infty$ . This in combination with (3.8) and (3.10) implies that ${lim}_{n \to \infty} I_{n} (B_{R}^{c}) = 0$ and the lemma is proved. □

As a consequence of the previous lemma, we have the following local compactness result:

Proposition 3.3.

Assume (1.1), $(f_{1})$ – $(f_{3})$ , $(V)$ and $(Q)$ . Then the functional J satisfies the ${(PS)}_{c}$ condition for every $c \in (0, α_{2} / 2 α_{0})$ .

Proof.

Let $(u_{n}) \subset E_{r}$ be a ${(PS)}_{c}$ . By Lemma 3.2 we can assume, up to a subsequence, that $u_{n} ⇀ u$ weakly in $E_{r}$ and for all $φ \in C_{0, rad}^{\infty} (R^{2})$ we have $\begin{matrix} (3.11) & \int_{R^{2}} (\nabla u_{n} \nabla φ + V (| x |) u_{n} φ) d x - \int_{R^{2}} Q (| x |) f (u_{n}) φ d x = o_{n} (1) ‖ φ ‖ . \end{matrix}$ Passing to the limit and using Lemma 3.2 we get $\begin{matrix} (3.12) & \int_{R^{2}} (\nabla u \nabla φ + V (| x |) u φ) d x - \int_{R^{2}} Q (| x |) f (u) φ d x = 0, for all φ \in C_{0, rad}^{\infty} (R^{2}) . \end{matrix}$ Next, we are going to check that $\begin{matrix} (3.13) & lim_{n \to \infty} \int_{R^{2}} Q (| x |) f (u_{n}) u_{n} d x = \int_{R^{2}} Q (| x |) f (u) u d x . \end{matrix}$ If this is true, from (3.11) and (3.12) it follows that $\begin{matrix} lim_{n \to \infty} ‖ u_{n} ‖^{2} = lim_{n \to \infty} \int_{R^{2}} Q (| x |) f (u_{n}) u_{n} d x = \int_{R^{2}} Q (| x |) f (u) u d x = ‖ u ‖^{2} \end{matrix}$ and this finishes the proof. Thus, it remains to prove (3.13). To this end, we first observe that by Lemma 3.2 $Q (| x |) F (u_{n}) \to Q (| x |) F (u)$ in $L^{1} (R^{2})$ and hence from the fact that $J (u_{n}) = c + o_{n} (1)$ we get $\begin{matrix} (3.14) & lim_{n \to \infty} ‖ u_{n} ‖^{2} = 2 (c + \int_{R^{2}} Q (| x |) F (u) d x) > 0 . \end{matrix}$ Defining $v_{n} : = u_{n} / ‖ u_{n} ‖$ , by the weak convergence of $(u_{n})$ we have $\begin{matrix} v_{n} ⇀ v : = u / lim_{n \to \infty} ‖ u_{n} ‖ weakly in E_{r}, \end{matrix}$ with $‖ v ‖ ⩽ 1$ . If $‖ v ‖ = 1$ we finish the proof. Otherwise, it follows from $(f_{2})$ and (3.12) that $\begin{matrix} J (u) = J (u) - \frac{1}{θ} J^{'} (u) u = (\frac{1}{2} - \frac{1}{θ}) ‖ u ‖^{2} + \int_{R^{2}} Q (| x |) (\frac{1}{θ} f (u) u - F (u)) d x ⩾ 0 . \end{matrix}$ Setting $A : = (c + \int_{R^{2}} Q (| x |) F (u) d x) (1 - ‖ v ‖^{2})$ , it follows from the definition of v that $A = c - J (u)$ . Thus, from (3.14) we reach $\begin{matrix} \frac{1}{2} lim_{n \to \infty} ‖ u_{n} ‖^{2} = \frac{A}{1 - ‖ v ‖^{2}} = \frac{c - J (u)}{1 - ‖ v ‖^{2}} ⩽ \frac{c}{1 - ‖ v ‖^{2}} < \frac{α_{2}}{2 α_{0} (1 - ‖ v ‖^{2})} . \end{matrix}$ Consequently, for large $n \in N$ there are $q_{1} > 1$ sufficiently close to 1, $α > α_{0}$ close to $α_{0}$ and $β > 0$ such that $q_{1} α ‖ u_{n} ‖^{2} ⩽ β < α_{2} {(1 - ‖ v ‖^{2})}^{- 1}$ . Therefore, by Corollary 2.6 there exists $C_{1} > 0$ such that $\begin{matrix} (3.15) & \int_{R^{2}} Q (| x |) Φ_{q_{1} α, j_{0}} (u_{n}) d x ⩽ C_{1} . \end{matrix}$ Now we observe that $\begin{matrix} | \int_{R^{2}} Q (| x |) (f (u_{n}) u_{n} - f (u) u) d x | ⩽ L_{1} (n) + L_{2} (n), \end{matrix}$ where $\begin{matrix} L_{1} (n) : = \int_{R^{2}} Q (| x |) | f (u_{n}) (u_{n} - u) | d x and L_{2} (n) : = \int_{R^{2}} Q (| x |) | f (u_{n}) - f (u) | | u | d x . \end{matrix}$ We are going to check that ${lim}_{n \to \infty} L_{i} (n) = 0$ for $i = 1, 2$ . To this, by inequality (2.9) with $q = 1$ , for every $ε > 0$ there exists $C_{2} >$ such that Hölder’s inequality implies $\begin{matrix} I_{1} (n) ⩽ ε ‖ u_{n} ‖_{L^{γ} (R^{2}; Q)}^{γ - 1} ‖ u_{n} - u ‖_{L^{γ} (R^{2}; Q)} + C_{2} \int_{R^{2}} Q (| x |) | u_{n} - u | Φ_{α, j_{0}} (u_{n}) d x . \end{matrix}$ Now, using (2.11) and Hölder’s inequality with exponents $1 / q_{1} + 1 / q_{2} = 1$ , $q_{1}$ close to 1 and $q_{2}$ sufficiently large we get $\begin{matrix} \int_{R^{2}} Q (| x |) Φ_{α, j_{0}} (u_{n}) | u_{n} - u | d x ⩽ ‖ u_{n} - u ‖_{L^{q_{2}} (R^{2}; Q)} {(\int_{R^{2}} Q (| x |) Φ_{q_{1} α, j_{0}} (u_{n}) d x)}^{1 / q_{1}} . \end{matrix}$ This combined with (3.15) and the compact embedding in Lemma 2.2 implies that ${lim}_{n \to \infty} L_{1} (n) = 0$ . Next we will check that ${lim}_{n \to \infty} L_{2} (n) = 0$ . For this purpose, since $C_{0, rad}^{\infty} (R^{2})$ is dense in $E_{r}$ , for each $ε > 0$ , there exists $v \in C_{0, rad}^{\infty} (R^{2})$ such that $‖ u - v ‖ < ε$ . Now, notice that $\begin{matrix} L_{2} (n) ⩽ & \int_{R^{2}} Q (| x |) | f (u_{n}) (u - v) | d x + \int_{R^{2}} Q (| x |) | f (u) (v - u) | d x \\ + ‖ v ‖_{L^{\infty} (R^{2})} \int_{supp v} Q (| x |) | f (u_{n}) - f (u) | d x . \end{matrix}$ Since $(u_{n})$ is bounded, applying (3.11) with $φ = u - v$ we find $C_{3} > 0$ such that $\begin{matrix} | \int_{R^{2}} Q (| x |) f (u_{n}) (u - v) d x | ⩽ o_{n} (1) ‖ u - v ‖ + ‖ u_{n} ‖ ‖ u - v ‖ ⩽ C_{3} ε . \end{matrix}$ In a similar way, from (3.12) we get $C_{4} > 0$ such that $\begin{matrix} \int_{R^{2}} Q (| x |) | f (u) (u - v) | d x < C_{4} ε \end{matrix}$ and according to Lemma 3.2 we have $\begin{matrix} ‖ v ‖_{L^{\infty} (R^{2})} \int_{supp v} Q (| x |) | f (u_{n}) - f (u) | d x = o_{n} (1) . \end{matrix}$ Therefore, ${lim}_{n \to \infty} L_{2} (n) = 0$ and this completes the proof. □

Next, we will obtain an estimate for the minimax level.

Lemma 3.4.

Assume $(f_{2})$ , $(f_{4})$ , $(V)$ and $(Q)$ . Then, there exists $n \in N$ such that $\begin{matrix} max_{t ⩾ 0} J (t {\tilde{M}}_{n}) = max_{t ⩾ 0} {\frac{t^{2}}{2} - \int_{R^{2}} Q (| x |) F (t {\tilde{M}}_{n}) d x} < \frac{α_{2}}{2 α_{0}}, \end{matrix}$ where ${\tilde{M}}_{n} (x, r_{0}) : = M_{n} (x, r_{0}) / ‖ M_{n} ‖$ with $r_{0}$ given in condition $(f_{4})$ .

Proof.

We argue towards a contradiction, by supposing that the conclusion of the lemma fails. Then, for every $n \in N$ , there exists $t_{n} > 0$ such that $\begin{matrix} \frac{t_{n}^{2}}{2} - \int_{R^{2}} Q (| x |) F (t_{n} {\tilde{M}}_{n}) d x ⩾ \frac{α_{2}}{2 α_{0}} . \end{matrix}$ Since $Q (| x |) > 0$ and $F (s) ⩾ 0$ , we have $t_{n}^{2} ⩾ α_{2} / α_{0}$ . Taking into account that $\frac{d}{d t} (J (t {\tilde{M}}_{n})) |_{t = t_{n}} = 0$ we infer that $\begin{matrix} (3.16) & t_{n}^{2} = \int_{R^{2}} Q (| x |) f (t_{n} {\tilde{M}}_{n}) t_{n} {\tilde{M}}_{n} d x . \end{matrix}$ Now we recall that by hypothesis $(f_{4})$ , for all $0 < ε < β_{0}$ there exists $R = R (ε) > 0$ such that $\begin{matrix} (3.17) & f (s) s ⩾ (β_{0} - ε) e^{α_{0} s^{2}} for all | s | ⩾ R, \end{matrix}$ where $β_{0} > 2 {(b_{0} + 2)}^{2} / D_{0} α_{0} r_{0}^{b_{0} + 2}$ with $r_{0}$ and $D_{0}$ satisfying $\begin{matrix} (3.18) & Q (| x |) ⩾ \frac{D_{0}}{2} | x |^{b_{0}} for all 0 < | x | ⩽ r_{0} . \end{matrix}$ Since $t_{n}^{2} ⩾ α_{2} / α_{0}$ , for large $n \in N$ we obtain $t_{n} {\tilde{M}}_{n} (x, r_{0}) ⩾ R$ for all $0 ⩽ | x | ⩽ r_{0} / n$ and hence by (3.17) we obtain $\begin{matrix} f (t_{n} {\tilde{M}}_{n} (x, r_{0})) t_{n} {\tilde{M}}_{n} (x, r_{0}) ⩾ (β_{0} - ε) e^{α_{0} {(t_{n} {\tilde{M}}_{n} (x, r_{0}))}^{2}} . \end{matrix}$ On the other hand, from estimate (2.6) with $r = r_{0}$ we have $\begin{matrix} {\tilde{M}}_{n}^{2} (x, r_{0}) ⩾ \frac{log n}{2 π (1 + o_{n} (1))} if | x | ⩽ r_{0} / n . \end{matrix}$ Then, from the above estimates we obtain $\begin{matrix} t_{n}^{2} ⩾ \frac{(β_{0} - ε) D_{0}}{2} \int_{| x | ⩽ r_{0} / n} | x |^{b_{0}} e^{\frac{α_{0} t_{n}^{2} log n}{2 π (1 + o_{n} (1))}} d x = \frac{(β_{0} - ε) D_{0} π}{(b_{0} + 2)} {(\frac{r_{0}}{n})}^{b_{0} + 2} e^{\frac{α_{0} t_{n}^{2} log n}{2 π (1 + o_{n} (1))}}, \end{matrix}$ which leads to $\begin{matrix} (3.19) & C_{0} t_{n}^{2} n^{b_{0} + 2} ⩾ n^{α_{0} t_{n}^{2} / 2 π (1 + o_{n} (1))} with C_{0} = (b_{0} + 2) / (β_{0} - ε) D_{0} π r_{0}^{b_{0} + 2} . \end{matrix}$ We claim that $(t_{n})$ is bounded. Indeed, suppose by contradiction that $t_{n} \to \infty$ . From (3.19) we get $\begin{matrix} (3.20) & log (C_{0}) + log (t_{n}^{2}) + (b_{0} + 2) log n ⩾ \frac{α_{0}}{2 π (1 + o_{n} (1))} t_{n}^{2} log n . \end{matrix}$ Thus, $\begin{matrix} \frac{log (C_{0})}{t_{n}^{2} log n} + \frac{log (t_{n}^{2})}{t_{n}^{2} log n} + \frac{(b_{0} + 2)}{t_{n}^{2}} ⩾ \frac{α_{0}}{2 π (1 + o_{n} (1))}, \end{matrix}$ and taking the limit we obtain a contradiction. Now we will show that $\begin{matrix} lim_{n \to \infty} t_{n}^{2} = \frac{α_{2}}{α_{0}} . \end{matrix}$ Otherwise, there exists some $δ > 0$ such that for $n \in N$ sufficiently large $t_{n}^{2} ⩾ α_{2} / α_{0} + δ$ . This, together with (3.20) implies $\begin{matrix} \frac{log (C_{0})}{log n} + \frac{log (t_{n}^{2})}{log n} + (b_{0} + 2) ⩾ \frac{α_{0} t_{n}^{2}}{2 π (1 + o_{n} (1))} ⩾ \frac{α_{2} + α_{0} δ}{2 π (1 + o_{n} (1))} . \end{matrix}$ Since $(t_{n})$ is bounded, taking the limit we obtain $2 π (b_{0} + 2) ⩾ (α_{2} + α_{0} δ)$ , which contradicts the fact that $α_{2} = 2 π (b_{0} + 2)$ . Finally, we estimate $β_{0}$ to get a contradiction. It follows from (3.16) that $\begin{matrix} (3.21) & t_{n}^{2} = \int_{A_{n}} Q (| x |) f (t_{n} {\tilde{M}}_{n}) t_{n} {\tilde{M}}_{n} d x + \int_{B_{n}} Q (| x |) f (t_{n} {\tilde{M}}_{n}) t_{n} {\tilde{M}}_{n} d x, \end{matrix}$ where $A_{n} : = {x \in B_{0} : | t_{n} {\tilde{M}}_{n} | ⩽ R}$ and $B_{n} : = {x \in B_{0} : | t_{n} {\tilde{M}}_{n} | ⩾ R}$ . Since ${\tilde{M}}_{n} \to 0$ a.e. in $R^{2}$ , by applying the Lebesgue Dominated Convergence Theorem we get $\begin{matrix} lim_{n \to \infty} \int_{A_{n}} Q (| x |) f (t_{n} {\tilde{M}}_{n}) t_{n} {\tilde{M}}_{n} d x = 0 . \end{matrix}$ Thus, taking the limit in (3.21) we obtain $\begin{matrix} (3.22) & \frac{α_{2}}{α_{0}} = lim_{n \to \infty} \int_{B_{n}} Q (| x |) f (t_{n} {\tilde{M}}_{n}) t_{n} {\tilde{M}}_{n} d x . \end{matrix}$ Using that $t_{n}^{2} ⩾ α_{2} / α_{0}$ , from (3.17) and (3.18) it follows that $\begin{matrix} \int_{B_{n}} Q (| x |) f (t_{n} {\tilde{M}}_{n}) t_{n} {\tilde{M}}_{n} d x ⩾ \frac{(β_{0} - ε) D_{0}}{2} \int_{B_{n}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x . \end{matrix}$ Now we observe that $\begin{matrix} \int_{B_{r_{0}}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x = \frac{2 π r_{0}^{b_{0} + 2}}{(b_{0} + 2)} + \int_{\frac{r_{0}}{n} ⩽ | x | ⩽ r_{0}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x . \end{matrix}$ Performing a straightforward computation and doing the change of variables $r = r_{0} e^{- ‖ {\tilde{M}}_{n} ‖ {(log n)}^{1 / 2} s}$ we get $\begin{matrix} \int_{\frac{r_{0}}{n} ⩽ | x | ⩽ r_{0}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x & = 2 π \int_{r_{0} / n}^{r_{0}} r^{b_{0}} e^{(b_{0} + 2) {[{(log n)}^{- 1 / 2} ‖ {\tilde{M}}_{n} ‖^{- 1} log (r_{0} / r)]}^{2}} r d r \\ = 2 π r_{0}^{b_{0} + 2} ‖ {\tilde{M}}_{n} ‖ \sqrt[2]{log n} \int_{0}^{‖ {\tilde{M}}_{n} ‖^{- 1} {(log n)}^{1 / 2}} e^{(b_{0} + 2) [s^{2} - ‖ {\tilde{M}}_{n} ‖ {(log n)}^{1 / 2} s]} d s . \end{matrix}$ Since $e^{(b_{0} + 2) s^{2}} ⩾ 1$ , after a simple computation we find $\begin{matrix} \int_{\frac{r_{0}}{n} ⩽ | x | ⩽ r_{0}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x ⩾ \frac{2 π r_{0}^{b_{0} + 2}}{(b_{0} + 2)} (1 + \frac{1}{n^{b_{0} + 2}}) . \end{matrix}$ Therefore, $\begin{matrix} (3.23) & \underset{n \to \infty}{lim inf} \int_{B_{r_{0}}} | x |^{b_{0}} e^{α_{0} t_{n}^{2} {\tilde{M}}_{n}^{2}} d x ⩾ \frac{4 π}{(b_{0} + 2)} r_{0}^{b_{0} + 2} . \end{matrix}$ On the other hand, observing that $\begin{matrix} \int_{B_{n}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x = \int_{B_{r_{0}}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x - \int_{A_{n}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x, \end{matrix}$ and by applying the Lebesgue Dominated Convergence Theorem we obtain $\begin{matrix} lim_{n \to \infty} \int_{A_{n}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x = \frac{2 π}{(b_{0} + 2)} r_{0}^{b_{0} + 2} . \end{matrix}$ Then, estimate (3.23) yields $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B_{n}} | x |^{b_{0}} e^{α_{2} {\tilde{M}}_{n}^{2}} d x ⩾ \frac{2 π}{(b_{0} + 2)} r_{0}^{b_{0} + 2} . \end{matrix}$ Therefore, from (3.21) and (3.22) we get $\begin{matrix} \frac{α_{2}}{α_{0}} ⩾ \frac{(β_{0} - ε) D_{0} π}{(b_{0} + 2)} r_{0}^{b_{0} + 2} . \end{matrix}$ Using that $α_{2} = 2 π (b_{0} + 2)$ and letting $ε \to 0$ we contradicts $(f_{4})$ , and this completes the proof. □

We shall also need a basic regularity result, which will be used to prove (1.4).

Lemma 3.5.

Let $R > 0$ and $u \in H_{0}^{1} (B_{R})$ be a weak solution of the semilinear elliptic problem $\begin{matrix} (3.24) & \{\begin{matrix} - Δ u = h (x, u), & in B_{R}, \\ u = 0, & on \partial B_{R}, \end{matrix} \end{matrix}$ where $h : B_{R} \times R \to R$ is a Carathéodory function satisfying $\begin{matrix} | h (x, s) | ⩽ C_{0} | x |^{b_{0}} e^{α s^{2}}, for a.e. x \in B_{R}, and s \in R, \end{matrix}$ with $C_{0} > 0$ , $b_{0} > - 2$ , and $α > 0$ . Then, $u \in C^{σ} ({\overline{B}}_{R})$ for some $σ \in (0, 1)$ .

Proof.

Since $b_{0} > - 2$ , there exists $p > 1$ such that $p b_{0} > - 2$ . Similarly, we can choose $q_{1}, q_{2} > 1$ satisfying $1 / q_{1} + 1 / q_{2} = 1$ and $q_{1} p b_{0} > - 2$ . Then, by Hölder’s inequality one has $\begin{matrix} \int_{B_{R}} {| h (x, u) |}^{p} d x ⩽ C_{0}^{p} {(\int_{B_{R}} | x |^{q_{1} p b_{0}} d x)}^{1 / q_{1}} {(\int_{B_{R}} e^{q_{2} p α u^{2}} d x)}^{1 / q_{2}} . \end{matrix}$ Taking into account that $| x |^{q_{1} p b_{0}} \in L^{1} (B_{R})$ , and by the classical Trudinger–Moser inequality (see [19, 31]) it holds that $e^{q_{2} p α u^{2}} \in L^{1} (B_{R})$ , we conclude that $| h (x, u) |^{p} \in L^{1} (B_{R})$ . Therefore, by classical elliptic regularity theory $u \in W^{2, p} (B_{R}) ↪ C^{σ} ({\overline{B}}_{R})$ for some $σ \in (0, 1)$ and this finishes the proof. □

We now present the proof of Theorem 1.2 with the aid of the previous results.

Proof of Theorem 1.2.

By Proposition 2.8 it is sufficient to show that J has a critical point. By Proposition 3.3, Lemma 3.4 and the Mountain Pass Theorem (see [23]), J has a nonzero critical point. Moreover, we can assume that $f (s) = 0$ for $s ⩽ 0$ and the above results are valid also for this modified nonlinearity. Thus, there exists $u \in E_{r} ∖ {0}$ such that $J^{'} (u) = 0$ . Since $u^{-} (x) : = max {- u (x), 0}$ one has $0 = J^{'} (u) u^{-} = - ‖ u^{-} ‖^{2}$ , which implies that $u ⩾ 0$ a.e. in $R^{2}$ . To conclude the proof it remains to prove (1.4). For this purpose, by the assumption $(V)$ for all $R_{0} > 0$ there exists $C_{0} > 0$ such that $V (| x |) ⩾ C_{0} | x |^{a}$ for all $| x | ⩾ R_{0}$ . Defining $ϕ (x) = e^{- c_{1} | x |^{(a + 2) / 4}}$ with $c_{1} = 2 \sqrt{C_{0}} / (a + 2) > 0$ and using a straightforward computation we see that $\begin{matrix} (3.25) & - Δ ϕ + V (| x |) ϕ ⩾ \frac{C_{0}}{4} | x |^{a} ϕ in | x | ⩾ R_{0} . \end{matrix}$ On the other hand, from assumption $(Q)$ there exists $C_{1} > 0$ such that $Q (| x |) ⩽ C_{1} | x |^{b}$ for all $| x | ⩾ R_{0}$ . Then, by inequality (2.5) and $(f_{1})$ , for $R_{0}$ sufficiently large we have $\begin{matrix} Q (| x |) f (u) ⩽ \frac{C_{0}}{4} | x |^{b} | u |^{γ - 2} u ⩽ \frac{C_{0}}{4} | x |^{b - (γ - 2) (a + 2) / 4} u for all | x | ⩾ R_{0} . \end{matrix}$ Taking into account that $b - \frac{(γ - 2) (a + 2)}{4} ⩽ a$ we get $\begin{matrix} Q (| x |) f (u) ⩽ \frac{C_{0}}{4} | x |^{a} u, for all | x | ⩾ R_{0} . \end{matrix}$ This combined with (3.25) (where $c_{0}$ is a positive constant such that $u ⩽ c_{0} ϕ$ on $| x | = R_{0}$ ) implies $\begin{matrix} \{\begin{matrix} - Δ (u - c_{0} ϕ) + (V (| x |) - \frac{C_{0}}{4} | x |^{a}) (u - c_{0} ϕ) ⩽ 0 & in | x | > R_{0}, \\ u - c_{0} ϕ ⩽ 0 & on | x | = R_{0} . \end{matrix} \end{matrix}$ Then, by the maximum principle we have that $u (x) ⩽ c_{0} ϕ (x)$ if $| x | ⩾ R_{0}$ . To complete the proof it is enough to show that $u \in C^{σ} ({\overline{B}}_{R_{0}})$ for some $σ \in (0, 1)$ . Defining $v (x) : = u (x) - u (R_{0})$ and using Lemma 2.1, $v \in H_{0}^{1} (B_{R_{0}})$ . By the behavior of V and Q at the origin, we can assume that $V (| x |) = | x |^{a_{0}}$ and $Q (| x |) = | x |^{b_{0}}$ and so v is a weak solution of problem (3.24) with $R = R_{0}$ and $h (x, v) = | x |^{b_{0}} [f (v + u (R_{0})) - | x |^{a_{0} - b_{0}} (v + u (R_{0}))]$ . Now using that $b_{0} ⩽ a_{0}$ (similarly to $a_{0} < b_{0}$ ), by (1.1) and the continuity of f we find $C_{2} = C_{2} (R) > 0$ such that for all $x | ⩽ R_{0}$ $\begin{matrix} | h (x, v) | ⩽ | x |^{b_{0}} [| f (v + u (R_{0})) | + R_{0}^{a_{0} - b_{0}} | v + u (R_{0}) |] ⩽ C_{2} | x |^{b_{0}} e^{α v^{2}} . \end{matrix}$ By applying Lemma 3.5, we conclude that $v \in C^{σ} ({\overline{B}}_{R_{0}})$ . Therefore, $u = v + u (R_{0}) \in C^{σ} ({\overline{B}}_{R_{0}})$ and this completes the proof of Theorem 1.2. □

In order to prove our multiplicity result we shall use the following version of the Symmetric Mountain Pass Theorem (see, e.g., [6, 23]).

Theorem 3.6.

Let E be a real Banach space and $I \in C^{1} (E, R)$ be an even functional satisfying $I (0) = 0$ and

there are constants $ρ, τ > 0$ such that $I (u) ⩾ τ$ , for all $‖ u ‖_{E} = ρ$ ;

there exist $D > 0$ and a finite-dimensional subspace S of E such that ${max}_{u \in S} I (u) ⩽ D$ .

If the functional I satisfies the

{(PS)}_{c}

condition for

0 < c < D

, then it possesses at least

dim S

pairs of nonzero critical points.

Proof of Theorem 1.3.

Since f is odd and satisfies $(f_{1})$ , then $J_{λ}$ is even and $J_{λ} (0) = 0$ . Moreover, we observe that all the results proved in the previous sections holds for $J_{λ}$ , for all $λ > 0$ . Arguing as in the proof of the first theorem, we obtain that $J_{λ}$ satisfies $(I_{1})$ . From $(f_{3})$ and the local condition $(f_{5})$ , there exists $C_{0} > 0$ such that $F (s) ⩾ \frac{C_{0}}{ν} | s |^{ν}$ , for all $s \in R$ . Consequently, $\begin{matrix} J_{λ} (u) ⩽ \frac{1}{2} ‖ u ‖^{2} - \frac{C_{0} λ}{ν} \int_{R^{2}} Q (| x |) | u |^{ν} d x . \end{matrix}$ We now observe that, for any k-dimensional subspace S of $E_{r}$ , the norms are equivalent and hence $\begin{matrix} max_{u \in S} J_{λ} (u) ⩽ max_{u \in S} [\frac{1}{2} ‖ u ‖^{2} - c_{k} \frac{C_{0} λ}{ν} ‖ u ‖^{ν}] = (\frac{1}{2} - \frac{1}{ν}) {(\frac{1}{c_{k} C_{0}})}^{\frac{2}{ν - 2}} λ^{\frac{2}{2 - ν}} = : D_{k} (λ) . \end{matrix}$ Since $2 / (2 - ν) < 0$ , we have that ${lim}_{λ \to \infty} D_{k} (λ) = 0$ . Thus, there exists $λ_{k} > 0$ such that $D_{k} (λ) < α_{2} / (2 α_{0})$ for any $λ > λ_{k}$ . Therefore, we can apply Proposition 3.3 and Theorem 3.6 to obtain k pairs of nonzero critical points of $J_{λ}$ , which concludes the proof. □

Footnotes

Acknowledgements

We would like to thank the anonymous referees for his/her valuable comments on the paper and by pointing out an additional reference.

The first author was partially supported by FAPITEC/CAPES and CNPq – Universal.

The second author was supported by CAPES/BRAZIL.

The third author was partially supported by Grant 2019/2014 Paraíba State Research Foundation (FAPESQ) and CNPq.

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