Stationary nonlinear Schrödinger equations in $\mathbb {R}^2$ with potentials vanishing at infinity

Abstract

We deal with a class of 2-D stationary nonlinear Schrödinger equations (NLS) involving potentials V and weights Q decaying to zero at infinity as $(1+|x|^\alpha )^{-1}$, $\alpha \in (0,2)$, and $(1+|x|^\beta )^{-1}$, $\beta \in (2, + \infty )$, respectively, and nonlinearities with exponential growth of the form $\exp {\gamma _0 s^2}$ for some $\gamma _0>0$. Working in weighted Sobolev spaces, we prove the existence of a bound state solution, i.e. a solution belonging to $H^1({{\mathrm{\mathbb {R}}}}^2)$. Our approach is based on a weighted Trudinger–Moser-type inequality and the classical mountain pass theorem.

1 Introduction

This paper concerns the existence of solutions of stationary nonlinear Schrödinger equations of the form

$$\begin{aligned} - \Delta u + V(x)u = Q(x)f(u) \quad \text { in } {{\mathrm{\mathbb {R}}}}^2 \end{aligned}$$

(NLS)

in the case when the potential V and the weight Q decay to zero at infinity as $(1 +|x|^\alpha )^{-1}$ with $\alpha \in (0,2)$ and $(1 +|x|^\beta )^{-1}$ with $\beta \in (2, + \infty )$, respectively, and the nonlinear term $f=f(s)$ has exponential growth.

Equation (NLS) is a particular case of the following more general class of two-dimensional problems

$$\begin{aligned} - \Delta u + V(x)u =g(x,u) \quad \text { in } {{\mathrm{\mathbb {R}}}}^2 , \end{aligned}$$

(1.1)

where $V=V(x)$ is positive and $g=g(x,s)$ has exponential growth at infinity with respect to the variable s, i.e.

$$\begin{aligned} \lim _{|s| \rightarrow + \infty } \frac{|g(x,s)|}{e^{\gamma s^2}}= {\left\{ \begin{array}{ll} 0 &{} \quad \text {if } \gamma > \gamma _0 ,\\ + \infty &{} \quad \text {if } \gamma < \gamma _0 , \end{array}\right. } \end{aligned}$$

for some $\gamma _0 \ge 0$.

We mention that, for bounded domains $\Omega \subset {{\mathrm{\mathbb {R}}}}^2$ and nonlinear terms $g=g(x,s)$ with exponential growth at infinity, a lot of work has been devoted to the study of corresponding elliptic equations of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta u =g(x, u) &{} \text { in } \, \Omega ,\\ u=0 &{} \text { on } \, \partial \Omega . \end{array}\right. } \end{aligned}$$

We limit ourselves to refer the reader to the following papers [2, 3, 17, 19–22, 38].

1.1 Potentials bounded away from zero

In the last decades, considerable attention has been paid to the study of equations of the form (1.1), under various assumptions on the potential V. However, to our knowledge, it is everywhere assumed (with the only exception of [5, 27]) that V is bounded away from zero by a positive constant, that is

• $(V_0)$ there exists $V_0>0$ such that $V(x) \ge V_0$ for any $x \in {{\mathrm{\mathbb {R}}}}^2$

Assuming, in addition to $(V_0)$,

$$\begin{aligned} \frac{1}{V} \in L^1({{\mathrm{\mathbb {R}}}}^2) \end{aligned}$$

(1.2)

or

$$\begin{aligned} \lim _{|x| \rightarrow + \infty } V(x)= + \infty \end{aligned}$$

(1.3)

results concerning the existence of solutions for problem (1.1) can be found in [4, 6, 24, 25, 30, 40, 41]. While, in the case when the potential V is constant

$$\begin{aligned} V(x) = V_0 \quad x \in {{\mathrm{\mathbb {R}}}}^2 \end{aligned}$$

the results available in the literature are [7, 16, 26, 28, 29, 37].

It is important to point out that $(V_0)$ ensures that the natural space for a variational study of (1.1) is a complete subspace E of the classical Sobolev space $H^1({{\mathrm{\mathbb {R}}}}^2)$, more precisely

$$\begin{aligned} E:= \Bigl \{\, u \in H^1({{\mathrm{\mathbb {R}}}}^2) \, \Big | \, \int _{{{\mathrm{\mathbb {R}}}}^2} V(x) u^2 \, \hbox {d}x < + \infty \,\Bigr \} \end{aligned}$$

and $E=H^1({{\mathrm{\mathbb {R}}}}^2)$ if the potential V is constant. Besides this property of the function space setting, there is a main difference between the above-mentioned classes of problems distinguished by the behavior of the potential at infinity: When the potential V is large at infinity [i.e. (1.2) or (1.3) holds], one gains compact embeddings of the subspace E of $H^1({{\mathrm{\mathbb {R}}}}^2)$ in $L^p$-spaces, while when V is constant, one has to deal with the loss of compact embeddings in $L^p({{\mathrm{\mathbb {R}}}}^2)$ given by the unboundedness of the domain ${{\mathrm{\mathbb {R}}}}^2$.

1.2 Vanishing potentials

The new aspect of the present paper is that we will consider a class of positive potentials vanishing at infinity, i.e.,

$$\begin{aligned} \lim _{|x| \rightarrow + \infty } V(x)=0 . \end{aligned}$$

Starting from the work by Ambrosetti et al. [8], various types of stationary nonlinear Schrödinger equations involving decaying potentials at infinity have been studied in the higher-dimensional case $N\ge 3$, and we refer the reader to [9–11, 13–15, 31, 32, 39] and the references therein, even if these references are far to be exhaustive.

In particular, the analysis developed in [9, 13–15, 31, 39] covers also the two-dimensional case but for nonlinearities with polynomial growth at infinity (more precisely, $g(x,s)=Q(x)s^p$) or asymptotically linear growth. Moreover, with the only exception of [31, 39], these results for the 2-D case concern the study of semiclassical states of (3.1). If we replace the operator $-\Delta$ by $- {{\mathrm{\varepsilon }}}^2 \Delta$ in (1.1), then a semiclassical state $u_{{{\mathrm{\varepsilon }}}}$ is a solution with ${{\mathrm{\varepsilon }}}<<1$ and the authors of [9, 13–15] constructed semiclassical states concentrating on some set S (i.e. tending uniformly to zero as ${{\mathrm{\varepsilon }}}\downarrow 0$, outside of a neighborhood of S) by means of the Lyapunov–Schmidt reduction method or penalization schemes.

As already mentioned, the only results available in the literature for 2-D stationary nonlinear Schrödinger equations with vanishing potentials and exponential growth nonlinearities are [5, 27], see Remark 2.2.

2 Main result

Inspired by Ambrosetti et al. [8], we will study the existence of solutions of (NLS) under the following growth conditions on the potential V and the weight Q:

• (V) $V \in {\mathcal {C}} ({{\mathrm{\mathbb {R}}}}^2)$, there exist $\alpha$, a, $A>0$ such that

  $$\begin{aligned} \frac{a}{1 + |x|^{\alpha }} \le V(x) \le A \end{aligned}$$

  and $V(x) \sim |x|^{- \alpha }$ as $|x| \rightarrow + \infty$;

• (Q) $Q \in {\mathcal {C}} ({{\mathrm{\mathbb {R}}}}^2)$, there exist $\beta$, $b>0$ such that

  $$\begin{aligned} 0 < Q(x) \le \frac{b}{1 + |x|^{\beta }} \end{aligned}$$

  and $Q(x) \sim |x|^{- \beta }$ as $|x| \rightarrow + \infty$.

In particular, we restrict our attention to the case when $\alpha$ and $\beta$ satisfy

$$\begin{aligned} \alpha \in (0,2) \quad \text { and } \quad \beta \in (2, + \infty ) . \end{aligned}$$

(2.1)

This choice will be motivated in Sect. 3, more precisely see Theorem 3.1 and Remark 3.4, and it is strictly related with the variational structure of (NLS). In fact, we aim to develop a variational approach to study the existence of solutions of (NLS) via the classical mountain pass theorem. To this aim, we will frame the variational study of (NLS) in the weighted space

$$\begin{aligned} H^1_V({{\mathrm{\mathbb {R}}}}^2):= \Bigl \{\, u \in L^1_{\text {loc}}({{\mathrm{\mathbb {R}}}}^2) \, \Big | \, |\nabla u| \in L^2({{\mathrm{\mathbb {R}}}}^2) \text { and } \int _{{{\mathrm{\mathbb {R}}}}^2} V(x) u^2 \, \hbox {d}x < + \infty \,\Bigr \} \end{aligned}$$

that will be discussed in some details in Sect. 3. When (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in [2, + \infty )$, it turns out that functions belonging to $H^1_V({{\mathrm{\mathbb {R}}}}^2)$ satisfy the following weighted exponential integrability condition with weight Q

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) (e^{\gamma u^2} -1) \, \hbox {d}x < + \infty \quad \text {for any } \, u \in H^1_V({{\mathrm{\mathbb {R}}}}^2), \, \gamma >0, \end{aligned}$$

and, in Sect. 4, we will obtain the corresponding uniform inequality of Trudinger–Moser type. This motivates the choice of nonlinear terms $f=f(s)$ with exponential growth at infinity of the form $e^{\gamma _0 s^2}$ for some $\gamma _0>0$, i.e. there exists $\gamma _0>0$ such that

We point out that, due to the presence of a potential V and a weight Q satisfying (V) and (Q), this is the maximal growth which can be treated variationally in the space $H^1_V({{\mathrm{\mathbb {R}}}}^2)$ (see Theorem 4.1).

We also assume that $f: \, {{\mathrm{\mathbb {R}}}}\rightarrow {{\mathrm{\mathbb {R}}}}$ is a continuous function satisfying $f(0)=0$ and

• there exists $\mu >2$ such that

  

• there exist $s_0$, $M_0 >0$ such that

  

To guarantee that the mountain pass level is inside the region of compactness of Palais–Smale sequences, we assume an additional growth condition on the nonlinearity f. In particular, we will consider two different type of growth conditions. The first one prescribes an asymptotic behavior at infinity, more precisely

where

$$\begin{aligned}&{\mathcal {M}} = {\mathcal {M}} (V,Q):= \inf _{r >0} \frac{4 e^{\frac{1}{2} \, r^2 \, V_{\text {max},r}}}{\gamma _0 r^2 \, Q_{\text {min},r}}, \quad V_{\text {max},r}:= \max _{|x| \le r} V(x) >0 \quad \text { and}\\&\quad Q_{\text {min},r}:= \min _{|x| \le r} Q(x) >0 . \end{aligned}$$

We recall that this condition was introduced in [2] and then refined in [21].

Remark 2.1

It is easy to see that if

$$\begin{aligned} V(x) = \frac{a}{1 + |x|^\alpha } \quad \text { and } \quad Q(x)= \frac{b}{1 +|x|^\beta } \end{aligned}$$

with $\alpha \in (0,2)$ and $\beta \in (2, + \infty )$ then ${\mathcal {M}}>0$. A typical example of nonlinear term satisfying ($f_{0}$)–($f_{3}$) is

$$\begin{aligned} f_\lambda (s):= \lambda s(e^{\gamma _0 s^2} -1) \quad s \in {{\mathrm{\mathbb {R}}}}\end{aligned}$$

(2.2)

with $\lambda >0$ and $\gamma _0>0$.

The second growth condition that we will take into account was introduced in [16] and prescribes the growth of f near the origin:

where

$$\begin{aligned} \lambda > \Bigl (\, \frac{\gamma _0}{4 \pi } \, \frac{p-2}{p} \,\Bigr )^{\frac{p-2}{2}} \, S_{p,V,Q}^{p/2} \end{aligned}$$

and

$$\begin{aligned} S_{p,V,Q}:= \inf _{u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) {\setminus } \{0\}} \, \frac{\int _{{{\mathrm{\mathbb {R}}}}^2}[\, |\nabla u|^2 + V(x)u^2 \, ] \, \hbox {d}x}{\Bigl ( \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) |u|^p \, \hbox {d}x\Bigr )^{2/p}} . \end{aligned}$$

Note that $S_{p,V,Q} >0$ for any $p \ge 2$, see Theorem 3.1. An example of nonlinear term satisfying ($f_{0}$)–($f_{2}$) and ($f_{3}'$) is given by the function $f_\lambda$ defined by (2.2) provided $\lambda >0$ is sufficiently large. As pointed out in [41, Proposition 2.9], there exist continuous functions such that ($f_{0}$)–($f_{2}$) and ($f_{3}'$) are satisfied but ($f_{3}$) is not satisfied.

Our main result is the following

Theorem 2.1

Assume (V) and (Q) hold with $\alpha$ and $\beta$ in the range (2.1). Let $f: \, {{\mathrm{\mathbb {R}}}}\rightarrow {{\mathrm{\mathbb {R}}}}$ be a continuous function satisfying $f(0)=0$, ($f_{0}$), ($f_{1}$) and ($f_{2}$). If in addition either ($f_{3}$) or ($f_{3}'$) holds then equation (NLS) admits a nontrivial mountain pass solution $u_0 \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$.

Remark 2.2

We recall that Fei and Yin [27] studied concentration properties of semiclassical states of (NLS) in the case when $f(u):=|u|^{p-2}u e^{\gamma _0 u^2}$ with $p>2$ and $\gamma _0>0$, i.e.

$$\begin{aligned} - {{\mathrm{\varepsilon }}}^2 \Delta u + V(x)u = Q(x)|u|^{p-2}u e^{\gamma _0 u^2} \quad \text { in } {{\mathrm{\mathbb {R}}}}^2 , \end{aligned}$$

(2.3)

under more general assumptions on V and Q. More precisely, it is just required that

• $\displaystyle V(x) \ge \frac{a}{1 +|x|^2}$ and $\displaystyle Q(x) \le b(1+|x|^{\beta })$ with a, b and $\beta >0$

• or $\displaystyle V(x) \ge \frac{a}{1 +|x|^\alpha }$ and $\displaystyle Q(x) \le be^{\beta |x|^{(2- \alpha )/2}}$ with a, b, $\beta >0$ and $\alpha \in (0,2)$

provided there exists a smooth bounded domain $\Lambda \subset {{\mathrm{\mathbb {R}}}}^2$ such that

$$\begin{aligned} \max _{x \in \overline{\Lambda }} \frac{V(x)}{Q(x)} < \frac{4 \pi }{\gamma _0}^{\frac{p}{2} -1} S_p^{- \frac{p}{2}} , \quad \text { where } \quad S_p:= \inf _{u \in H^1({{\mathrm{\mathbb {R}}}}^2) {\setminus } \{0\}} \frac{\int _{{{\mathrm{\mathbb {R}}}}^2} [\, |\nabla u|^2 + u^2\,] \, \hbox {d}x}{\Bigl (\int _{{{\mathrm{\mathbb {R}}}}^2} |u|^p \, \hbox {d}x \Bigr )^{2/p}}, \end{aligned}$$

and the ground energy functional associated to the limit problem has local minimum points.

In this framework, the authors of [27] constructed semiclassical states $u_{{{\mathrm{\varepsilon }}}}$ of (2.3) belonging to $H^1({{\mathrm{\mathbb {R}}}}^2)$ and concentrating around some point $x_0 \in \Lambda$ by means of a penalization method. However, it should be pointed out that the existence result in [27] only works for ${{\mathrm{\varepsilon }}}<1$ sufficiently small. In the present work, we consider the case ${{\mathrm{\varepsilon }}}=1$ and, in fact, we can deal with any fixed ${{\mathrm{\varepsilon }}}>0$.

More recently, Albuquerque et al. [5] considered the existence of radial solutions of (NLS) when the nonlinear term f has exponential growth at infinity (i.e. f satifies ($f_{0}$)) and, V and Q are unbounded or decaying radial potentials. Besides the restriction to the radial case, the growth conditions on V and Q in [5] are less restrictive than (V) and (Q) with $\alpha$ and $\beta$ in the range (2.1), but a rigorous interpretation of the function space setting considered in [5] is needed (see for instance Remark 3.1). With the help of a weighted Trudinger–Moser inequality for radial functions, the authors in [5] obtained the existence of a positive radial solution in $H^1({{\mathrm{\mathbb {R}}}}^2)$ with exponential decay outside of a neighborhood of the origin.

Note that here, we do not require V and Q to be radial and, the vanishing behavior of V seems to prevent a reduction of the problem to the radial case.

Of particular interest are solutions of (NLS) which have finite $L^2$-norm, i.e. bound state solutions. The mountain pass solution $u_0 \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ obtained in Theorem 2.1 is a weak solution of (NLS) in the sense that

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} (\, \nabla u_0 \cdot \nabla v + V(x) u_0 v \,) \, \hbox {d}x - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_0) v \, \hbox {d}x =0 \quad \text { for any } \, v \in H^1_V({{\mathrm{\mathbb {R}}}}^2)\nonumber \\ \end{aligned}$$

(2.4)

and we will show that $u_0 \in L^2({{\mathrm{\mathbb {R}}}}^2)$, hence $u_0 \in H^1({{\mathrm{\mathbb {R}}}}^2)$. In fact, we will prove that any weak solution in the sense expressed by (2.4) is a bound state solution of (NLS).

Proposition 2.2

Assume (V) and (Q) hold with $\alpha$ and $\beta$ in the range (2.1). Let $f: \, {{\mathrm{\mathbb {R}}}}\rightarrow {{\mathrm{\mathbb {R}}}}$ be a continuous function satisfying $f(0)=0$, ($f_{0}$), ($f_{1}$) and ($f_{2}$). If (NLS) admits a weak solution $u_0 \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ (i.e. $u_0$ satisfies (2.4)) then $u_0 \in L^2({{\mathrm{\mathbb {R}}}}^2)$ and hence $u_0 \in H^1({{\mathrm{\mathbb {R}}}}^2)$.

2.1 Open question

Assume V, Q and f satisfy the assumptions of Theorem 2.1. The arguments of the proofs of Theorem 2.1 and Proposition 2.2 can be easily adapted to obtain, for any $\varepsilon >0$, the existence of a nontrivial mountain pass solution $u_\varepsilon \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ of the problem

$$\begin{aligned} - {{\mathrm{\varepsilon }}}^2 \Delta u_{{{\mathrm{\varepsilon }}}} + V(x)u_{{{\mathrm{\varepsilon }}}} = Q(x) f(u_{{{\mathrm{\varepsilon }}}}) \quad \text { in } {{\mathrm{\mathbb {R}}}}^2 , \end{aligned}$$

and $u_{{{\mathrm{\varepsilon }}}} \in H^1({{\mathrm{\mathbb {R}}}}^2)$. To study the concentration behavior of such solutions $\{u_{{{\mathrm{\varepsilon }}}}\}_{{{\mathrm{\varepsilon }}}>0}$ when ${{\mathrm{\varepsilon }}}\downarrow 0$, some sharp pointwise decay estimates and appropriate bounds of the energy are needed, uniformly with respect to ${{\mathrm{\varepsilon }}}>0$. This problem is still unsolved.

2.2 Notations

Let $w: \, {{\mathrm{\mathbb {R}}}}\rightarrow [0,+\infty )$ be a weight function, we denote by $L^p_w({{\mathrm{\mathbb {R}}}}^2)$ with $p \in [1, + \infty ]$ the corresponding weighted $L^p$-space, i.e. $L^p_w({{\mathrm{\mathbb {R}}}}^2)$ is the space consisting of all measurable functions $u: \, {{\mathrm{\mathbb {R}}}}^2 \rightarrow {{\mathrm{\mathbb {R}}}}$ with

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} w(x) |u|^p \, \hbox {d}x < + \infty \quad \text { when } p \in [1, + \infty ) \end{aligned}$$

and

$$\begin{aligned} \inf \{\, C \ge 0 \, | \, w(x)|u(x)| \le C \text { a.e. in } {{\mathrm{\mathbb {R}}}}^2 \,\} < + \infty \quad \text { when } p = + \infty . \end{aligned}$$

We also denote by $B(x,R) \subset {{\mathrm{\mathbb {R}}}}^2$ the closed ball of radius $R>0$ centered at $x \in {{\mathrm{\mathbb {R}}}}^2$ and, to simplify notations, we set

$$\begin{aligned} B_R:=B(0,R) \quad \text { and } \quad B^{\text {c}}_{R}:= {{\mathrm{\mathbb {R}}}}^2 {\setminus } B_R . \end{aligned}$$

3 The functional space setting

In order to develop a variational approach to study the existence of solutions of (NLS), a key step is to identify a suitable function space setting. Since we are interested in vanishing potentials at infinity, this basic step turns out to be a priori not obvious. The difficulty is due to the peculiar features of the two-dimensional case and can be seen comparing our situation with the higher-dimensional case $N \ge 3$. In fact, let us consider a nonlinear Schrödinger equation of the form

$$\begin{aligned} - \Delta u + V(x)u = g(x,u) \quad \text { in } {{\mathrm{\mathbb {R}}}}^N, \; N \ge 2, \end{aligned}$$

(3.1)

where $g: \, {{\mathrm{\mathbb {R}}}}^N \times {{\mathrm{\mathbb {R}}}}\rightarrow {{\mathrm{\mathbb {R}}}}$ is a suitable nonlinear term and V is continuous, positive and vanishing at infinity, i.e.

$$\begin{aligned} V \in {\mathcal {C}}({{\mathrm{\mathbb {R}}}}^N) , \quad V>0 \text { in } {{\mathrm{\mathbb {R}}}}^N \quad \text { and } \quad V(x) \rightarrow 0 \text { as } |x| \rightarrow + \infty . \end{aligned}$$

(3.2)

Since we deal with a potential V which decays to zero at infinity, the variational theory in $H^1({{\mathrm{\mathbb {R}}}}^N)$ cannot be used. Moreover, under the above conditions (3.2) on V, the space

$$\begin{aligned} \biggl \{\, u \in H^1({{\mathrm{\mathbb {R}}}}^N) \, \bigg | \, \int _{{{\mathrm{\mathbb {R}}}}^N} V(x) u^2 \, \hbox {d}x < + \infty \, \biggr \} \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert u\Vert ^2 : = \Vert \nabla u\Vert _2^2 + \int _{{{\mathrm{\mathbb {R}}}}^N} V(x) u^2 \, \hbox {d}x \end{aligned}$$

is not complete in general. In the higher-dimensional case $N \ge 3$, this leads to frame the variational study of problem (3.1) in the space

$$\begin{aligned} H^1_V({{\mathrm{\mathbb {R}}}}^N):= {\mathcal {D}}^{1,2}({{\mathrm{\mathbb {R}}}}^N) \cap L^2_V({{\mathrm{\mathbb {R}}}}^N) , \; N \ge 3, \end{aligned}$$

which is a Banach space with respect to the norm $\Vert \cdot \Vert$.

Remark 3.1

The situation in the two-dimensional case is more delicate, due to the fact that the completion ${\mathcal {D}}^{1,2}({{\mathrm{\mathbb {R}}}}^2)$ of the space of smooth compactly supported functions with respect to the Dirichlet norm $\Vert \nabla \cdot \Vert _2$ is not directly comparable with the space $L^2_V({{\mathrm{\mathbb {R}}}}^2)$ and it does not make sense to consider the intersection

$$\begin{aligned} {\mathcal {D}}^{1,2}({{\mathrm{\mathbb {R}}}}^2) \cap L^2_V({{\mathrm{\mathbb {R}}}}^2) , \end{aligned}$$

unless a rigorous interpretation is specified.

In analogy with the higher-dimensional case, when $N=2$, the natural framework for a variational approach of problem (3.1) is given by the space

$$\begin{aligned} H^1_V({{\mathrm{\mathbb {R}}}}^2):= \bigl \{\, u \in L^2_V ({{\mathrm{\mathbb {R}}}}^2) \, \big | \, |\nabla u| \in L^2({{\mathrm{\mathbb {R}}}}^2) \, \bigr \} . \end{aligned}$$

Actually, $H^1_V({{\mathrm{\mathbb {R}}}}^2)$ endowed with the norm

$$\begin{aligned} \Vert u\Vert ^2 : = \Vert \nabla u\Vert _2^2 + \int _{{{\mathrm{\mathbb {R}}}}^2} V(x) u^2 \, \hbox {d}x \end{aligned}$$

(3.3)

is a Banach space. In fact, as a consequence of (3.2), we have

$$\begin{aligned} H^1_V({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow H^1_{\text {loc}}({{\mathrm{\mathbb {R}}}}^2) \end{aligned}$$

and this continuous embedding, together with the definition of Cauchy sequences and Fatou Lemma, enables to show that $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \Vert \cdot \Vert )$ is complete. Note also that the norm $\Vert \cdot \Vert$ comes from the inner product

$$\begin{aligned} <u,v>:= \int _{{{\mathrm{\mathbb {R}}}}^2} \bigl [\nabla u \cdot \nabla v + V(x)u v \bigr ] \, \hbox {d}x . \end{aligned}$$

(3.4)

Remark 3.2

If $V \in L^1({{\mathrm{\mathbb {R}}}}^2)$ then any constant function $u\equiv c$ in ${{\mathrm{\mathbb {R}}}}^2$, with $c \in {{\mathrm{\mathbb {R}}}}$, belongs to $H^1_V({{\mathrm{\mathbb {R}}}}^2)$. However, under the assumption (V) and since $\alpha \in (0,2)$, our potential $V \notin L^1({{\mathrm{\mathbb {R}}}}^2)$ and in this case the only constant function that belongs to $H^1_V({{\mathrm{\mathbb {R}}}}^2)$ is the trivial one, i.e. $u \equiv 0$ in ${{\mathrm{\mathbb {R}}}}^2$.

In conclusion, we frame the variational study of (NLS) in the Hilbert space $H^1_V({{\mathrm{\mathbb {R}}}}^2)$ with inner product $<\cdot , \cdot >$ and norm $\Vert \cdot \Vert$ given respectively by (3.4) and (3.3).

Remark 3.3

In view of (V), the potential V is positive and uniformly bounded on ${{\mathrm{\mathbb {R}}}}^2$, therefore we have

$$\begin{aligned} H^1({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow H^1_V({{\mathrm{\mathbb {R}}}}^2) . \end{aligned}$$

Moreover, the space ${\mathcal {C}}^\infty _0({{\mathrm{\mathbb {R}}}}^2)$ of smooth compactly supported functions is dense in $(H^1_V({{\mathrm{\mathbb {R}}}}^2),$ $\Vert \cdot \Vert )$. This can be proved by standard arguments and using, for instance, the property

$$\begin{aligned} \lim _{|x| \rightarrow + \infty } |x|^2 V(x) >0 \end{aligned}$$

which follows directly from (V) and the range of $\alpha$ given by (2.1).

Similarly to the higher-dimensional case $N \ge 3$, the vanishing behavior of the potential V (i.e. $V(x) \rightarrow 0$ as $|x| \rightarrow + \infty$) implies that

(3.5)

As a consequence, this rules out exponential integrability and hence any kind of Trudinger–Moser-type inequality on $H^1_V({{\mathrm{\mathbb {R}}}}^2)$, unless one introduces some suitable weight in the target space. This remark justifies the choice a nonlinear term of the form

$$\begin{aligned} g(x,u):= Q(x) f(u) \end{aligned}$$

in equation (NLS). In fact, for a variational study of (NLS) in the function space $H^1_V({{\mathrm{\mathbb {R}}}}^2)$, some suitable integrability condition on the nonlinearity is needed: the validity of (3.5) leads to introduce a weight Q(x) and look for appropriate assumptions on Q(x) in such a way that

(3.6)

at least for some $p \ge 1$. In particular, the vanishing behavior of Q given by assumption (Q) guarantees that the embeddings

$$\begin{aligned} L^p_Q({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow L^p({{\mathrm{\mathbb {R}}}}^2) \quad \text { for any } \, p \in [1,+\infty ) \end{aligned}$$

(3.7)

do not hold. Note that, in view of (3.5), the validity of (3.7) would be against the embedding (3.6).

The embedding (3.6) is a particular case of embeddings of weighted spaces discussed in [34], where the following result is proved.

Theorem 3.1

([34], Example 20.6) Suppose that (V) and (Q) hold with $\alpha \in (0,2]$ and $\beta \in [2, + \infty )$. Then

$$\begin{aligned} H^1_V ({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow L^p_Q({{\mathrm{\mathbb {R}}}}^2) \quad \text { for any } \, p \in [2, + \infty ) \end{aligned}$$

(3.8)

and there exists $C_{p}>0$ such that

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) |u|^p \, \mathrm{d}x \le C_{p} \Vert u\Vert ^p \quad \text { for any } \, u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) . \end{aligned}$$

Moreover, if $\beta \ne 2$ then the above embeddings are compact.

Note that if $V(x) \sim (1+ |x|^{\alpha })^{-1}$ with $\alpha \in (0,2]$ and $Q(x) \sim (1 +|x|^{\beta })^{-1}$ then the growth restriction $\beta \in [2, + \infty )$ on the weight Q is a necessary condition for the embedding (3.8), as proved in [34].

Remark 3.4

If (V) and (Q) hold with $\alpha \in (0,2]$ and $\beta =2$ then the embeddings (3.8) are continuous but not compact. For this reason, we can say that the case $\beta =2$ should correspond to the critical case. Since we confine our attention to the study of problem (NLS) when (V) and (Q) hold with $\alpha$ and $\beta$ satisfying (2.1), in particular $\beta \ne 2$ and in this respect problem (NLS) can be seen as subcritical. Note also that assuming (2.1), we also require that $\alpha \ne 2$: this is just a technical restriction due to the method of proof that we use to obtain the corresponding weighted Trudinger–Moser inequality (see Sect. 4).

In view of Theorem 3.1 and Remark 3.4, in what follows, we will assume that (V) and (Q) hold with $\alpha$ and $\beta$ satisfying (2.1). In this framework, since

$$\begin{aligned} H^1_0(B_1) \hookrightarrow H^1({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow H^1_V({{\mathrm{\mathbb {R}}}}^2) , \end{aligned}$$

(3.9)

we infer that

In fact, it is well known that there exists $\overline{u} \in H_0^1(B_1)$ such that $\overline{u} \notin L^\infty (B_1)$. Therefore, $\overline{u} \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ but $\overline{u} \notin L^\infty _Q({{\mathrm{\mathbb {R}}}}^2)$ and, it is natural to look for a weighted Trudinger–Moser inequality on $H^1_V({{\mathrm{\mathbb {R}}}}^2)$. Due to the embedding (3.9) and the uniform boundedness of the weight Q, it turns out to be reasonable to consider an exponential growth function $\phi$ of the form

$$\begin{aligned} \phi (t):= e^{\gamma t^2} -1, \quad \gamma >0. \end{aligned}$$

4 A subcritical Trudinger–Moser-type inequality in weighted spaces

In this Section we will prove the following weighted Trudinger–Moser inequality on the space $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \, \Vert \cdot \Vert )$

Theorem 4.1

Suppose that (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in [2, + \infty )$. For any $\gamma >0$ and any $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$, we have

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) \bigl (e^{\gamma u^2} -1\bigr ) \, \mathrm{d}x < + \infty . \end{aligned}$$

(4.1)

Moreover, if we consider the supremum

$$\begin{aligned} S_{\gamma }= S_{\gamma }(V,Q):= \sup _{u \in H^1_V({{\mathrm{\mathbb {R}}}}^2), \, \Vert u\Vert \le 1} \, \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) \bigl (e^{\gamma u^2} -1\bigr ) \, \mathrm{d}x , \quad \gamma >0 , \end{aligned}$$

then, for any $\gamma \in (0, 4 \pi )$, there exists a constant $C=C(\gamma , V, Q)>0$ such that

$$\begin{aligned} S_{\gamma } \le C \end{aligned}$$

(4.2)

and

$$\begin{aligned} S_{\gamma }= + \infty \quad \text { for any } \, \gamma > 4 \pi . \end{aligned}$$

(4.3)

Note that the inequality that we obtain is subcritical, in the sense that the range of the exponent is the open interval $(0, 4 \pi )$. This is essentially due to the technical difficulties arising from the decay of the potential V at infinity. In fact, the vanishing behavior of V seems to prevent a reduction of the problem to radial case. For instance, it is not possible to apply classical symmetrization methods and this forces to look for a rearrangement-free argument.

Even if our proof does not cover the critical case $\gamma =4 \pi$, the subcritical inequality expressed by Theorem 4.1 will enable us to obtain the existence of a nontrivial solution for the nonlinear Schrödinger equation (NLS).

To prove (4.2), we will combine the ideas of Kufner and Opic [34] with the argument by Yang and Zhu [42]. More precisely, we will obtain the desired uniform estimate by means of a suitable covering lemma and the classical Trudinger–Moser inequality on balls, i.e.

Theorem 4.2

([33]) Let $\Omega \subset {{\mathrm{\mathbb {R}}}}^2$ be a bounded domain. There exists a constant $C>0$ such that

$$\begin{aligned} \sup _{u \in H_0^1(\Omega ), \, \Vert \nabla u \Vert _2 \le 1} \, \int _{\Omega } e^{\alpha u^2} \, \mathrm{d}x {\left\{ \begin{array}{ll} \le C |\Omega | &{} \text {if } 0<\alpha \le 4 \pi ,\\ = + \infty &{} \text {if } \alpha > 4 \pi . \end{array}\right. } \end{aligned}$$

(4.4)

In particular, inspired by Yang et al. [42], we will mainly make use of the following local estimate that can be derived directly from (4.4) with the aid of the scaling $\tilde{u}:= u/ \Vert \nabla u\Vert _2$

Lemma 4.3

([42], Lemma 2.1) There exists a constant $C>0$ such that for any $y \in {{\mathrm{\mathbb {R}}}}^2$, $R>0$ and any $u \in H_0^1(B(y,R))$ with $\Vert \nabla u \Vert _2 \le 1$, we have

$$\begin{aligned} \int _{B(y,R)} (e^{4 \pi u^2} -1) \, \mathrm{d}x \le C R^2 \int _{B(y, R)} |\nabla u|^2 \, \mathrm{d}x . \end{aligned}$$

(4.5)

In view of the fact that V and Q are bounded away from zero by positive constants on compact subsets of ${{\mathrm{\mathbb {R}}}}^2$, the sharpness (4.3) is a direct consequence of the sharpness of the following Trudinger–Moser inequality due to Ruf [36] (see also [4] and [18, Remark 6.1]; in addition, we refer to [1] for a scale invariant form of the result in [36]).

Theorem 4.4

([36]) Let $\Omega \subseteq {{\mathrm{\mathbb {R}}}}^2$ be a domain (possibly unbounded) and let $\tau >0$. For any $\gamma \in [0, 4 \pi ]$ there exists a constant $C_{\tau }>0$ such that

$$\begin{aligned} R_{\gamma }(\tau , \Omega ):= \sup _{u \in H^1_0(\Omega ), \, \Vert \nabla u\Vert _2^2 + \tau \Vert u\Vert _2^2 \le 1} \, \int _{\Omega } (e^{\gamma u^2} -1) \, \mathrm{d}x \le C_{\tau } \end{aligned}$$

and the above inequality is sharp, i.e.

$$\begin{aligned} R_{\gamma }(\tau , \Omega )=+ \infty \quad \text { for any } \, \gamma > 4 \pi . \end{aligned}$$

First, we set

$$\begin{aligned} \tilde{V}:= \max _{x \in B_1} \, V(x) \quad \text { and } \quad \tilde{Q}:= \min _{x \in B_1} \, Q(x) . \end{aligned}$$

Since V and Q are continuous and positive, we have that $\tilde{V}$, $\tilde{Q} >0$. Therefore recalling (3.9), we may estimate

$$\begin{aligned} S_{\gamma } \ge \sup _{u \in H_0^1(B_1), \, \Vert u\Vert \le 1} \, \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) \bigl (e^{\gamma u^2} -1\bigr ) \, \hbox {d}x \ge \tilde{Q} \, \sup _{u \in H_0^1(B_1), \, \Vert u\Vert \le 1} \, \int _{B_1} (e^{\gamma u^2} -1) \, \hbox {d}x . \end{aligned}$$

Inasmuch as

$$\begin{aligned} \Vert u\Vert ^2 = \Vert \nabla u\Vert _2^2 + \int _{B_1} V(x)u^2 \, \hbox {d}x \le \Vert \nabla u\Vert _2^2 + \tilde{V} \Vert u\Vert _2^2 \quad \text { for any } \, u \in H_0^1(B_1), \end{aligned}$$

we get

$$\begin{aligned} S_{\gamma } \ge \tilde{Q} \, \sup _{u \in H_0^1(B_1), \, \Vert \nabla u\Vert _2^2 + \tilde{V} \Vert u\Vert _2^2 \le 1} \, \int _{B_1} (e^{\gamma u^2} -1) \, \hbox {d}x =R_\gamma (\tilde{V}, B_1) . \end{aligned}$$

Hence, for $\gamma > 4 \pi$, we have

$$\begin{aligned} S_{\gamma } \ge R_{\gamma } (\tilde{V}, B_1) = + \infty . \end{aligned}$$

Next, we will derive (4.1) from (4.2) whose proof will be carried out in essentially two steps. In what follows, $\gamma \in (0, 4 \pi )$ is fixed and we set

$$\begin{aligned} \gamma = 4 \pi (1- \varepsilon ) \end{aligned}$$

for a suitable $\varepsilon \in (0,1)$.

4.1 Uniform estimate on a large ball

Let $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ be such that $\Vert u \Vert \le 1$ and let us estimate

$$\begin{aligned} \int _{B_R} Q(x) \bigl (e^{\gamma u^2} -1\bigr ) \, \hbox {d}x \end{aligned}$$

for some $R>0$ to be chosen during the proof independently of u. First, note that using (Q) we have

$$\begin{aligned} \int _{B_{R}} Q(x) \bigl (e^{\gamma u^2} -1\bigr ) \, \hbox {d}x \le b \, \int _{B_R} (e^{ \gamma u^2} -1) \, \hbox {d}x = b \int _{B_R} (e^{4 \pi (1- \varepsilon ) u^2} -1) \, \hbox {d}x . \end{aligned}$$

Next, we follow the argument in [42] and we introduce a cutoff function $\varphi \in {\mathcal {C}}_0^{\infty }(B_{2R})$ such that

$$\begin{aligned} 0 \le \varphi \le 1 \text { in } B_{2R}, \quad \varphi \equiv 1 \text { in } B_R \quad \text { and } \quad |\nabla \varphi | \le \frac{C}{R} \, \text { in } B_{2R} \end{aligned}$$

for some universal constant $C>0$. Then $\varphi u \in H^1_0(B_{2R})$ and by Young’s inequality

$$\begin{aligned} \int _{B_{2R}} |\nabla (\varphi u)|^2 \, \hbox {d}x&\le (1+ \varepsilon ) \int _{B_{2R}} \varphi ^2 |\nabla u|^2 \, \hbox {d}x + \Bigl (1 + \frac{1}{\varepsilon }\Bigr ) \int _{B_{2R}} |\nabla \varphi |^2 u^2 \, \hbox {d}x\\&\le (1+ \varepsilon ) \int _{B_{2R}} |\nabla u|^2 \, \hbox {d}x + \Bigl (1 + \frac{1}{\varepsilon }\Bigr ) \, \frac{C^2}{R^2}\int _{B_{2R}} u^2 \, \hbox {d}x . \end{aligned}$$

In view of (V)

$$\begin{aligned} V(x) \ge \frac{a}{1 +|x|^\alpha } \ge \frac{a}{1 + (2R)^\alpha } \,, \end{aligned}$$

and hence

$$\begin{aligned} \int _{B_{2R}} |\nabla (\varphi u)|^2 \, \hbox {d}x \le (1+ \varepsilon ) \int _{B_{2R}} |\nabla u|^2 \, \hbox {d}x + \Bigl (1 + \frac{1}{\varepsilon }\Bigr ) \, \frac{C^2}{a} \, \frac{1 +(2R)^\alpha }{R^2} \int _{B_{2R}} V(x) u^2 \, \hbox {d}x . \end{aligned}$$

Since by assumption $\alpha \in (0,2)$, we can choose $\overline{R}>0$ sufficiently large so that

$$\begin{aligned} \Bigl (1 + \frac{1}{\varepsilon }\Bigr ) \, \frac{C^2}{a} \, \frac{1 +(2R)^\alpha }{R^2} \le 1 + \varepsilon \quad \text { for any } \, R \ge \overline{R} . \end{aligned}$$

We remark that the choice of $\overline{R}$ is independent of u, $\overline{R} = \overline{R}(\varepsilon , a, \alpha )$, and by construction

$$\begin{aligned} \int _{B_{2R}} |\nabla (\varphi u)|^2 \, \hbox {d}x \le (1 + \varepsilon ) \Vert u\Vert ^2 \le 1 + \varepsilon . \end{aligned}$$

Therefore, if we define

$$\begin{aligned} v := \sqrt{1 - \varepsilon } \, \varphi u \in H_0^1(B_{2R}) \end{aligned}$$

we have that $\Vert \nabla v \Vert _2^2 \le 1 - \varepsilon ^2 \le 1$, and by applying the classical Trudinger–Moser inequality (4.4), we can conclude

$$\begin{aligned} \int _{B_R} \bigl (e^{4 \pi (1- \varepsilon ) u^2} -1\bigr ) \, \hbox {d}x = \int _{B_R} (e^{4 \pi (1- \varepsilon ) (\varphi u)^2} -1) \, \hbox {d}x \le \int _{B_{2R}} e^{4 \pi v^2} \, \hbox {d}x \le C R^2 . \end{aligned}$$

What we proved so far shows the existence of $\overline{R}= \overline{R} (\varepsilon , a, \alpha )>0$ such that for any $R \ge \overline{R}$ we have

$$\begin{aligned} \int _{B_R} Q(x) \bigl (e^{\gamma u^2} -1\bigr ) \, \hbox {d}x \le C R^2 \quad \text { for any } \, u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) \text { with } \Vert u\Vert \le 1 \,. \end{aligned}$$

(4.6)

4.2 Uniform estimate in the exterior of a large ball

Let $\tilde{n}>>1$ to be chosen later during the proof. For any fixed $n \ge \tilde{n}$, we consider the exterior $B^{\text {c}}_{n}$ of the ball $B_n$ and we introduce the covering of $B^{\text {c}}_{n}$ consisting of all annuli $A^\sigma _n$ with $\sigma >n$ defined by

$$\begin{aligned} A^\sigma _n:= \bigl \{x \in B^{\text {c}}_{n} \, \big | \, |x| < \sigma \bigr \} = \bigl \{ x \in {{\mathrm{\mathbb {R}}}}^2 \, \big | \, n < |x| < \sigma \bigr \} . \end{aligned}$$

For any $\sigma >\tilde{n}$, in view of the Besicovitch covering lemma (see for instance [23]), there exist a sequence of points $\{x_k\}_k \in A_{\tilde{n}}^\sigma$ and a universal constant $\theta >0$ such that

• $A_{\tilde{n}}^\sigma \subseteq \bigcup _k \, U_k^{1/2}$, where $U_k^{1/2}:= B\Bigl (x_k, \frac{1}{2} \, \frac{|x_k|}{3}\Bigr )$;

• $\sum _k \chi _{U_k} (x) \le \theta$ for any $x \in {{\mathrm{\mathbb {R}}}}^2$, where $\chi _{U_k}$ is the characteristic function of $U_k:= B\Bigl (x_k, \frac{|x_k|}{3}\Bigr )$.

Actually, the classical version of the Besicovitch covering lemma states that

$$\begin{aligned} \sum _k \chi _{U_k^{1/2}} (y) \le \eta \quad \text { for any } \, y \in {{\mathrm{\mathbb {R}}}}^2 \end{aligned}$$

(4.7)

for some universal constant $\eta >0$, and $U_k^{1/2} \subset U_k$. However, it is possible to show that (4.7) implies

$$\begin{aligned} \sum _k \chi _{U_k} (y) \le \theta \quad \text { for any } \, y \in {{\mathrm{\mathbb {R}}}}^2 \end{aligned}$$

(4.8)

where $\theta = \theta (\eta )>0$. To prove that (4.8) holds, we recall the statement of Besicovitch covering lemma.

Let E be a subset of ${{\mathrm{\mathbb {R}}}}^N$. A collection $\mathcal {F}$ of nontrivial closed balls in ${{\mathrm{\mathbb {R}}}}^N$ is a Besicovitch covering for E if each $x\in E$ is the center of a nontrivial ball belonging to $\mathcal {F}$.

Lemma 4.5

([12]) Let E be a bounded subset of ${{\mathrm{\mathbb {R}}}}^N$ and let $\mathcal {F}$ be a Besicovitch covering for E. There exist a countable collection $\{x_k\}_k$ of points in E and a corresponding collection of balls $\{B_k\}_k$ in $\mathcal {F}$, where $B_k:=B(x_k, \rho _k)$, with $E\subset \bigcup _k B_k$. Moreover, there exists a positive integer $c_N$ (depending only on the dimension N and independent of E and the covering $\mathcal {F}$) such that the balls $\{B_k\}_k$ can be organized into at most $c_N$ subcollections $\mathcal {B}_j:=\{B_{j_k}\}_k$, $j=1,2,\dots ,c_N$ in such a way that the balls $\{B_{j_k}\}_k$ of each subcollection $\mathcal {B}_j$ are disjoint.

Proof of (4.8)

We recall that, by Lemma 4.5, $A_{\tilde{n}}^\sigma \subseteq \bigcup _k \, U_k^{1/2}$ and there exists a positive integer $\eta$ such that the balls $\{U_k^{1/2}\}_k$ can be organized into at most $\eta$ subcollections $\mathcal {B}_j:=\{U_{j_k}^{1/2}\}_k$, $j=1,2,\dots ,\eta$ where the balls $\{U_{j_k}^{1/2}\}_k$ of each subcollection $\mathcal {B}_j$ are disjoint. Then

$$\begin{aligned} \sum _k \chi _{U_k^{1/2}} (y) \le \eta \quad \text { for any } \, y \in {{\mathrm{\mathbb {R}}}}^2. \end{aligned}$$

Next, we show that

$$\begin{aligned} \sum _k \chi _{U_k} (y) \le 196 \eta \quad \text { for any } \, y \in {{\mathrm{\mathbb {R}}}}^2. \end{aligned}$$

Assume that $y\in U_{j_k}$ for some $j\in \{ \, 1,2,\dots ,\eta \, \}$ and $k \ge 1$. Then $\frac{2}{3}|x_{j_k}|<|y|<\frac{4}{3}|x_{j_k}|$ and it follows that

$$\begin{aligned} U_{j_k}^{1/2}\subset B(0,\frac{7}{4}|y|) . \end{aligned}$$

Note that the ball $B(0,\frac{7}{4}|y|)$ contains at most 196 disjoint balls $B(x,\frac{1}{2}\frac{|x|}{3})$ with $\frac{3}{4}|y|<|x|<\frac{3}{2}|y|$. Thus, for any $j=1,2,\dots ,\eta$,

$$\begin{aligned} \sum _k \chi _{U_{j_k}} (y) \le 196 \end{aligned}$$

and

$$\begin{aligned} \sum _k \chi _{U_k} (y)=\sum _{j=1}^{\eta }\sum _k \chi _{U_{j_k}} (y) \le 196 \eta . \end{aligned}$$

The proof is completed. $\square$

Let $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ be such that $\Vert u\Vert \le 1$ and let us estimate the weighted exponential integral of u on $A_{3n}^\sigma$ with $n \ge \tilde{n}$ and $\sigma >n$. To do this, following [34], we introduce the set of indices

$$\begin{aligned} K_{n, \sigma }:= \{k \in \mathbb N \, \big | \, U_k^{1/2} \cap B^{\text {c}}_{3n} \ne \varnothing \} . \end{aligned}$$

From the definition of $K_{n, \sigma }$ and recalling that

$$\begin{aligned} A_{3n}^\sigma \subset A_{\tilde{n}}^\sigma \subseteq \bigcup _k \, U_k^{1/2} , \end{aligned}$$

we deduce that

$$\begin{aligned} A_{3n}^\sigma \subseteq \bigcup _{k \in K_{n, \sigma }} \, U_k^{1/2} \end{aligned}$$

and hence

$$\begin{aligned} \int _{A_{3n}^\sigma } Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x \le \sum _{k \in K_{n, \sigma }} \, \int _{U_k^{1/2}} Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x . \end{aligned}$$

Next, we estimate the single terms of the series on the right hand side. In this respect, the choice of the balls $U_k^{1/2}$ and $U_k$ will play a crucial role to overcome the difficulties arising from the vanishing behavior of the potential V and the weight Q.

Remark 4.1

We have

$$\begin{aligned} \frac{2}{3} |x_k| \le |y| \le \frac{4}{3} |x_k| \quad \text { for any } \, y \in U_k . \end{aligned}$$

Consequently, in view of the assumptions (V) and (Q), we get

$$\begin{aligned} V(y) \ge \frac{a}{1 + |y|^\alpha } \ge \frac{a}{1+ C_\alpha |x_k|^\alpha } \quad \text { for any } \, y \in U_k \end{aligned}$$

(4.9)

where $C_\alpha :=(4/3)^\alpha$, and

$$\begin{aligned} Q(y) \le \frac{b}{1+ |y|^\beta } \le \frac{b}{1 + C_\beta |x_k|^\beta } \quad \text { for any } \, y \in U_k \end{aligned}$$

(4.10)

where $C_\beta := (2/3)^\beta$.

Moreover, it is easy to prove that if $U_k \cap B^{\text {c}}_{3n} \ne \varnothing$ then $U_k \subset B^{\text {c}}_{n}$ and this entails

$$\begin{aligned} \bigcup _{k \in K_{n, \sigma }} U_k^{1/2} \subseteq \bigcup _{k \in K_{n,\sigma }} U_k \subseteq B^{\text {c}}_{n} \subseteq B^{\text {c}}_{\tilde{n}} . \end{aligned}$$

(4.11)

Properties (4.9) and (4.10) together with (4.11) will be useful in the proof to obtain some suitable uniform estimates.

Let us fix $k \in K_{n, \sigma }$. In view of (4.10),

$$\begin{aligned} \int _{U_k^{1/2}} Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x \le \frac{b}{1 + C_\beta |x_k|^\beta } \int _{U_k^{1/2}} (e^{ \gamma u^2} -1) \, \hbox {d}x \end{aligned}$$

and

$$\begin{aligned} \int _{U_k^{1/2}} (e^{ \gamma u^2} -1) \, \hbox {d}x = \int _{U_k^{1/2}} (e^{4 \pi (1- \varepsilon ) u^2} -1) \, \hbox {d}x . \end{aligned}$$

Following [42], the idea is to estimate the integral on the right hand side by means of the local Trudinger–Moser inequality (4.5) on $U_k$. To this aim, we consider the cutoff function $\varphi _k \in {\mathcal {C}}^\infty _0(U_k)$ satisfying

$$\begin{aligned} 0 \le \varphi _k \le 1 \text { in } U_k, \quad \varphi _k \equiv 1 \text { in } U_k^{1/2} \quad \text { and } \quad |\nabla \varphi | \le \frac{C}{|x_k|} \, \text { in } U_k \end{aligned}$$

for some universal constant $C>0$. Then $\varphi _k u \in H_0^1(U_k)$ and we may estimate

$$\begin{aligned} \int _{U_k} |\nabla (\varphi _k u)|^2 \, \hbox {d}x&\le (1+ \varepsilon ) \int _{U_k} |\nabla u|^2 \, \hbox {d}x + \Bigl (1+ \frac{1}{\varepsilon }\Bigr ) \, \frac{C^2}{|x_k|^2} \, \int _{U_k} u^2 \, \hbox {d}x\\&\le (1+ \varepsilon ) \int _{U_k} |\nabla u|^2 \, \hbox {d}x + \Bigl (1+ \frac{1}{\varepsilon }\Bigr ) \, \frac{C^2}{a} \, \frac{1 + C_\alpha |x_k|^\alpha }{|x_k|^2} \, \int _{U_k} V(x) u^2 \, \hbox {d}x \end{aligned}$$

where we also used (4.9). Recalling that $k \in K_{n, \sigma }$, in view of (4.11), we have that $x_k \in B^{\text {c}}_{\tilde{n}}$. Since $\alpha \in (0,2)$, we can choose $\tilde{n} = \tilde{n} (\varepsilon , a, \alpha )$ sufficiently large so that

$$\begin{aligned} \Bigl (1+ \frac{1}{\varepsilon }\Bigr ) \, \frac{C^2}{a} \, \frac{1 + C_\alpha |x_k|^\alpha }{|x_k|^2} \le 1 + \varepsilon \quad \text { for any } \, k \in K_{n, \sigma }, \, n \ge \tilde{n} . \end{aligned}$$

In this way, we get

$$\begin{aligned} \int _{U_k} |\nabla (\varphi _k u)|^2 \, \hbox {d}x \le (1+ \varepsilon ) \int _{U_k}(|\nabla u|^2 + V(x) u^2) \, \hbox {d}x \le (1+ \varepsilon ) . \end{aligned}$$

(4.12)

If we let

$$\begin{aligned} v_k:= \sqrt{1- \varepsilon } \, \varphi _k \overline{u} \in H_0^1(U_k) \end{aligned}$$

then $\Vert \nabla v_k \Vert _2^2 \le 1- \varepsilon ^2 \le 1$ and we can apply Lemma 4.3 to $v_k$ obtaining

$$\begin{aligned} \int _{U_k^{1/2}} (e^{4 \pi (1- \varepsilon ) u^2} -1) \, \hbox {d}x= & {} \int _{U_k^{1/2}} (e^{4 \pi (1- \varepsilon ) (\varphi _k u)^2} -1) \, dx \le \int _{U_k} (e^{4 \pi v_k^2} -1) \, \hbox {d}x\\\le & {} C |x_k|^2 \int _{U_k} |\nabla v_k|^2 \, \hbox {d}x . \end{aligned}$$

Finally, from (4.12), we deduce

$$\begin{aligned} \int _{U_k^{1/2}} (e^{4 \pi (1- \varepsilon ) u^2} -1) \, \hbox {d}x\le & {} C|x_k|^2 (1- \varepsilon ) \int _{U_k} |\nabla (\varphi _k u)|^2 \, \hbox {d}x\\\le & {} C |x_k|^2 (1-\varepsilon ^2) \int _{U_k}( |\nabla u|^2 + V(x) u^2) \, \hbox {d}x . \end{aligned}$$

Combining the above estimates, we get

$$\begin{aligned} \int _{A_{3n}^\sigma } Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x&\le b C (1-\varepsilon ^2) \sum _{k \in K_{n, \sigma }} \frac{|x_k|^2}{1 + C_\beta |x_k|^\beta } \, \int _{U_k} ( |\nabla u|^2 + V(x) u^2) \, \hbox {d}x\\&\le b C (1-\varepsilon ^2) \sum _{k \in K_{n, \sigma }} \frac{|x_k|^2}{1 + C_\beta |x_k|^\beta } \, \int _{B^{\text {c}}_{n}} ( |\nabla u|^2 + V(x) u^2) \chi _{U_k} (x) \, \hbox {d}x \end{aligned}$$

where the last inequality follows from (4.11). Using again (4.11), we have

$$\begin{aligned} \frac{|x_k|^2}{1 + C_\beta |x_k|^\beta } \le {\mathcal {B}}_n:= \sup _{x \in B^{\text {c}}_{n}} \frac{|x|^2}{1 + C_\beta |x|^\beta } \quad \text { for any } \, k \in K_{n, \sigma } . \end{aligned}$$

Hence

$$\begin{aligned} \int _{A_{3n}^\sigma } Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x \le b C (1-\varepsilon ^2) {\mathcal {B}}_n \sum _{k \in K_{n, \sigma }} \int _{B^{\text {c}}_{n}} ( |\nabla u|^2 + V(x) u^2) \chi _{U_k}(x) \, \hbox {d}x \end{aligned}$$

and, in view of the Besicovitch covering lemma,

$$\begin{aligned} \int _{A_{3n}^\sigma } Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x \le b C (1-\varepsilon ^2) \theta {\mathcal {B}}_n \int _{B^{\text {c}}_{n}} ( |\nabla u|^2 + V(x) u^2) \, \hbox {d}x . \end{aligned}$$

Letting $\sigma \rightarrow + \infty$, we can conclude the existence of $\tilde{n}=\tilde{n}(\varepsilon , a, \alpha ) >>1$ such that for any $n \ge \tilde{n}$ we have

$$\begin{aligned} \int _{B^{\text {c}}_{3n}} Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x \le b C \theta {\mathcal {B}}_n \int _{B^{\text {c}}_{n}} ( |\nabla u|^2 + V(x) u^2) \, \hbox {d}x . \end{aligned}$$

(4.13)

Note that

$$\begin{aligned} \lim _{n \rightarrow + \infty } {\mathcal {B}}_n = \lim _{n \rightarrow + \infty } \frac{n^2}{1 + C_\beta n^\beta } = {\left\{ \begin{array}{ll} 0 &{} \text { if } \beta >2\\ 1 / (1 + C_2) &{} \text { if } \beta =2 \end{array}\right. } \end{aligned}$$

therefore, in particular, we have also the following estimate that can be seen as an analogue of [8, Proposition 11] for the two-dimensional case

Proposition 4.6

Suppose that (V) and (Q) hold with $\alpha$ and $\beta$ satisfying (2.1), i.e. $\alpha \in (0,2)$ and $\beta \in (2,+ \infty )$, and let $0< \gamma < 4 \pi$. Then for any $\eta >0$ there exists $\tilde{n}= \tilde{n} (\gamma , a, \alpha ) >1$ such that for any $n \ge \tilde{n}$

$$\begin{aligned} \int _{B^{\text {c}}_{3n}} Q(x) \bigl ( e^{\gamma u^2} -1 \bigr ) \, \hbox {d}x \le \eta \int _{B^{\text {c}}_{n}} ( |\nabla u|^2 + V(x) u^2) \, \hbox {d}x \quad \text { for any } \, u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) \text { with } \Vert u\Vert \le 1 . \end{aligned}$$

The above Proposition will be useful to prove the existence of a bound state solution of (NLS), see Sect. 6.

4.3 Proof of Theorem 4.1 completed

To conclude the proof of (4.2), it is sufficient to combine (4.6) with (4.13).

Now, we show that (4.1) holds. This follows from (4.2) and the density of $C^{\infty }_0({{\mathrm{\mathbb {R}}}}^2)$ in $H^1_V({{\mathrm{\mathbb {R}}}}^2)$ (see Remark 3.3). In fact, let $\gamma >0$ and $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$. Then by density, there exists $u_0 \in {\mathcal {C}}^\infty _0({{\mathrm{\mathbb {R}}}}^2)$ such that

$$\begin{aligned} \Vert u - u_0\Vert \le \sqrt{\frac{1}{\gamma }} \end{aligned}$$

and, we may estimate

$$\begin{aligned} u^2=(u-u_0 + u_0)^2 \le 2(u-u_0)^2 + 2 u_0^2 . \end{aligned}$$

Let $R>0$ be such that ${{\mathrm{\text {supp }}}}u_0 \subseteq B_R$. Recalling the elementary inequality

$$\begin{aligned} ab-1 \le \frac{1}{2} (a^2 -1) + \frac{1}{2} (b^2 -1) \quad \text { for any } \, a, b \ge 0 \end{aligned}$$

we get

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) (e^{\gamma u^2} -1) \, \hbox {d}x&\le \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x)(e^{2 \gamma (u-u_0)^2} e^{2 \gamma u_0^2} -1) \, \hbox {d}x\\&\le \frac{1}{2} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) (e^{4 \gamma (u-u_0)^2} -1) \, \hbox {d}x + \frac{1}{2} \int _{B_R} Q(x) (e^{4 \gamma u_0^2} -1) \, \hbox {d}x\\&\le \frac{1}{2} S_4 + \frac{b}{2} \, |B_R| \, e^{4 \gamma \Vert u_0\Vert _\infty ^2} < + \infty , \end{aligned}$$

which completes the proof of Theorem 4.1.

5 Existence result

This section is devoted to the proof of Theorem 2.1 which is based on the classical mountain pass theorem.

First, we introduce the functional setting for a variational approach to problem (NLS). Since the nonlinear term f satisfies $f(0)=0$, ($f_{0}$) and ($f_{1}$), for fixed $\gamma > \gamma _0$, $q \ge 1$ and for any $\sigma >0$ we have

$$\begin{aligned} |f(s)| \le \sigma |s| + C(\gamma , q, \sigma ) |s|^{q-1} (e^{\gamma s^2} -1) \quad \text { for any } \, s \in {{\mathrm{\mathbb {R}}}}. \end{aligned}$$

(5.1)

Hence, the Ambrosetti–Rabinowitz condition ($f_{1}$) yields

$$\begin{aligned} |F(s)| \le \sigma |s|^2 + C(\gamma , q, \sigma ) |s|^{q} (e^{\gamma s^2} -1) \quad \text { for any } \, s \in {{\mathrm{\mathbb {R}}}}. \end{aligned}$$

(5.2)

Given $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$, we can use (5.2) with $\gamma > \gamma _0$, $q\ge 2$ and $\sigma >0$ to obtain the following estimate

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u) \, \hbox {d}x&\le \sigma \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) u^2 \, \hbox {d}x + C(\gamma , q, \sigma ) \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) u^q (e^{\gamma u^2} -1) \, \hbox {d}x\nonumber \\&\le \sigma \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) u^2 \, \hbox {d}x\nonumber \\&\quad + C(\gamma , q, \sigma ) \, \Bigl ( \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) u^{qp} \, \hbox {d}x \Bigr )^{\frac{1}{p}} \, \Bigl ( \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x)(e^{\gamma p'u^2} -1) \, \hbox {d}x \Bigr )^{\frac{1}{p'}} \end{aligned}$$

(5.3)

where we also applied Hölder’s inequality with $p>1$ and $\frac{1}{p} + \frac{1}{p'}=1$. Since $\alpha$ and $\beta$ satisfy (2.1), we have the continuous embeddings (3.8) and also the Trudinger–Moser estimate (4.1) and this enables us to conclude that

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u) \, \hbox {d}x < + \infty \quad \text { for any } \, u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) . \end{aligned}$$

(5.4)

Therefore, if we introduce the functional

$$\begin{aligned} I(u):= \frac{1}{2} \, \Vert u\Vert ^2 - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u) \, \hbox {d}x \end{aligned}$$

from (5.4) it follows that I is well defined on $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \Vert \cdot \Vert )$. Moreover, I is of class ${\mathcal {C}}^1$ with

$$\begin{aligned} I'[u](v):= <u,v> - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u) v \, \hbox {d}x \quad \text { for any } \, u, v \in H^1_V({{\mathrm{\mathbb {R}}}}^2) . \end{aligned}$$

In particular, any critical point $u_{0}$ of I is a weak solution of (NLS).

Lemma 5.1

The functional I has a mountain pass geometry on $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \Vert \cdot \Vert )$. More precisely

• (i) there exist $\tau >0$ and $\varrho >0$ such that $I(u) \ge \tau$ provided $\Vert u\Vert = \varrho$;

• (ii) there exists $e_*\in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ with $\Vert e_*\Vert > \varrho$ such that $I(e_*)<0$.

Proof

Let $\gamma >\gamma _0$ $q>2$ and $p>1$ with $\frac{1}{p} + \frac{1}{p'} =1$. It is easy to see that (5.3) implies that for any $\sigma >0$

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x)F(u) \, \hbox {d}x \le C_1 \sigma \Vert u\Vert ^2 + C_2(\gamma , q, \sigma ) \Vert u\Vert ^q \quad \text { for any } \, u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) \text { with } \Vert u\Vert =\varrho \end{aligned}$$

where $\varrho >0$ satisfies

$$\begin{aligned} \gamma p' \varrho ^2 < 4 \pi . \end{aligned}$$

In fact, due to the choice of $\alpha$ and $\beta$ in the range (2.1), it suffices to use the continuous embeddings given by Theorem 3.1 and the Trudinger–Moser inequality (4.2).

Therefore, if $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ and $\Vert u\Vert =\varrho$ then

$$\begin{aligned} I(u) \ge \Bigl (\, \frac{1}{2} - C_1 \sigma \,\Bigr ) \Vert u\Vert ^2 - C_2(\gamma , q, \sigma ) \Vert u\Vert ^q = \Bigl (\, \frac{1}{2} - C_1 \sigma \,\Bigr ) \varrho ^2 - C_2(\gamma , q, \sigma )\varrho ^q \end{aligned}$$

and, choosing $\sigma >0$ sufficiently small,

$$\begin{aligned} I(u) \ge \tilde{C}_1 \varrho ^2 - C_2(\gamma , q, \sigma ) \varrho ^q . \end{aligned}$$

Since $q >2$, for $\varrho >0$ small enough, there exists $\tau >0$ such that

$$\begin{aligned} I(u) \ge \tau \quad \text { for any } \, u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) \text { with } \Vert u\Vert =\varrho . \end{aligned}$$

To prove (ii), first note that, from ($f_{1}$),

$$\begin{aligned} F(s) \ge A |s|^\mu - B \quad \text { for any } \, s \in {{\mathrm{\mathbb {R}}}}\end{aligned}$$

for some $A,B >0$. If $u \in {\mathcal {C}}^\infty _0 ({{\mathrm{\mathbb {R}}}}^2)$ with ${{\mathrm{\text {supp }}}}u \subseteq B_R$, for some $R>0$, then for any $t >0$

$$\begin{aligned} I(tu)= & {} \frac{1}{2} t^2 \Vert u\Vert ^2 - \int _{B_R} Q(x) F(tu) \, \hbox {d}x \\\le & {} \frac{1}{2} t^2 \Vert u\Vert ^2 - A t^\mu \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) |u|^\mu \, \hbox {d}x + B \int _{B_R} Q(x) \, \hbox {d}x \end{aligned}$$

and, since $\mu >2$, $I(tu) \rightarrow - \infty$ as $t \rightarrow + \infty$. $\square$

In view of the mountain pass geometry of I on $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \Vert \cdot \Vert )$, we can consider the mountain pass level

$$\begin{aligned} c:= \inf _{ \gamma \in \Gamma } \, \sup _{t \in [0,1]} \, I(\gamma (t)) \ge \tau >0 \end{aligned}$$

where

$$\begin{aligned} \Gamma := \{\, \gamma \in {\mathcal {C}}([0,1], H^1_V({{\mathrm{\mathbb {R}}}}^2)) \, | \, \gamma (0)=0 \text { and } I(\, \gamma (1)\,) <0 \,\} . \end{aligned}$$

5.1 Estimate of the mountain pass level

As a consequence of ($f_{3}$) or ($f_{3}'$), using standard arguments, we will obtain the following upper bound for the mountain pass level

$$\begin{aligned} c < \frac{2 \pi }{\gamma _0} . \end{aligned}$$

(5.5)

We start assuming that the nonlinear term f satisfies the growth condition at infinity ($f_{3}$), i.e.

where

$$\begin{aligned}&{\mathcal {M}} = {\mathcal {M}} (V,Q):= \inf _{r >0} \frac{4 e^{\frac{1}{2} \, r^2 \, V_{\text {max},r}}}{\gamma _0 r^2 \, Q_{\text {min},r}},\\&V_{\text {max},r}:= \max _{|x| \le r} V(x) >0 \quad \text { and } \quad Q_{\text {min},r}:= \min _{|x| \le r} Q(x) >0. \end{aligned}$$

In this case, for fixed $r>0$, we consider Moser’s sequence of functions (see [33])

$$\begin{aligned} \tilde{w}_n(x):= \frac{1}{\,\sqrt{2 \pi } \,} {\left\{ \begin{array}{ll} \sqrt{\log n} &{} \quad \text {if } |x| \le \frac{r}{n} ,\\ \displaystyle {\frac{\log \frac{r}{|x|}}{\,\sqrt{\log n}\,}} &{} \quad \text {if } \frac{r}{n} \le |x| \le r ,\\ 0 &{} \quad \text {if } |x| \ge r . \end{array}\right. } \end{aligned}$$

It is well known that $\tilde{w}_n \in H_0^1(B_r) \subset H^1_V({{\mathrm{\mathbb {R}}}}^2)$ and one can easily prove (see for instance [30, Equation (3.5)] or [40, Lemma 3.2]) that

$$\begin{aligned} 1 \le \Vert \tilde{w}_n\Vert ^2 \le 1 + \frac{d_n(r)}{\log n} \, V_{\text {max},r} \end{aligned}$$

(5.6)

where

$$\begin{aligned} d_n(r):= \frac{r^2}{4} + o_n(1) \quad \text { and } \quad o_n(1) \rightarrow 0 \text { as } n \rightarrow + \infty . \end{aligned}$$

Let

$$\begin{aligned} w_n:= \frac{\tilde{w}_n}{\Vert \tilde{w}_n\Vert } \in H_0^1(B_r) \subset H^1_V({{\mathrm{\mathbb {R}}}}^2) \end{aligned}$$

so that $\Vert w_n\Vert =1$ and, in view of (5.6), when $|x| \le \frac{r}{n}$ we have

$$\begin{aligned} w_n^2(x) = \frac{1}{2 \pi } \, \log n \, \Bigl ( \frac{1}{\Vert \tilde{w}_n\Vert ^2} \pm 1 \Bigr )\ge & {} \frac{1}{2 \pi } \, \biggl ( \log n - \, \frac{\, d_n(r) \, V_{\text {max},r} \,}{\Vert \tilde{w}_n\Vert ^2} \biggr )\nonumber \\\ge & {} \frac{1}{2 \pi } \, \Bigl ( \log n - \, d_n(r) \, V_{\text {max},r} \Bigr ) . \end{aligned}$$

(5.7)

Note that, from ($f_{3}$), we deduce the existence of $r>0$ such that

$$\begin{aligned} \beta _0 > \frac{4 e^{\frac{1}{2} \, r^2 \, V_{\text {max},r}}}{\gamma _0 r^2 \, Q_{\text {min},r}} \end{aligned}$$

(5.8)

and, with this choice of $r>0$, we will prove the following

Lemma 5.2

There exists $n \in \mathbb N$ such that

$$\begin{aligned} \max _{t \ge 0} I(t w_n) < \frac{2 \pi }{\gamma _0} . \end{aligned}$$

Proof

The arguments of the proof are standard (see for instance [30, Lemma 3.6] or [40, Lemma 3.3]) but for the convenience of the reader we will sketch the main steps.

We argue by contradiction assuming that for any $n \in \mathbb N$

$$\begin{aligned} \max _{t \ge 0} I(t w_n) \ge \frac{2 \pi }{\gamma _0} . \end{aligned}$$

Let $t_n >0$ be such that

$$\begin{aligned} I(t_n w_n)= \max _{t \ge 0} I(t w_n) \end{aligned}$$

then

$$\begin{aligned} t_n^2 \ge 2 I(t_n w_n) \ge \frac{4 \pi }{\gamma _0} \end{aligned}$$

(5.9)

and, since

$$\begin{aligned} \frac{\hbox {d}}{{\hbox {d}}t} I(t w_n) \bigg |_{t = t_n}=0, \end{aligned}$$

we have also

$$\begin{aligned} t_n^2= \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(t_n w_n) t_n w_n \, \hbox {d}x . \end{aligned}$$

(5.10)

Note that, as a consequence of ($f_{3}$), for any $\varepsilon >0$ there exists $R_\varepsilon >0$ such that

$$\begin{aligned} sf(s) \ge (\beta _0 - \varepsilon ) e^{\gamma _0 s^2} \quad \text { for any } \, |s| \ge R_\varepsilon . \end{aligned}$$

(5.11)

Let $x_n \in B_{r/n}$ be the minimum point of the weight Q on $B_{r/n}$, i.e.

$$\begin{aligned} Q(x_n)=\min _{|x| \le r/n} Q(x), \end{aligned}$$

then

$$\begin{aligned} \lim _{n \rightarrow + \infty } Q(x_n)=Q(0) >0 . \end{aligned}$$

Therefore, using (5.10) and recalling (5.7), we get

$$\begin{aligned} t_n^2 \ge (\beta _0- \varepsilon ) \int _{B_{r/n}} Q(x)e^{\gamma _0 (t_n w_n)^2} \, \hbox {d}x \ge (\beta _0 - \varepsilon ) Q(x_n) \, \Bigl (\frac{r}{n}\Bigr )^2 \, e^{\frac{\gamma _0}{2 \pi } t_n^2 [\, \log n - d_n(r)V_{\text {max}, r} \,]} \end{aligned}$$

and, from this inequality, we deduce not only that the sequence $\{t_n\}_n$ is bounded but, in view of (5.9),

$$\begin{aligned} \lim _{n \rightarrow + \infty } t_n^2= \frac{4 \pi }{\gamma _0} . \end{aligned}$$

To reach a contradiction, we try to obtain an estimate of $\beta _0$ from above. From (5.10) and (5.11), it follows that

Since $w_n \rightarrow 0$ a.e. in ${{\mathrm{\mathbb {R}}}}^2$, we can apply the Lebesgue dominated convergence theorem obtaining

$$\begin{aligned} \lim _{n \rightarrow + \infty } \int _{\{t_nw_n < R_\varepsilon \}} f(t_n w_n) t_n w_n \, \hbox {d}x = 0 \quad \text { and } \quad \lim _{n \rightarrow + \infty } \int _{\{t_nw_n < R_\varepsilon \}} e^{\gamma _0 (t_n w_n)^2} \, \hbox {d}x = \pi r^2 . \end{aligned}$$

Moreover, (5.9) yields

$$\begin{aligned} \int _{B_r} e^{\gamma _0 (t_n w_n)^2} \, \hbox {d}x \ge \int _{B_{r/n}} + \int _{B_r {\setminus } B_{r_n}} e^{4 \pi w_n^2} \, \hbox {d}x . \end{aligned}$$

On one hand, using (5.7), we get

$$\begin{aligned} \int _{B_{r/n}} e^{4 \pi w_n^2} \, \hbox {d}x \ge \pi r^2 \, e^{- 2 d_n(r) \, V_{\text {max},r}} . \end{aligned}$$

On the other hand, using the definition of $w_n$ and the change of variable $s=r e^{-\Vert \tilde{w}_n\Vert \sqrt{\log n} \, t}$,

$$\begin{aligned}&\int _{B_r {\setminus } B_{r_n}} e^{4 \pi w_n^2} \, \hbox {d}x\\&\quad = 2 \pi \int _{r/n}^r e^{2 \frac{\log ^2 r/s}{\Vert \tilde{w}_n \Vert ^2 \log n}} \, s \, \hbox {d}s = 2 \pi r^2 \Vert \tilde{w}_n\Vert \sqrt{\log n} \, \int _{0}^{\frac{\sqrt{\log n}}{\Vert \tilde{w}_n\Vert }} \, e^{2(\,t^2 - \Vert \tilde{w}_n\Vert \sqrt{\log n} \, t \,)} \, \hbox {d}t\\&\quad \ge 2 \pi r^2 \Vert \tilde{w}_n\Vert \sqrt{\log n} \, \int _{0}^{\frac{\sqrt{\log n}}{\Vert \tilde{w}_n\Vert }} \, e^{-2\Vert \tilde{w}_n\Vert \sqrt{\log n} \, t} \, \hbox {d}t = \pi r^2 \bigl ( 1 - e^{-2 \log n}\bigr ) . \end{aligned}$$

In conclusion,

$$\begin{aligned} \frac{4 \pi }{\gamma _0} = \lim _{n \rightarrow + \infty } t_n^2 \ge (\beta _0 - \varepsilon ) \, Q_{\text {min},r} \, \pi r^2 e^{- \frac{1}{2} \, r^2 \, V_{\text {max},r} } \end{aligned}$$

and, from the arbitrary choice of $\varepsilon >0$, we deduce that

$$\begin{aligned} \beta _0 \le \frac{4 e^{\frac{1}{2} \, r^2 \, V_{\text {max},r}}}{\gamma _0 r^2 \, Q_{\text {min},r}} \end{aligned}$$

which contradicts (5.8). $\square$

Next we consider the case when the nonlinear term f satisfies the growth condition ($f_{3}'$), i.e.

where

$$\begin{aligned} \lambda > \Bigl (\, \frac{\gamma _0}{4 \pi } \, \frac{p-2}{p} \,\Bigr )^{\frac{p-2}{2}} \, S_{p,V,Q}^{p/2} \end{aligned}$$

and

$$\begin{aligned} S_{p,V,Q}:= \inf _{u \in H^1_V({{\mathrm{\mathbb {R}}}}^2) {\setminus } \{0\}} \, \frac{\Vert u\Vert ^2}{\Bigl ( \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) |u|^p \, \hbox {d}x\Bigr )^{2/p}} . \end{aligned}$$

In view of Theorem 3.1, the embedding $H^1_V({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow L^p_Q({{\mathrm{\mathbb {R}}}}^2)$ is compact and hence, there exists $\overline{u} \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ such that

$$\begin{aligned} \Vert \overline{u}\Vert ^2 =S_{p,V,Q} \quad \text { and } \quad \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) |\overline{u}|^p \, \hbox {d}x =1 . \end{aligned}$$

Therefore, we may estimate

$$\begin{aligned} c \le \max _{t \ge 0} I(t \overline{u}) = \max _{t \ge 0} \, \biggl \{\, \frac{1}{2} t^2 S_{p,V,Q} - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(t \overline{u}) \, \hbox {d}x \,\biggr \} \end{aligned}$$

and, using ($f_{3}'$), we get

$$\begin{aligned} c \le \max _{t \ge 0} \, \Bigl \{\, \frac{1}{2} t^2 S_{p,V,Q} - \frac{1}{p} \lambda t^p \,\Bigr \} = \frac{p-2}{2p} \, \frac{S_{p,V,Q}^{p/(p-2)}}{\lambda ^{2/(p-2)}} < \frac{2 \pi }{\gamma _0} . \end{aligned}$$

5.2 Palais–Smale sequences

Applying the mountain pass theorem without the Palais–Smale compactness condition, we get the existence of a Palais–Smale sequence $\{u_n\}_n \subset H^1_V({{\mathrm{\mathbb {R}}}}^2)$ at the level c (for short (PS)$_{c}$-sequence), i.e.

$$\begin{aligned} I (u_n) \rightarrow c \quad \text { and } \quad I'[u_n] \rightarrow 0 \,, \quad \text { as }\quad n \rightarrow + \infty . \end{aligned}$$

(5.12)

Lemma 5.3

Any (PS)$_{c}$-sequence $\{u_n\}_n$ for I is bounded in $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \Vert \cdot \Vert )$ and satisfies

$$\begin{aligned} \sup _n \, \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n)u_n \, \hbox {d}x < + \infty . \end{aligned}$$

(5.13)

Proof

Since $\{u_n\}_n$ is a (PS)$_{c}$-sequence for I, we have

$$\begin{aligned} I (u_n) \rightarrow c \quad \text { as } \quad n \rightarrow + \infty \end{aligned}$$

(5.14)

and

$$\begin{aligned} \Bigl | \, <u_n,v> - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n) v \, \hbox {d}x \, \Bigr | \le \varepsilon _n \Vert v\Vert \quad \text { for any } \, v \in H^1_V({{\mathrm{\mathbb {R}}}}^2) \end{aligned}$$

(5.15)

where $\varepsilon _n \downarrow 0$ as $n \rightarrow + \infty$.

From (5.14), we deduce that $\{I(u_n)\}_n \subset {{\mathrm{\mathbb {R}}}}$ is bounded and hence, there exists a constant $C>0$ such that

$$\begin{aligned} \frac{1}{2} \, \Vert u_n\Vert ^2 \le C + \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_n) \, \hbox {d}x \quad \text { for any } \, n \ge 1 . \end{aligned}$$

In view of the Ambrosetti–Rabinowitz condition ($f_{1}$),

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_n) \, \hbox {d}x \le \frac{1}{\mu }\, \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n) u_n \, \hbox {d}x \end{aligned}$$

and, using (5.15) with $v=u_n$,

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n)u_n \, \hbox {d}x \le \Vert u_n\Vert ^2 + \varepsilon _n \Vert u_n\Vert . \end{aligned}$$

(5.16)

Therefore

$$\begin{aligned} \frac{1}{2} \, \Vert u_n\Vert ^2 \le C + \frac{1}{\mu }\, \Vert u_n\Vert ^2 + \frac{\varepsilon _n}{\mu }\, \Vert u_n\Vert \end{aligned}$$

and, since $\mu >2$,

$$\begin{aligned} 0 \le \Bigl (\frac{1}{2} - \frac{1}{\mu }\Bigr ) \, \Vert u_n\Vert ^2 \le C + \frac{\varepsilon _n}{\mu }\, \Vert u_n\Vert \end{aligned}$$

from which we deduce that $\{u_n\}_n$ must be bounded in $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \Vert \cdot \Vert )$.

Finally, the boundedness of $\{u_n\}_n$ in $(H^1_V({{\mathrm{\mathbb {R}}}}^2), \Vert \cdot \Vert )$ together with (5.16) gives (5.13). $\quad \square$

Without loss of generality, we may assume that

$$\begin{aligned} u_n \rightharpoonup u_{0} \quad \text { in } H^1_V({{\mathrm{\mathbb {R}}}}^2) . \end{aligned}$$

(5.17)

Moreover, in view of (5.13), we may apply [21, Lemma 2.1] obtaining

$$\begin{aligned} Q(x)f(u_n) \rightarrow Q(x) f(u_0) \quad \text { in } L^1_{\text {loc}}({{\mathrm{\mathbb {R}}}}^2) . \end{aligned}$$

Hence,

$$\begin{aligned} <u_0, \varphi > - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_0) \varphi \, \hbox {d}x =0 \quad \text { for any } \, \varphi \in {\mathcal {C}}_0^\infty ({{\mathrm{\mathbb {R}}}}^2) \end{aligned}$$

and $u_0$ is a weak solution of (NLS). To prove that $u_0 \ne 0$ and complete the proof of Theorem 2.1, we will use the following convergence result

Lemma 5.4

If $\{u_n\}_n$ is a (PS)$_{c}$-sequence for I, with $u_n \rightharpoonup u_{0}$ in $H^1_V({{\mathrm{\mathbb {R}}}}^2),$ then

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_n) \, \mathrm{d}x \rightarrow \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_{0}) \, \mathrm{d}x \quad \text { as } n \rightarrow + \infty . \end{aligned}$$

Proof

This result is essentially a consequence of the compact embedding

$$\begin{aligned} H^1_V({{\mathrm{\mathbb {R}}}}^2)\hookrightarrow \hookrightarrow L^2_Q({{\mathrm{\mathbb {R}}}}^2) \end{aligned}$$

(5.18)

and the generalized Lebesgue dominated convergence theorem (see for instance [35, Chapter 4, Theorem 17]). Recall that (5.18) holds in view of Theorem 3.1 and the assumptions (V) and (Q) with $\alpha$ and $\beta$ satisfying (2.1).

First note that from ($f_{1}$) and ($f_{2}$), it follows that

$$\begin{aligned} 0 \le \lim _{|s| \rightarrow + \infty } \frac{F(s)}{s f(s)} \le \lim _{|s| \rightarrow + \infty } \frac{M_0}{|s|}=0 \end{aligned}$$

and for any $\varepsilon >0$ there exists $\overline{s}= \overline{s}(\varepsilon ) >0$ such that

$$\begin{aligned} F(s) \le \varepsilon sf(s) \quad \text { for any } \, |s| \ge \overline{s} . \end{aligned}$$

Since $u_{0} \in H^1_{V}({{\mathrm{\mathbb {R}}}}^2)$ and recalling the uniform bound (5.13), we have also

$$\begin{aligned} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_{0}) u_{0} \, \hbox {d}x \le C \quad \text { and } \quad \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n)u_n \, \hbox {d}x \le C \quad \text { for any } \, n \ge 1 \end{aligned}$$

for some constant $C>0$.

Consequently, for fixed $\varepsilon >0$, we get

$$\begin{aligned} \int _{\{ \,|u_{0}| \ge \overline{s} \,\}} Q(x) F(u_{0}) \, \hbox {d}x \le \varepsilon \int _{\{ \,|u_{0}| \ge \overline{s} \,\}} Q(x) f(u_{0}) u_{0} \, \hbox {d}x \le C \varepsilon \end{aligned}$$

and

$$\begin{aligned} \int _{\{ \,|u_n| \ge \overline{s} \,\}} Q(x) F(u_n) \, \hbox {d}x \le \varepsilon \int _{\{ \,|u_n| \ge \overline{s} \,\}} Q(x) f(u_{n}) u_{n} \, \hbox {d}x \le C \varepsilon . \end{aligned}$$

Now, we let

$$\begin{aligned} h_n(x):= Q(x) \chi _{\{\, |u_n| < \overline{s} \,\}} F(u_n) \quad \text { and } \quad h(x):= Q(x) \chi _{\{\, |u_{0}| < \overline{s} \,\}} F(u_{0}) . \end{aligned}$$

Then $\{h_n\}_n$ is a sequence of measurable functions and

$$\begin{aligned} h_n(x) \rightarrow h(x) \quad \text { for a.e. } x \in {{\mathrm{\mathbb {R}}}}^2 \,, \end{aligned}$$

as a consequence of the fact that $u_n \rightarrow u_{0}$ a.e. in ${{\mathrm{\mathbb {R}}}}^2$. Using (5.2) with $\gamma > \gamma _0$, $q=2$ and $\sigma >0$, we may estimate for any $|s| \le \overline{s}$

$$\begin{aligned} F(s) \le \sigma s^2 + C(\gamma , \sigma ) s^2(e^{\gamma s^2} -1) \le C(\gamma , \sigma , \overline{s}) s^2 . \end{aligned}$$

Then, letting

$$\begin{aligned} g_n(x):= C(\gamma , \sigma , \overline{s}) Q(x) u_n^2 \quad \text { and } \quad g(x):= C(\gamma , \sigma , \overline{s}) Q(x) u_{0}^2 \,,\end{aligned}$$

we get

$$\begin{aligned} 0 \le h_n(x) \le g_n(x) \quad x \in {{\mathrm{\mathbb {R}}}}^2 . \end{aligned}$$

Note that $\{g_n\}_n$ is a sequence of measurable functions, $g_n(x) \rightarrow g(x)$ a.e. in ${{\mathrm{\mathbb {R}}}}^2$ and, in view of the compact embedding (5.18),

$$\begin{aligned} \lim _{n \rightarrow + \infty } \int _{{{\mathrm{\mathbb {R}}}}^2} g_n(x) \, \hbox {d}x = \int _{{{\mathrm{\mathbb {R}}}}^2} g(x) \, \hbox {d}x . \end{aligned}$$

Therefore, applying the generalized Lebesgue dominated convergence theorem, we get

$$\begin{aligned} \lim _{n \rightarrow + \infty } \int _{{{\mathrm{\mathbb {R}}}}^2} h_n(x) \, \hbox {d}x = \int _{{{\mathrm{\mathbb {R}}}}^2} h(x) \, \hbox {d}x . \end{aligned}$$

In conclusion, for any fixed $\varepsilon >0$, we have

$$\begin{aligned} L_n&:= \Bigl |\, \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_n) \, \hbox {d}x - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_{0}) \, \hbox {d}x \,\Bigr | \le \int _{\{\, |u_n| \ge \overline{s} \,\}} Q(x)F(u_n) \, \hbox {d}x\\&\quad + \int _{\{\, |u_{0}| \ge \overline{s} \,\}} Q(x)F(u_{0}) \, \hbox {d}x + \Bigl |\, \int _{\{\, |u_n| < \overline{s} \,\}} Q(x)F(u_n) \, \hbox {d}x - \int _{\{\, |u_{0}| < \overline{s} \,\}} Q(x)F(u_{0}) \, \hbox {d}x \,\Bigr |\\&\le 2 C \varepsilon + \Bigl |\, \int _{{{\mathrm{\mathbb {R}}}}^2} h_n(x) \, \hbox {d}x - \int _{{{\mathrm{\mathbb {R}}}}^2} h(x) \, \hbox {d}x \,\Bigr | \end{aligned}$$

and, passing to the limit as $n \rightarrow + \infty$,

$$\begin{aligned} 0 \le \lim _{n \rightarrow + \infty } L_n \le 2 C \varepsilon . \end{aligned}$$

Since $\varepsilon >0$ is arbitrarily fixed, letting $\varepsilon \downarrow 0$, we obtain the desired convergence result. $\square$

5.3 Nontrivial mountain pass solution

In order to complete the proof of Theorem 2.1, we have simply to show that the weak limit $u_0$ given by (5.17) is nontrivial, i.e. $u_0 \ne 0$. To this aim, we argue by contradiction assuming that $u_0 = 0$.

Since $\{u_n\}_n$ is a (PS)$_{c}$-sequence, (5.12) holds. In particular

$$\begin{aligned} \lim _{n \rightarrow + \infty } \Vert u_n\Vert ^2 = \lim _{n \rightarrow + \infty } 2 \Bigl (\, \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_n) \, \hbox {d}x + c \,\Bigr ) \end{aligned}$$

(5.19)

and

$$\begin{aligned} \lim _{n \rightarrow + \infty } \Vert u_n\Vert ^2 = \lim _{n \rightarrow + \infty } \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n) u_n \, \hbox {d}x . \end{aligned}$$

(5.20)

From the convergence result expressed by Lemma 5.4, we deduce that

$$\begin{aligned} \lim _{n \rightarrow + \infty } \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) F(u_n) \, \hbox {d}x =0 . \end{aligned}$$

This together with (5.19) yields

$$\begin{aligned} \lim _{n \rightarrow + \infty } \Vert u_n\Vert ^2 =2c >0 . \end{aligned}$$

(5.21)

In view of (5.5),

$$\begin{aligned} c < \frac{2 \pi }{\gamma _0} \end{aligned}$$

and we deduce the existence of $\varepsilon >0$ and $\overline{n} \ge 1$ such that

$$\begin{aligned} \Vert u_n\Vert ^2 \le \frac{4 \pi }{\gamma _0}(1 - \varepsilon ) \quad \text { for any } \, n \ge \overline{n} . \end{aligned}$$

Therefore, we can choose $\gamma > \gamma _0$ sufficiently close to $\gamma _0$ and $p>1$ sufficiently close to 1 in such a way that

$$\begin{aligned} \gamma p \Vert u_n\Vert ^2 < 4 \pi (1- \varepsilon ^4) \quad \text { for any } \, n \ge \overline{n} . \end{aligned}$$

(5.22)

With this choice of $\gamma > \gamma _0$ and $p>1$, we apply (5.1) with $q=2$ and Hölder’s inequality with $\frac{1}{p} + \frac{1}{p'}=1$ obtaining

$$\begin{aligned}&\int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n) u_n \, \hbox {d}x \\&\quad \le C_1 \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) u_n^2 \, \hbox {d}x + C_2 \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x)u_n^2(e^{\gamma u_n^2} -1) \, \hbox {d}x\\&\quad \le C_1 \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) u_n^2 \, \hbox {d}x + C_2 \Bigl (\int _{{{\mathrm{\mathbb {R}}}}^2} Q(x)u_n^{2p'} \, \hbox {d}x \Bigr )^{\frac{1}{p'}} \Bigl (\int _{{{\mathrm{\mathbb {R}}}}^2} Q(x)(e^{\gamma p u_n^2} -1) \, \hbox {d}x \Bigr )^{\frac{1}{p}} . \end{aligned}$$

Note that $2p' >2$ and, in view of Theorem 3.1 and the assumptions (V) and (Q) with $\alpha$ and $\beta$ in the range (2.1), we have the compact embeddings

$$\begin{aligned} H^1_V({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow \hookrightarrow L^2_Q({{\mathrm{\mathbb {R}}}}^2) \quad \text { and } \quad H^1_V({{\mathrm{\mathbb {R}}}}^2) \hookrightarrow \hookrightarrow L^{2p'}_Q({{\mathrm{\mathbb {R}}}}^2) . \end{aligned}$$

Moreover, from (5.22),

$$\begin{aligned} \sup _{n \ge \overline{n}} \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x)(e^{\gamma p u_n^2} -1) \, \hbox {d}x \le S_{4 \pi (1- \varepsilon ^4)} (V,Q) \end{aligned}$$

where $S_{4 \pi (1- \varepsilon ^4)} (V,Q) < + \infty$ is the supremum of the Trudinger–Moser inequality given by Theorem 4.1.

Therefore

$$\begin{aligned} \lim _{n \rightarrow + \infty } \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_n) u_n \, \hbox {d}x =0 \end{aligned}$$

and, from (5.20), we get

$$\begin{aligned} \lim _{n \rightarrow + \infty } \Vert u_n\Vert ^2 =0 \end{aligned}$$

which contradicts (5.21).

6 Bound state solutions

This Section is devoted to the proof of Proposition 2.2. In particular, we will prove that if $u_0 \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ is a weak solution of (NLS), i.e.

$$\begin{aligned} <u_0,v> - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_0) v \, \hbox {d}x =0 \quad \text { for any } \, v \in H^1_V({{\mathrm{\mathbb {R}}}}^2) \end{aligned}$$

then $u_0 \in L^2({{\mathrm{\mathbb {R}}}}^2)$ and hence $u_0 \in H^1({{\mathrm{\mathbb {R}}}}^2)$.

We will follow almost the same arguments introduced in [8, Lemma 17 and Lemma 18], see also [31, Section 3].

Lemma 6.1

Suppose that (V) and (Q) hold with $\alpha$ and $\beta$ satisfying (2.1), i.e. $\alpha \in (0,2)$ and $\beta \in (2,+ \infty )$. Let $\gamma >0$ and $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$. Then for any $\varepsilon >0$ there exists $\overline{R}= \overline{R} (u, \gamma , a, \alpha ) >1$ such that for any $R \ge \overline{R}$

$$\begin{aligned} \int _{B^{\text {c}}_{R}} Q(x) (e^{\gamma u^2} -1) \, \mathrm{d}x \le \varepsilon \, \gamma \, \int _{B^{\text {c}}_{R}} (|\nabla u|^2 + V(x) u^2) \, \mathrm{d}x . \end{aligned}$$

Proof

Let $R>1$ and let $\tilde{\psi }_R \, : \; {{\mathrm{\mathbb {R}}}}^+ \rightarrow [0,1]$ be a smooth nondecreasing function such that

$$\begin{aligned} \tilde{\psi }_R(r):= {\left\{ \begin{array}{ll} 0 &{} \quad \text {if } 0 \le r \le R-R^{\alpha /2}\\ 1 &{} \quad \text {if } r \ge R \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} |\tilde{\psi }_R' (r)| \le \frac{2}{R^{\alpha /2}} . \end{aligned}$$

In polar coordinates $(r, \theta ) \in [0,+ \infty ) \times \mathbb S^1$, we define

$$\begin{aligned} \tilde{u}_R(r, \theta ):= {\left\{ \begin{array}{ll} 0 &{} \quad \text {if } 0 \le r \le R - R^{\alpha /2} ,\\ \tilde{\psi }_R(r) u(2R-r, \theta ) &{} \quad \text {if } R - R^{\alpha /2} \le r \le R ,\\ u(r, \theta ) &{} \quad \text {if } r \ge R . \end{array}\right. } \end{aligned}$$

Arguing as in [8, Proposition 11], we can prove the following estimate

$$\begin{aligned} \int _{A_R} (\, |\nabla \tilde{u}_R|^2 + V(x) \tilde{u}_R^2 \,) \, \hbox {d}x \le C \int _{B^{\text {c}}_{R}} (\, |\nabla u|^2 + V(x) u^2 \,) \, \hbox {d}x \end{aligned}$$

where $A_R$ is the annulus

$$\begin{aligned} A_R:= \{\, x \in {{\mathrm{\mathbb {R}}}}^2 \, | \, R- R^{\alpha /2} \le |x| \le R \,\} . \end{aligned}$$

Recalling that $\tilde{u}_R \equiv 0$ when $|x| \le R-R^{\alpha /2}$ and $\tilde{u}_R \equiv u$ when $|x| \ge R$, we get

$$\begin{aligned} \Vert \tilde{u}_R\Vert ^2 = \int _{B^{\text {c}}_{R - R^{\alpha /2}}} (\, |\nabla \tilde{u}_R|^2 + V(x) \tilde{u}_R^2 \,) \, \hbox {d}x \le (1+C) \int _{B^{\text {c}}_{R}} (\, |\nabla u|^2 + V(x) u^2 \,) \, \hbox {d}x . \end{aligned}$$

Since $u \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$, there exists $\overline{R}=\overline{R}(u, \gamma )>1$ such that

$$\begin{aligned} \int _{B^{\text {c}}_{\overline{R}}} (\, |\nabla u|^2 + V(x) u^2 \,) \, \hbox {d}x \le \, \frac{1}{1+C} \, \frac{1}{\gamma }\end{aligned}$$

and in particular

$$\begin{aligned} \Vert \sqrt{\gamma } \, \tilde{u}_R \Vert \le 1 \quad \text { for any } \, R \ge \overline{R} . \end{aligned}$$

Therefore, we may estimate

$$\begin{aligned} \int _{B^{\text {c}}_{R}} Q(x)(e^{\gamma u^2} -1) \, \hbox {d}x&= \int _{B^{\text {c}}_{R}} Q(x)(e^{\gamma \tilde{u}_R^2} -1) \, \hbox {d}x \le \int _{B^{\text {c}}_{R- R^{\alpha /2}}} Q(x)(e^{(\sqrt{\gamma } \, \tilde{u}_R)^2} -1) \, \hbox {d}sx\\&\le \eta \Vert \sqrt{\gamma } \, \tilde{u}_R\Vert ^2 = \eta \, \gamma \, \Vert \tilde{u}_R\Vert ^2 \end{aligned}$$

where $\eta >0$ is arbitrarily fixed and we used Proposition 4.6. This is possible provided $\overline{R} - \overline{R}^{\alpha /2} \ge 3 \tilde{n}$ where $\tilde{n}= \tilde{n} (\gamma , a, \alpha )>1$ is given by Proposition 4.6. $\square$

From now on, $u_0 \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ will denote a weak solution of (NLS).

Lemma 6.2

There exists $\tilde{R} >0$ such that for any $n \in \mathbb N$ satisfying $R_{n}:=n^{2/(2- \alpha )} \ge \tilde{R}$ we have

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n+1}}} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \mathrm{d}x \le \frac{3}{4} \int _{B^{\text {c}}_{R_n}} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \mathrm{d}x . \end{aligned}$$

Proof

Let $\chi _n \, : \, {{\mathrm{\mathbb {R}}}}^2 \rightarrow [0,1]$ be a piecewise affine function such that

$$\begin{aligned} \chi _n(x):= {\left\{ \begin{array}{ll} 0 &{} \text {if } |x| \le R_n , \\ 1 &{} \text {if } |x| \ge R_{n+1} . \end{array}\right. } \end{aligned}$$

Arguing as in [8, Lemma 17], we can prove that

$$\begin{aligned} |\nabla \chi _n(x)|^2 \le V(x) . \end{aligned}$$

By construction $\chi _n u_0 \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$,

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n+1}}} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x \le \int _{B^{\text {c}}_{R_{n}}} \chi _n(\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x \end{aligned}$$

and we can compute

$$\begin{aligned} <u_0, \chi _n u_0> \,= \int _{B^{\text {c}}_{R_{n}}} \chi _n(\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x + \int _{B^{\text {c}}_{R_n}} u_0 \nabla u_0 \cdot \nabla \chi _n \, \hbox {d}x . \end{aligned}$$

Moreover, if we use $\chi _n u_0 \in H^1_V({{\mathrm{\mathbb {R}}}}^2)$ as test function, we obtain

$$\begin{aligned} <u_0, \chi _n u_0> - \int _{{{\mathrm{\mathbb {R}}}}^2} Q(x) f(u_0) \chi _n u_0 \, \hbox {d}x =0 . \end{aligned}$$

Therefore, we may estimate

$$\begin{aligned}&\int _{B^{\text {c}}_{R_{n}}} \chi _n(\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x \\&\quad = \int _{B^{\text {c}}_{R_{n}}} Q(x)f(u_0) \chi _n u_0 \, \hbox {d}x - \int _{B^{\text {c}}_{R_n}} u_0 \nabla u_0 \cdot \nabla \chi _n \, \hbox {d}x\\&\quad \le \int _{B^{\text {c}}_{R_{n}}} Q(x)f(u_0) u_0 \, \hbox {d}x + \frac{1}{2} \Bigl (\, \int _{B^{\text {c}}_{R_{n}}} |\nabla u_0|^2 \, \hbox {d}x + \int _{B^{\text {c}}_{R_{n}}} |\nabla \chi _n|^2 u_0^2 \, \hbox {d}x \, \Bigr )\\&\quad \le \int _{B^{\text {c}}_{R_{n}}} Q(x)f(u_0) u_0 \, \hbox {d}x + \frac{1}{2} \, \int _{B^{\text {c}}_{R_{n}}} (\,|\nabla u_0|^2 \, \hbox {d}x +V(x) u_0^2 \,) \, \hbox {d}x . \end{aligned}$$

To complete the proof, it is sufficient to prove the existence of $\tilde{R}>0$ such that for any $n \in \mathbb N$ with $R_n \ge \tilde{R}$

$$\begin{aligned} \int _{B^{\text {c}}_{R_n}} Q(x) f(u_0) u_0 \, \hbox {d}x \le \frac{1}{4} \, \int _{B^{\text {c}}_{R_{n}}} (\,|\nabla u_0|^2 \, dx +V(x) u_0^2 \,) \, \hbox {d}x . \end{aligned}$$

To this aim, arguing as in (5.3), for fixed $\gamma > \gamma _0$ and $\sigma >0$ we have

$$\begin{aligned} \int _{B^{\text {c}}_{R_n}} Q(x) f(u_0) u_0 \, \hbox {d}x&\le \sigma \int _{B^{\text {c}}_{R_n}} Q(x) u_0^2 \, \hbox {d}x\\&\quad + C(\gamma , \sigma ) \, \Bigl ( \int _{B^{\text {c}}_{R_n}} Q(x) u_0^2 \, \hbox {d}x \Bigr )^{\frac{1}{2}} \, \Bigl ( \int _{B^{\text {c}}_{R_n}} Q(x) (e^{2 \gamma u_0^2} -1) \, \hbox {d}x \Bigr )^{\frac{1}{2}} . \end{aligned}$$

Let $\overline{R}>1$ be as in Lemma 6.1. If $\tilde{R} \ge \overline{R}$ then, for any $n \in \mathbb N$ satisfying $R_{n} \ge \tilde{R}$, we may apply Lemma 6.1 obtaining

$$\begin{aligned} \int _{B^{\text {c}}_{R_n}} Q(x) (e^{2 \gamma u_0^2} -1) \, \hbox {d}x \le C \gamma \int _{B^{\text {c}}_{R_n}} (\, |\nabla u_0|^2 +V(x)u_0^2 \,) \, \hbox {d}x . \end{aligned}$$

(6.1)

Moreover, if $\tilde{R} >0$ is sufficiently large then, for any $n \in \mathbb N$ satisfying $R_{n}\ge \tilde{R}$, we have

$$\begin{aligned} \sup _{x \in B^{\text {c}}_{R_n}} \frac{Q(x)}{V(x)} \le \sup _{x \in B^{\text {c}}_{\tilde{R}}} \frac{Q(x)}{V(x)} \le \sup _{x \in B^{\text {c}}_{\tilde{R}}} \frac{b}{a} \, \frac{1 + |x|^\alpha }{1 + |x|^\beta } \le \frac{b}{a} \, \frac{1 + |\tilde{R}|^\alpha }{1 + |\tilde{R}|^\beta } =: {\mathcal {B}}(\tilde{R}) \end{aligned}$$

where we used assumptions (V) and (Q) with $\alpha$ and $\beta$ in the range (2.1). Therefore, when $R_n \ge \tilde{R}$,

$$\begin{aligned} \int _{B^{\text {c}}_{R_n}} Q(x) u_0^2 \, \hbox {d}x \le {\mathcal {B}}(\tilde{R}) \, \int _{B^{\text {c}}_{R_n}} V(x) u_0^2 . \end{aligned}$$

(6.2)

Combining (6.1) and (6.2), we obtain

$$\begin{aligned} \int _{B^{\text {c}}_{R_n}} Q(x) f(u_0) u_0 \, \hbox {d}x \le \Bigl [ \, \sigma \, [{\mathcal {B}}(\tilde{R})]^{\frac{1}{2}} + \tilde{C}(\gamma , \sigma ) \, \Bigr ] \, [{\mathcal {B}}(\tilde{R}) ]^{ \frac{1}{2}} \, \int _{B^{\text {c}}_{R_n}} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x . \end{aligned}$$

Due to the range (2.1) of the parameters $\alpha$ and $\beta$, we point out that

$$\begin{aligned} \lim _{\tilde{R} \rightarrow + \infty } {\mathcal {B}}(\tilde{R})= \lim _{\tilde{R} \rightarrow + \infty } \frac{b}{a} \, \tilde{R}^{\alpha - \beta } =0 \end{aligned}$$

and, since $\sigma >0$ and $\gamma > \gamma _0$ are fixed, we can choose $\tilde{R}>0$ sufficiently large so that

$$\begin{aligned} \Bigl [ \, \sigma \, [{\mathcal {B}}(\tilde{R})]^{\frac{1}{2}} + \tilde{C}(\gamma , \sigma ) \, \Bigr ] \, [{\mathcal {B}}(\tilde{R}) ]^{ \frac{1}{2}}\le \frac{1}{4} . \end{aligned}$$

$\square$

Lemma 6.3

There exists $\tilde{R} >0$ and a constant $C>0$ such that for any $\varrho > 2 \tilde{R}$

$$\begin{aligned} \int _{B^{\text {c}}_{\varrho }} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \mathrm{d}x \le C e^{- \bigl | \log \frac{3}{4} \bigr |\, \varrho ^{(2-\alpha )/2}} . \end{aligned}$$

Proof

Let $\tilde{R}$ and $\{R_n\}_n$ be as in Lemma 6.2 and let $\varrho > 2 \tilde{R}$. Then there exist two positive integers $\overline{n} > \tilde{n}$ such that

$$\begin{aligned} R_{\tilde{n}} \le \tilde{R} \le R_{\tilde{n} +1} \quad \text { and } \quad R_{\overline{n} -1} \le \varrho \le R_{\overline{n}} \end{aligned}$$

and it is easy to see that

$$\begin{aligned} \overline{n} - \tilde{n} \ge \varrho ^{(2- \alpha )/2} - \tilde{R}^{(2- \alpha )/2} > \tilde{R}^{(2- \alpha )/2}(2^{(2- \alpha )/2} -1) >2 \end{aligned}$$

provided $\tilde{R} >0$ is sufficiently large. Therefore $\overline{n} - \tilde{n} \ge 3$, in particular

$$\begin{aligned} R_{\overline{n} -2} \ge R_{\tilde{n} +1} \ge \tilde{R} \end{aligned}$$

and we may estimate, using Lemma 6.2,

$$\begin{aligned} \int _{B^{\text {c}}_{\varrho }} (\, |\nabla u_0|^2&+ V(x) u_0^2 \,) \, \hbox {d}x \le \int _{B^{\text {c}}_{R_{\overline{n} -1}}} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x\\&\le \Bigl ( \frac{3}{4}\Bigr )^{\overline{n} - \tilde{n} -2} \int _{B^{\text {c}}_{\tilde{R}}} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x\\&\le \Bigl (\frac{4}{3}\Bigr )^2 \, e^{- \bigl | \log \frac{3}{4} \bigr | (\varrho ^{(2-\alpha )/2} - \tilde{R}^{(2- \alpha )/2})} \, \int _{B^{\text {c}}_{\tilde{R}}} (\, |\nabla u_0|^2 + V(x) u_0^2 \,) \, \hbox {d}x . \end{aligned}$$

$\square$

In order to conclude that $u_0 \in L^2({{\mathrm{\mathbb {R}}}}^2)$, it is enough to prove that

$$\begin{aligned} \int _{B^{\text {c}}_{2}} u_0^2 \, \hbox {d}x < + \infty . \end{aligned}$$

First, for fixed $r \ge 2$ and $|y| \ge 2r$, note that

$$\begin{aligned} \sup _{x \in B(y,r)} \frac{1+ |x|^\alpha }{a |y|^\alpha } \le \frac{1 +(r +|y|)^\alpha }{a |y|^\alpha } \le \frac{1+ \bigl (\frac{3}{2} \, |y|\bigr )^\alpha }{a|y|^\alpha } \le \sup _{y \in B^{\text {c}}_{4}} \frac{1+ \bigl (\frac{3}{2} \, |y|\bigr )^\alpha }{a|y|^\alpha } =: C(\alpha ) < + \infty . \end{aligned}$$

Hence, in view of (V), we have

$$\begin{aligned} \int _{B(y,r)} u_0^2 \, \hbox {d}x \le \int _{B(y,r)} \frac{1 + |x|^\alpha }{a} \, V(x) u_0^2 \, \hbox {d}x \le C(\alpha ) \, |y|^\alpha \int _{B^{\text {c}}_{|y|/2}} V(x) u_0^2 \, \hbox {d}x \end{aligned}$$

where we also used the inclusion $B(y,r) \subseteq B^{\text {c}}_{|y|/2}$. If $r > 2 \tilde{R}$ then we may apply Lemma 6.3 and get

$$\begin{aligned} \int _{B(y,r)} u_0^2 \, \hbox {d}x \le \tilde{C}(\alpha ) \, |y|^\alpha \, e^{- \bigl | \log \frac{3}{4} \bigr |\, \bigl (\frac{|y|}{2}\bigr )^{(2-\alpha )/2}} . \end{aligned}$$

(6.3)

Next, let $m \in \mathbb N$ and $|y_i| \ge 2$ with $i \in \{1, \dots , m\}$ be such that

$$\begin{aligned} B_5 {\setminus } B_2 \subset \bigcup _{i=1}^m B(y_i,1) \end{aligned}$$

and let $y_{i,k}:= 2^k y_i$. If $K_0$ is a positive integer such that $2^{K_0} > 2 \tilde{R}$ then, using (6.3) with $r=2^k$ and $y=y_{i,k}$,

$$\begin{aligned} \int _{B(y_{i,k}, 2^k)} u_0^2 \, \hbox {d}x\le \tilde{C}(\alpha ) \, |y_{i,k}|^\alpha \, e^{- \bigl | \log \frac{3}{4} \bigr |\, \bigl (\frac{|y_{i,k}|}{2}\bigr )^{(2-\alpha )/2}} \quad \text { for any } \, k \ge K_0 \end{aligned}$$

and

$$\begin{aligned} \int _{B^{\text {c}}_{2}} u_0^2 \, \hbox {d}x&\le \sum _{k=0}^{+ \infty } \int _{2^k B_5 {\setminus } B_2} u_0^2 \, \hbox {d}x \le \sum _{i=1}^{m} \sum _{k=0}^{+ \infty } \int _{B(y_{i,k}, 2^k)} u_0^2 \, \hbox {d}x\\&\le \sum _{i=1}^{m} \sum _{k=0}^{K_0-1} \int _{B(y_{i,k}, 2^k)} u_0^2 \, \hbox {d}x + \tilde{C}(\alpha )\, \sum _{i=1}^{m} \sum _{k=K_0}^{+ \infty } |y_{i,k}|^\alpha \, e^{- \bigl | \log \frac{3}{4} \bigr |\, \bigl (\frac{|y_{i,k}|}{2}\bigr )^{(2-\alpha )/2}} < + \infty \end{aligned}$$

since $\alpha \in (0,2)$. This completes the proof of Proposition 2.2.