Combined effects in planar quasilinear Schrödinger equations with superlinear reaction

Abstract

In this paper, we prove the existence of nontrivial solutions for the following planar quasilinear Schrödinger equation: $\begin{matrix} - Δ u + V (x) u - Δ (u^{2}) u = g (u), x \in R^{2}, \end{matrix}$ where $V \in C (R^{2}, [0, \infty))$ and $g \in C (R, R)$ is of subcritical exponential growth satisfying some mild conditions. In particular, by means of the Trudinger–Moser inequality, we give a different method from the one of the polynomial growth nonlinearities to prove the Brézis–Lieb split property when f has subcritical exponential growth. Our result extends and complements the one of Chen–Rădulescu–Tang–Zhang (Rev. Mat. Iberoam. 36 (2020) 1549–1570) dealing with the higher dimensions $N ⩾ 3$ to the dimension $N = 2$ .

Keywords

Quasilinear Schrödinger equation nontrivial solution exponential growth

1. Introduction and main results

In the last decades, there has been growing interests in the study of the existence of standing wave solutions for quasilinear Schrödinger equations of the type: $\begin{matrix} (1.1) & i \partial_{t} z = - Δ z + V (x) z - h (| z |^{2}) z - k Δ l (| z |^{2}) l^{'} (| z |^{2}) z, \end{matrix}$ where V is a given potential, k is a real constant, l and h are suitable functions. This kind of quasilinear Schrödinger equation plays an important role in physics research, which is mainly derived from plasma physics, fluid mechanics, dissipative quantum mechanics and condensed matter theory. For more details physical aspects, we refer the readers to [4,11–13,17,19]. Semilinear Schrödinger problems corresponding to $k = 0$ have been widely studied with a huge variety of conditions on the potential and the nonlinearity in recent years, see for example [1,2,23,24]. Considering $k > 0$ and $l (s) = s$ , (1.1) can be reduced to the following quasilinear Schrödinger equations of the form $\begin{matrix} (1.2) & - Δ u + V (x) u - Δ (u^{2}) u = g (u), x \in R^{N}, \end{matrix}$ where $N ⩾ 2$ , $V : R^{N} \to R$ and $g : R \to R$ .

Compared with semilinear Schrödinger problems, quasilinear Schrödinger problems like (1.2) are more complicated since the presence of quasilinear term $Δ (u^{2}) u$ makes the energy functional not well defined in $H^{1} (R^{N})$ . Moreover, one needs to overcome the difficulties caused by the lack of compactness thanks to the unboundedness of the domain. As far as we know, mathematical studies have focused on problem (1.2) in the case of $N ⩾ 3$ at present. In this case, Poppenberg, Schmitt and Wang [20] and Liu and Wang [16] obtained the existence results of solutions by using constrained minimization argument, which gives a solution of (1.2) with an unknown Lagrange multiplier in front of the nonlinear term. In [14], by using a change of variable to transform the quasilinear Schrödinger problem into the corresponding semilinear Schrödinger problem and selecting Orlicz space as the workspace, Liu, Wang and Wang obtained the existence of positive solutions of (1.2) via mountain pass theorem. Colin and Jeanjean [7] also used a change of variable (dual approach) in order to simplify (1.2) into semilinear equation and reached similar conclusions, but the working space is the usual Sobolev space $H^{1} (R^{N})$ , not Orlicz space. Liu, Wang and Wang [15] proved the existence of ground state solution for (1.2) by Nehari type constrained minimization method, yet their argument has nothing to do with a change of variables. In [21], still based on the idea of constrained minimization, Ruiz and Siciliano employed Nehari–Pohoz˘aev manifold, which is a combination of the Nehari equation and the Pohoz˘aev equality, to get the existence of ground state solution for (1.2). Zhang, Tang and Zhang [27] proved that (1.2) has infinitely many nontrivial solutions by using dual approach and mountain pass theorem, if V satisfies coercive condition and

$V \in C (R^{N}, R)$ ; ${inf}_{x \in R^{N}} V (x) > - \infty$ .

Later, imposing some new conditions on V and more general assumptions on g, with the help of Nehari–Pohoz˘aev manifold, Chen, Rădulescu, Tang and Zhang [6] obtained the existence of ground state solutions for (1.2), where V satisfies

$V \in C (R^{N}, [0, \infty))$ and $V_{\infty} : = {lim}_{| y | \to \infty} V (y) ⩾ V (x)$ for all $x \in R^{N}$ .

We would like to point out that Brézis–Lieb type Lemma and Lions’ concentration compactness principle are used to restore compactness in these works on (1.2) with

N ⩾ 3

Although there have been extensive existence results on (1.2) with polynomial growth nonlinearities in the case of $N ⩾ 3$ , much less can be found for $N = 2$ . In [7], Colin and Jeanjean first dealt with (1.2) with $N = 2$ where g has subcritical exponential growth, namely, g satisfies

there exists $α > 0$ such that ${lim}_{t \to \infty} \frac{g (t)}{exp (α t^{4})} = 0$ ,

which is motivated by Trudinger–Moser inequality [5,8,18,25], that is, for all

u \in H^{1} (R^{2})

and

α > 0

\begin{matrix} exp (α u^{2}) - 1 \in L^{1} (R^{2}), \end{matrix}

moreover, if

α < 4 π

and

‖ u ‖_{2}^{2} ⩽ M

, there exists a constant

C (M, α)

such that

\begin{matrix} (1.3) & sup_{‖ \nabla u ‖_{2}^{2} ⩽ 1} \int_{R^{2}} [exp (α u^{2}) - 1] d x < C (M, α) . \end{matrix}

Precisely, by using classical results on scalar field equations due to Berestycki-Gallouët-Kavian [3] when

N = 2

, they proved the existence of a nontrivial solution for (1.2) with

N = 2

, where V is a positive constant and g satisfies the assumption: for any

α > 0

there exists a positive constant

C_{α}

such that

\begin{matrix} (1.4) & | g (t) | ⩽ C_{α} exp (α t^{2}), \forall t ⩾ 0 . \end{matrix}

Clearly, the assumption (1.4) indicates that g satisfies (G1). Combined with Trudinger–Moser inequality, do Ó, Miyagaki and Soares [9] proved the existence of a positive solution for (1.2) with

N = 2

by using mountain pass theorem when g has exponential growth. do Ó and Severo [10] established the existence, concentration behavior and exponential decay of positive ground state solutions for singularly perturbed quasilinear Schrödinger equation in a nonstandard Orlicz space by using penalization technique and mountain pass arguments. Later, Severo and Carvalho [22] also used mountain pass theorem and gained the existence of a nonnegative ground state solution for quasilinear Schrödinger equation with vanishing potentials involving exponential growth. In these references, the authors assume that

V (x)

is a positive constant or is a periodic potential, or is a locally Hölder continuous function, and

g (t)

satisfies the following hypothesis of Ambrosetti–Rabinowitz type:

there exists $μ > 4$ such that $g (t) t ⩾ μ G (t) ⩾ 0$ $\forall t \in R$ , where $G (t) : = \int_{0}^{t} g (s) d s$ ,

or satisfies 4-superlinear at

t = \infty

and monotonicity condition as follows:

${lim}_{| t | \to \infty} G (t) / | t |^{4} = + \infty$ ;

$g (t) / | t |^{3}$ is strictly increasing for $t \in R ∖ {0}$ .

These conditions are very crucial to verify the mountain pass geometry or the boundedness of the (PS) sequences or Cerami sequences for the corresponding energy functional.

Now, a natural question is whether equation (1.2) with $N = 2$ has a nontrivial solution when ${lim}_{| t | \to \infty} \frac{| g (u) |}{| u |^{3}} = 0$ . To our knowledge, there has not been any work in the literature on this subject, even for the simpler nonlinearity $g (u) \sim | u |^{q - 2} u$ with $q \in (2, 4)$ . In the present paper, we shall give a positive answer for the above question. More precisely, we will further study the existence of a nontrivial solution to the following planar quasilinear Schrödinger equation: $\begin{matrix} (1.5) & - Δ u + V (x) u - Δ (u^{2}) u = g (u), x \in R^{2}, \end{matrix}$ where $V : R^{2} \to R$ satisfies

$V \in C (R^{2}, [0, \infty))$ and $V_{\infty} : = {lim}_{| y | \to \infty} V (y) ⩾ V (x)$ for all $x \in R^{2}$ ;

$V \in C^{1} (R^{2}, R)$ and $t^{4 / (p + 2)} V (t^{1 / (p + 2)} x)$ is concave on $t \in (0, \infty)$ ,

and g satisfy (G1) and the following conditions:

$g \in C (R, R)$ and $g (t) = o (t)$ as $t \to 0$ ;

there exists $p > 2$ such that $[g (t) t + 2 G (t)] / | t |^{p - 1} t$ is nondecreasing on both $(- \infty, 0)$ and $(0, \infty)$ ;

${lim}_{| t | \to \infty} G (t) / | t |^{2} = + \infty$ .

Obviously, our assumptions can cover the case when

{lim}_{| t | \to \infty} \frac{| g (t) |}{| t |^{3}} = 0

It is worth noting that the condition (V2) is equivalent to the following assumption which is easier to verify:

$V \in C^{1} (R^{2}, R)$ , and $t \mapsto [4 V (t x) + \nabla V (t x) \cdot (t x)] / t^{p - 2}$ is nonincreasing on $[0, \infty)$ for any $x \in R^{2}$ .

There are many functions satisfying (V1) and (V2); for example,

\begin{matrix} V (x) = a - \frac{b}{| x |^{β} + 1} with a > b > 0 and β > 0 . \end{matrix}

Clearly, the associated energy functional of (1.5) is defined formally by $\begin{matrix} (1.6) & Φ (u) = \frac{1}{2} \int_{R^{2}} (1 + 2 u^{2}) | \nabla u |^{2} d x + \int_{R^{2}} V (x) u^{2} d x - \int_{R^{2}} G (u) d x . \end{matrix}$ Since the appearance of quasilinear term $Δ (u^{2}) u$ makes Φ is not well defined in $H^{1} (R^{2})$ , we utilize a dual approach developed by Liu, Wang and Wang [14] and Colin and Jeanjean [7]. Let $v = f^{- 1} (u)$ , where f is defined by $\begin{array}{l} f^{'} (t) = \frac{1}{\sqrt{1 + 2 f^{2} (v)}} on [0, + \infty), \\ f (t) = - f (- t) on (- \infty, 0] . \end{array}$ After this change of variables from Φ, we obtain the following functional: $\begin{array}{rcl} (1.7) & I (v) = \frac{1}{2} \int_{R^{2}} [| \nabla v |^{2} + V (x) f^{2} (v)] d x - \int_{R^{2}} G (f (v)) d x, \end{array}$ Under hypothetical conditions (V1), (G1) and (G2), It is standard to verify that $I \in C^{1} (H^{1} (R^{2}), R)$ and $\begin{array}{l} ⟨ I^{'} (v), w ⟩ = \int_{R^{2}} [\nabla v \nabla w + V (x) f (v) f^{'} (v) w] d x - \int_{R^{2}} g (f (v)) f^{'} (v) w d x, \\ (1.8) & \forall v, w \in H^{1} (R^{2}) . \end{array}$ Based on variational method, the critical points of I are solutions of the semilinear equation $\begin{matrix} (1.9) & - Δ v + V (x) f (v) f^{'} (v) = g (f (v)) f^{'} (v), x \in R^{2} . \end{matrix}$ Moreover, v is a solution of (1.9) if and only if $u = f (v)$ solves (1.5), see [7,14].

To state our result, we define the Pohoz˘aev type functional of (1.9): $\begin{matrix} (1.10) & \begin{matrix} P (v) : = \frac{1}{2} \int_{R^{2}} [2 V (x) + \nabla V (x) \cdot x] f^{2} (v) d x - 2 \int_{R^{2}} G (f (v)) d x, \forall v \in H^{1} (R^{2}) . \end{matrix} \end{matrix}$ It is well-known that any solution v of (1.9) satisfies $P (v) = 0$ and $⟨ I^{'} (v), f (v) / f^{'} (v) ⟩ = 0$ . A solution u of (1.9) is called a Nehari–Pohoz˘aev type ground state solution if $v = f^{- 1} (u)$ satisfies $I (v) = {inf}_{M} I$ , where $M$ is the Nehari–Pohoz˘aev manifold of I by $\begin{matrix} (1.11) & M : = {v \in H^{1} (R^{2}) ∖ {0} : J (v) = 0} \end{matrix}$ and $\begin{matrix} (1.12) & \begin{matrix} J (v) : = & ⟨ I^{'} (v), f (v) / f^{'} (v) ⟩ + P (v) \\ = & ‖ \nabla v ‖_{2}^{2} + \int_{R^{2}} \frac{2 f^{2} (v)}{1 + 2 f^{2} (v)} | \nabla v |^{2} d x + \frac{1}{2} \int_{R^{2}} [4 V (x) + \nabla V (x) \cdot x] f^{2} (v) d x \\ - \int_{R^{2}} [g (f (v)) f (v) + 2 G (f (v))] d x . \end{matrix} \end{matrix}$ Obviously, every nontrivial solution of (1.9) is contained in $M$ . Our main result is as follows:

Theorem 1.1.
Assume that (V1), (V2) and (G1)–(G4) hold. Then equation ( 1.5 ) has a nontrivial solution $u = f (v) \in H^{1} (R^{2})$ such that $\begin{matrix} I (v) = inf_{M} I = inf_{v \in H^{1} (R^{2}) ∖ {0}} max_{t > 0} I (v_{t}) > 0, \end{matrix}$ where $v_{t} (x) = f^{- 1} (t f (v (t^{- 1} x))$ .

To achieve our goal, we must overcome two difficulties: 1) the lack of the Ambrosetti–Rabinowitz type condition (AR) prevents us to show the boundedness of the minimizing sequences following standard schemes; 2) many methods dealing with the polynomial growth nonlinearities may be not applied directly for exponential growth nonlinearities, for example, the proof of the Brézis–Lieb split property for the polynomial growth nonlinearities does not work. These difficulties enforce the implementation of new ideas and tricks. In particular, we give a different method from the one of the polynomial growth nonlinearities to prove the Brézis–Lieb split property when f has subcritical exponential growth (see Lemma 3.2 below).

Applying Theorem 1.1 to the “limit equation” of (1.5): $\begin{matrix} (1.13) & - Δ u + V_{\infty} u - Δ (u^{2}) u = g (u), x \in R^{2}, \end{matrix}$ we have the following corollary:
Corollary 1.2.
Assume that (G1)–(G4) hold. Then equation ( 1.13 ) has a ground state solution $u^{\infty} = f (v^{\infty}) \in H^{1} (R^{2})$ such that $\begin{matrix} I^{\infty} (v^{\infty}) = inf_{M^{\infty}} I^{\infty} = inf_{v \in H^{1} (R^{2}) ∖ {0}} max_{t > 0} I^{\infty} (v_{t}) > 0, \end{matrix}$ where $\begin{array}{l} (1.14) & I^{\infty} (v) = \frac{1}{2} \int_{R^{2}} [| \nabla v |^{2} + V_{\infty} f^{2} (v)] d x - \int_{R^{2}} G (f (v)) d x, \forall v \in H^{1} (R^{2}), \\ (1.15) & \begin{matrix} J^{\infty} (v) : = & ‖ \nabla v ‖_{2}^{2} + \int_{R^{2}} \frac{2 f^{2} (v)}{1 + 2 f^{2} (v)} | \nabla v |^{2} d x + 2 V_{\infty} {‖ f (v) ‖}_{2}^{2} \\ - \int_{R^{2}} [g (f (v)) f (v) + 2 G (f (v))] d x, \forall v \in H^{1} (R^{2}) \end{matrix} \end{array}$ and $\begin{matrix} (1.16) & M^{\infty} : = {v \in H^{1} (R^{2}) ∖ {0} : J^{\infty} (v) = 0} . \end{matrix}$

Throughout the paper we make use of the following notations:

∙ $H^{1} (R^{2})$ denotes the usual Sobolev space equipped with the inner product and norm $\begin{matrix} (u, v) = \int_{R^{2}} (\nabla u \cdot \nabla v + u v) d x, ‖ u ‖ = {(u, u)}^{1 / 2}, \forall u, v \in H^{1} (R^{2}); \end{matrix}$

∙ $L^{s} (R^{N}) (1 ⩽ s < \infty)$ denotes the Lebesgue space with the norm $‖ u ‖_{s} = {(\int_{R^{N}} | u |^{s} d x)}^{1 / s}$ ;

∙ For any $v \in E ∖ {0}$ , $v_{t} (x) = f^{- 1} (t f (v (t^{- 1} x)))$ for $t > 0$ ;

∙ For any $x \in R^{2}$ and $r > 0$ , $B_{r} (x) : = {y \in R^{2} : | y - x | < r}$ ;

∙ $C_{1}, C_{2}, \dots$ denote positive constants possibly different in different places.

Under assumptions (G1) and (G2), fix $α > 0$ , we know that for any $ε > 0$ and $q ⩾ 1$ , there exists $C_{α, ε, q} > 0$ such that $\begin{matrix} (1.17) & | g (t) | ⩽ ε | t | + C_{α, ε, q} | t |^{q} [exp (α t^{4}) - 1], \forall t \in R, \end{matrix}$ Moreover, using (1.17) with $q = 4$ , we deduce that for any $ε > 0$ , there exists $C_{α, ε} > 0$ such that $\begin{matrix} (1.18) & | G (t) | ⩽ ε | t |^{2} + C_{α, ε} | t |^{3} [exp (α t^{4}) - 1], \forall t \in R . \end{matrix}$

The paper is organized as follows. In Section 2, we give the variational setting and preliminary results. Based on Nehari–Pohoz˘aev manifold $M$ , we consider the existence of a nontrivial solution for (1.5) and give the proof of Theorem 1.1 in Section 3.
2. Variational setting and preliminary results

Lemma 2.1 (see [7,10,14]).

The following properties involving f and its derivative hold:

$| f^{'} (t) | ⩽ 1$ for all $t \in R$ ;

$| f (t) | ⩽ | t |$ for all $t \in R$ ;

$f (t) / 2 ⩽ t f^{'} (t) ⩽ f (t)$ for all $t ⩾ 0$ ;

$| f (t) | ⩽ 2^{1 / 4} | t |^{1 / 2}$ for all $t \in R$ ;

$| f (t) | / \sqrt{t} \to 2^{1 / 4}$ as $t \to + \infty$ ;

there exists a positive constant θ such that $\begin{array}{l} | f (t) | ⩾ \{\begin{matrix} θ | t |, & | t | ⩽ 1, \\ θ | t |^{1 / 2}, & | t | ⩾ 1 . \end{matrix} \end{array}$

Lemma 2.2.
The following inequality holds: $\begin{matrix} (2.1) & H (t, v) : = \int_{R^{2}} {[1 - \frac{t^{2} (1 + 2 t^{2} f^{2} (v))}{1 + 2 f^{2} (v)}] - \frac{1 - t^{2 + p}}{2 + p} [2 + \frac{4 f^{2} (v)}{1 + 2 f^{2} (v)}]} | \nabla v |^{2} d x > 0; \end{matrix}$ Moreover (V2) and (G3) respectively imply that the following inequality hold: $\begin{matrix} (2.2) & V (x) - t^{4} V (t x) - \frac{1 - t^{2 + p}}{2 + p} [4 V (x) + \nabla V (x) \cdot x] ⩾ 0, \forall t ⩾ 0, x \in R^{2}; \end{matrix}$ and $\begin{matrix} (2.3) & t^{2} G (t τ) - G (τ) + \frac{1 - t^{2 + p}}{2 + p} [g (τ) τ + 2 G (τ)] ⩾ 0, \forall t ⩾ 0, τ \in R . \end{matrix}$
Lemma 2.3.
Assume that (V1), (V2) and (G1)–(G3) hold. Then $\begin{array}{rcl} (2.4) & I (v) ⩾ I (v_{t}) + \frac{1 - t^{2 + p}}{2 + p} J (v) + \frac{1}{2} H (t, v), \forall v \in H^{1} (R^{2}) ∖ {0}, t > 0 . \end{array}$
Proof.
By Lemma 2.2, the proof is the same as the one in [6, Lemma 2.2], so we omit it. □

From Lemma 2.3, we have the following corollary.
Corollary 2.4.
Assume that (V1), (V2) and (G1)–(G3) hold. Then for $v \in M$ $\begin{matrix} (2.5) & I (v) = max_{t > 0} I (v_{t}) . \end{matrix}$

For any $v \in M$ , it easy to very that $I (t v) < 0$ for large $t > 0$ by (G4). Hence, we can prove the following lemma by a similar fashion as [6, Lemma 2.4].
Lemma 2.5.
Assume that (V1), (V2) and (G1)–(G4) hold. Then for any $v \in E ∖ {0}$ , there exists a unique $t_{v} > 0$ such that $v_{t_{v}} \in M$ .

From Corollary 2.4 and Lemma 2.5, we have $M \neq \emptyset$ and the following lemma. Lemma 2.6.
Assume that (V1), (V2) and (G1)–(G4) hold. Then $\begin{matrix} inf_{v \in M} I (v) = m = inf_{v \in H^{1} (R^{2}) ∖ {0}} max_{t > 0} I (v_{t}) . \end{matrix}$
Lemma 2.7.
Assume that (V1) and (V2) hold. Then $\begin{matrix} (2.6) & | \nabla V (x) \cdot x | \to 0 as | x | \to \infty . \end{matrix}$

Moreover, there exist constants $ρ_{1}, ρ_{2} > 0$ such that $\begin{array}{l} (2.7) & (p - 2) V (x) - \nabla V (x) \cdot x ⩾ ρ_{1}, \forall x \in R^{2}, \\ (2.8) & 4 V (x) + \nabla V (x) \cdot x ⩾ ρ_{2}, \forall x \in R^{2} . \end{array}$
Proof.
Arguing by contradition. We suppose that there exist a sequence ${x_{n}} \subset R^{2}$ and a number $δ > 0$ such that $\begin{matrix} | x | \to \infty and either \nabla V (x_{n}) \cdot x_{n} ⩾ δ or \nabla V (x_{n}) \cdot x_{n} ⩽ - δ for all n \in N . \end{matrix}$ Let us distinguish two cases: 1) $\nabla V (x_{n}) \cdot x_{n} ⩾ δ$ for all $n \in N$ and 2) $\nabla V (x_{n}) \cdot x_{n} ⩽ - δ$ for all $n \in N$ .

Case 1) $\nabla V (x_{n}) \cdot x_{n} ⩾ δ$ for all $n \in N$ . In this case, for all $0 < t < 1$ , $\begin{array}{rcl} δ & ⩽ & \nabla V (x_{n}) \cdot x_{n} \\ ⩽ & \frac{2 + p}{1 - t^{2 + p}} [V (x_{n}) - t^{4} V (t x_{n})] - 4 V (x_{n}) \\ (2.9) & = & \frac{(2 + p) t^{4}}{1 - t^{2 + p}} [V (x_{n}) - V (t x_{n})] + [\frac{(2 + p) (1 - t^{4})}{1 - t^{2 + p}} - 4] V (x_{n}) . \end{array}$ Since $\begin{array}{rcl} (2.10) & lim_{t \to 1} \frac{(2 + p) (1 - t^{4})}{1 - t^{2 + p}} = 4, \end{array}$ there exists $0 < t_{1} < 1$ such that $\begin{array}{rcl} (2.11) & [\frac{(2 + p) (1 - t_{1}^{4})}{1 - t_{1}^{2 + p}} - 4] V_{\infty} < \frac{δ}{2} . \end{array}$ Then (V1), (2.9) and (2.11) imply that $\begin{matrix} δ ⩽ \frac{(2 + p) t_{1}^{4}}{1 - t_{1}^{2 + p}} [V (x_{n}) - V (t_{1} x_{n})] + [\frac{(2 + p) (1 - t_{1}^{4})}{1 - t_{1}^{2 + p}} - 4] V_{\infty} ⩽ \frac{δ}{2} + o (1), \end{matrix}$ which is a obvious contradiction.

Case 2) $\nabla V (x_{n}) \cdot x_{n} ⩽ - δ for all n \in N$ . In this case, for all $t > 1$ , $\begin{array}{rcl} - δ & ⩾ & \nabla V (x_{n}) \cdot x_{n} \\ ⩾ & \frac{2 + p}{1 - t^{2 + p}} [V (x_{n}) - t^{4} V (t x_{n})] - 4 V (x_{n}) \\ (2.12) & = & \frac{(2 + p) t^{4}}{1 - t^{2 + p}} [V (x_{n}) - V (t x_{n})] + [\frac{(2 + p) (1 - t^{4})}{1 - t^{2 + p}} - 4] V (x_{n}) . \end{array}$ From (2.10), there exists $t_{2} > 1$ such that $\begin{array}{rcl} (2.13) & [\frac{(2 + p) (1 - t_{2}^{4})}{1 - t_{2}^{2 + p}} - 4] V_{\infty} > - \frac{δ}{2} . \end{array}$ Then (V1), (2.12) and (2.13) imply that $\begin{matrix} - δ ⩾ \frac{(2 + p) t_{2}^{4}}{1 - t_{2}^{2 + p}} [V (x_{n}) - V (t_{2} x_{n})] + [\frac{(2 + p) (1 - t_{2}^{4})}{1 - t_{2}^{2 + p}} - 4] V_{\infty} ⩾ - \frac{δ}{2} + o (1), \end{matrix}$ which is again a obvious contradiction. Therefore (2.6) holds.

By (V1), there exists $V_{0} > 0$ such that $\begin{matrix} V (x) ⩾ V_{0} for all x \in R^{2} . \end{matrix}$ Using (V1), (V2) and (2.6), it is easy to see that there exists a constant $V_{1} > 0$ such that $\begin{matrix} (2.14) & | \nabla V (x) \cdot x | ⩽ V_{1} for all x \in R^{2} . \end{matrix}$ Then it follows from (V1), (V2), (2.2) and (2.14) that for all $x \in R^{2}$ and $t > 0$ , $\begin{array}{rcl} (p - 2) V (x) - \nabla V (x) \cdot x & ⩾ & (2 + p) t^{4} V (t x) - t^{2 + p} [4 V (x) + \nabla V (x) \cdot x] \\ ⩾ & (2 + p) t^{4} {V (t x) - t^{p - 2} [V (x) + \nabla V (x) \cdot x]} \\ (2.15) & ⩾ & (2 + p) t^{4} [V_{0} - t^{p - 2} (V_{\infty} + V_{1})] . \end{array}$ Choose $t_{3} = {(\frac{V_{0}}{V_{\infty} + V_{1} + 1})}^{1 / (p - 2)}$ , then (2.15) implies that for all $x \in R^{2}$ , $\begin{array}{rcl} (p - 2) V (x) - \nabla V (x) \cdot x & ⩾ & (2 + p) t_{3}^{4} [V_{0} - t_{3}^{p - 2} (V_{\infty} + V_{1})] \\ = & (2 + p) {(\frac{V_{0}}{V_{\infty} + V_{1} + 1})}^{(2 + p) / (p - 2)} \\ (2.16) & : = & ρ_{1} . \end{array}$ Note that (2.2) implies that for all $x \in R^{2}$ and $t > 1$ , $\begin{array}{rcl} 4 V (x) + \nabla V (x) \cdot x & ⩾ & \frac{2 + p}{t^{2 + p} - 1} [t^{4} V (t x) - V (x)] \\ (2.17) & ⩾ & \frac{2 + p}{t^{2 + p} - 1} (t^{4} V_{0} - V_{\infty}) . \end{array}$ Choose $t_{4} = {(\frac{V_{\infty} + 1}{V_{0}})}^{1 / 4} > 1$ , then (2.17) implies that for all $x \in R^{2}$ , $\begin{array}{rcl} 4 V (x) + \nabla V (x) \cdot x & ⩾ & \frac{2 + p}{t_{4}^{2 + p} - 1} (t_{4}^{4} V_{0} - V_{\infty}) \\ = & \frac{(2 + p) V_{0}^{(2 + p) / 4}}{{(V_{\infty} + 1)}^{(2 + p) / 4} - V_{0}^{(2 + p) / 4}} \\ (2.18) & : = & ρ_{2} . \end{array}$ This completes the proof of the lemma. □

3. The proof of Theorem 1.1

Lemma 3.1.

Assume that (V1), (V2) and (G1)–(G4) hold. Then $m = {inf}_{M} I > 0$ .

Proof.

By (G1) and (G2), there exists a constant $C_{1} > 0$ such that $\begin{matrix} (3.1) & \int_{R^{2}} [| g (t) t + 2 G (t) |] d x ⩽ \frac{ρ_{2}}{4} t^{2} + C_{1} | t |^{3} [exp (α t^{4}) - 1], \forall t \in R . \end{matrix}$ Let $t = 0$ in (2.3), one has $\begin{matrix} (3.2) & g (τ) τ - p G (τ) ⩾ 0, \forall τ \in R . \end{matrix}$ Since $J (v) = 0$ for $v \in M$ , then (1.7) and (1.12) give $\begin{array}{l} I (v) = & I (v) - \frac{1}{p + 2} J (v) \\ = & \frac{p - 2}{2 (p + 2)} ‖ \nabla v ‖_{2}^{2} + \frac{1}{p + 2} {‖ \nabla f (v) ‖}_{2}^{2} \\ + \frac{1}{2 (p + 2)} \int_{R^{2}} [(p - 2) V (x) - \nabla V (x) \cdot x] f^{2} (v) d x \\ (3.3) & + \frac{1}{p + 2} \int_{R^{2}} [g (f (v)) f (v) - p G (f (v))] d x, \forall v \in M . \end{array}$ To obtain the desired conclusion, we distinguish two cases: i) $‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2} > \frac{π}{2 α}$ ; ii) $‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2} ⩽ \frac{π}{2 α}$ .

Case i) $‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2} > \frac{π}{2 α}$ . In this case, by (2.7), (3.2) and (3.3), we have $\begin{array}{l} I (v) & ⩾ \frac{p - 2}{2 (p + 2)} ‖ \nabla v ‖_{2}^{2} + \frac{ρ_{1}}{2 (p + 2)} {‖ f (v) ‖}_{2}^{2} \\ ⩾ \frac{min {p - 2, ρ_{1}}}{2 (p + 2)} (‖ \nabla v ‖_{2}^{2} + {‖ f (v) ‖}_{2}^{2}) \\ (3.4) & ⩾ \frac{π min {p - 2, ρ_{1}}}{4 α (p + 2)}, \forall v \in M with ‖ \nabla v ‖_{2}^{2} + {‖ f (v) ‖}_{2}^{2} > \frac{π}{2 α} . \end{array}$

Case ii) $‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2} ⩽ \frac{π}{2 α}$ . By (3.1), (f4), Hölder inequality, Trudinger–Moser inequality (1.3) and Sobolev embedding theorem, we have $\begin{array}{l} \int_{R^{2}} [g (f (v)) f (v) + 2 G (f (v))] d x \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v) ‖}_{2}^{2} + C_{1} \int_{R^{2}} {| f (v) |}^{3} [exp (α {| f (v) |}^{4}) - 1] d x \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v) ‖}_{2}^{2} + C_{1} {‖ f (v) ‖}_{6}^{3} {\int_{R^{2}} [exp (2 α {| f (v) |}^{4}) - 1] d x}^{1 / 2} \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v) ‖}_{2}^{2} + C_{1} {‖ f (v) ‖}_{6}^{3} {\int_{R^{2}} [exp (4 α v^{2}) - 1] d x}^{1 / 2} \\ = \frac{ρ_{2}}{4} {‖ f (v) ‖}_{2}^{2} + C_{1} {‖ f (v) ‖}_{6}^{3} \\ \times {\int_{R^{2}} [exp (4 α (‖ \nabla v ‖_{2}^{2} + {‖ f (v) ‖}_{2}^{2}) {| \frac{v}{\sqrt{‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2}}} |}^{2}) - 1] d x}^{1 / 2} \\ (3.5) & ⩽ \frac{ρ_{2}}{4} {‖ f (v) ‖}_{2}^{2} + C_{2} {‖ f (v) ‖}^{3}, \forall ‖ \nabla v ‖_{2}^{2} + {‖ f (v) ‖}_{2}^{2} ⩽ \frac{π}{2 α}, \end{array}$ where $ρ_{2}$ is given by (2.8). Here in order to apply Trudinger–Moser inequality (1.3), we need to check that $\frac{‖ v ‖_{2}^{2}}{‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2}} ⩽ C_{3}$ . To this end, for $‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2} ⩽ \frac{π}{2 α}$ , by (f2), (f6) and the Sobolev embedding theorem, we have $\begin{array}{l} \frac{‖ v ‖_{2}^{2}}{‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2}} & = \frac{\int_{| v | ⩽ 1} v^{2} d x + \int_{| v | > 1} v^{2} d x}{‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2}} \\ ⩽ \frac{\frac{1}{θ^{2}} \int_{| v | ⩽ 1} f^{2} (v) d x + \frac{1}{θ^{4}} \int_{| v | > 1} f^{4} (v) d x}{‖ \nabla v ‖_{2}^{2} + ‖ f (v) ‖_{2}^{2}} \\ ⩽ \frac{1}{θ^{2}} + \frac{‖ f (v) ‖^{4}}{θ^{4} ‖ f (v) ‖^{2}} \\ ⩽ \frac{1}{θ^{2}} + \frac{π}{2 α θ^{4}} \\ (3.6) & : = C_{3}, \end{array}$ as desired. Since $J (v) = 0$ for $v \in M$ , it follows from (1.12), (2.8), (3.5) and (f2) that $\begin{array}{l} {‖ \nabla f (v) ‖}_{2}^{2} + \frac{ρ_{2}}{2} {‖ f (v) ‖}_{2}^{2} \\ ⩽ ‖ \nabla v ‖_{2}^{2} + \int_{R^{2}} \frac{2 f^{2} (v)}{1 + 2 f^{2} (v)} | \nabla v |^{2} d x + \frac{1}{2} \int_{R^{2}} [4 V (x) + \nabla V (x) \cdot x] f^{2} (v) d x \\ = \int_{R^{2}} [g (f (v)) f (v) + 2 G (f (v))] d x \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v) ‖}_{2}^{2} + C_{2} {‖ f (v) ‖}^{3}, \forall v \in M with ‖ \nabla v ‖_{2}^{2} + {‖ f (v) ‖}_{2}^{2} ⩽ \frac{π}{2 α}, \end{array}$ which implies $\begin{matrix} (3.7) & ‖ f (v) ‖ ⩾ \frac{min {4, ρ_{2}}}{4 C_{2}}, \forall v \in M with ‖ \nabla v ‖_{2}^{2} + {‖ f (v) ‖}_{2}^{2} ⩽ \frac{π}{2 α} . \end{matrix}$ Then in this case, by (2.7), (3.2), (3.3) and (3.7), we have $\begin{array}{l} I (v) & ⩾ \frac{1}{p + 2} {‖ \nabla f (v) ‖}_{2}^{2} + \frac{ρ_{1}}{2 (p + 2)} {‖ f (v) ‖}_{2}^{2} \\ (3.8) & ⩾ \frac{min {2, ρ_{1}} min {16, ρ_{2}^{2}}}{32 C_{2}^{2} (p + 2)}, \forall v \in M with ‖ \nabla v ‖_{2}^{2} + {‖ f (v) ‖}_{2}^{2} ⩽ \frac{π}{2 α} . \end{array}$ Hence, (3.4) and (3.8) show that $m = {inf}_{M} I > 0$ , as stated. □

Lemma 3.2.

Assume that (G1) and (G2) hold. If $v_{n} ⇀ \bar{v}$ in $H^{1} (R^{2})$ , then $\begin{matrix} (3.9) & \int_{R^{2}} g (f (v_{n})) f (v_{n}) d x = \int_{R^{2}} g (f (\bar{v})) f (\bar{v}) d x + \int_{R^{2}} g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) d x + o (1) \end{matrix}$ and $\begin{matrix} (3.10) & \int_{R^{2}} G (f (v_{n})) d x = \int_{R^{2}} G (f (\bar{v})) d x + \int_{R^{2}} G (f (v_{n}) - f (\bar{v})) d x + o (1) . \end{matrix}$

Proof.

Let us first prove (3.9). Set $w_{n} = v_{n} - \bar{v}$ . Then $w_{n} ⇀ 0$ in $H^{1} (R^{2})$ , $w_{n} \to 0$ in $L_{loc}^{s} (R^{2})$ for $2 ⩽ s < \infty$ and $w_{n} \to 0$ a.e. on $R^{2}$ . Moreover, there exists a constant $C_{4} > 0$ such that $‖ w_{n} ‖ + ‖ v_{n} ‖ ⩽ C_{4}$ . For any $a > 0$ , let us define $\begin{matrix} A_{n}^{a} = {x \in R^{2} : | w_{n} | ⩽ a} and B_{n}^{a} = R^{2} ∖ A_{n}^{a} . \end{matrix}$ Since ${w_{n}}$ is bounded in $H^{1} (R^{2})$ , then there exists a constant $C_{5} > 0$ such that $\begin{matrix} (3.11) & | B_{n}^{a} | ⩽ \frac{1}{a^{2}} \int_{B_{n}^{a}} | w_{n} |^{2} d x ⩽ \frac{C_{5}}{a^{2}} \to 0 as a \to \infty . \end{matrix}$ Let $\begin{matrix} h (t) : = \{\begin{matrix} g (t) / t, & t \neq 0, \\ 0, & t = 0 . \end{matrix} \end{matrix}$ Then (G1) and (G2) imply that $h \in C (R, R)$ and there exists a constant $C_{6} > 0$ such that $\begin{matrix} (3.12) & | h (f (t)) | ⩽ 1 + C_{6} [exp (α {| f (v) |}^{4}) - 1] ⩽ 1 + C_{6} [exp (2 α v^{2}) - 1], \forall t \in R . \end{matrix}$ Let $α \in (0, π / 2 C_{4}^{2})$ . For any measurable set $Ω \subset R^{2}$ , Hölder inequality and Trudinger–Moser inequality (1.3) give $\begin{array}{l} (3.13) & \begin{matrix} \int_{Ω} [exp (2 α v_{n}^{2}) - 1] w_{n}^{2} d x & ⩽ {\int_{R^{2}} {[exp (2 α v_{n}^{2}) - 1]}^{2} d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}} \\ ⩽ {\int_{R^{2}} [exp (4 α v_{n}^{2}) - 1] d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}} \\ = {\int_{R^{2}} [exp (4 α ‖ v_{n} ‖^{2} {| \frac{v_{n}}{‖ v_{n} ‖} |}^{2}) - 1] d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}} \\ ⩽ C_{7} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}}, \end{matrix} \\ (3.14) & \begin{matrix} \int_{Ω} [exp (2 α w_{n}^{2}) - 1] w_{n}^{2} d x & ⩽ {\int_{R^{2}} {[exp (2 α w_{n}^{2}) - 1]}^{2} d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}} \\ ⩽ {\int_{R^{2}} [exp (4 α w_{n}^{2}) - 1] d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}} \\ = {\int_{R^{2}} [exp (4 α ‖ w_{n} ‖^{2} {| \frac{w_{n}}{‖ w_{n} ‖} |}^{2}) - 1] d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}} \\ ⩽ C_{7} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{2}}, \end{matrix} \\ \int_{Ω} [exp (2 α v_{n}^{2}) - 1] | w_{n} | | \bar{v} | d x \\ ⩽ {\int_{R^{2}} {[exp (2 α v_{n}^{2}) - 1]}^{2} d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{4}} {(\int_{Ω} {\bar{v}}^{4} d x)}^{\frac{1}{4}} \\ ⩽ {\int_{R^{2}} [exp (4 α v_{n}^{2}) - 1] d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{4}} {(\int_{Ω} {\bar{v}}^{4} d x)}^{\frac{1}{4}} \\ = {\int_{R^{2}} [exp (4 α ‖ v_{n} ‖^{2} {| \frac{v_{n}}{‖ v_{n} ‖} |}^{2}) - 1] d x}^{\frac{1}{2}} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{4}} {(\int_{Ω} {\bar{v}}^{4} d x)}^{\frac{1}{4}} \\ (3.15) & ⩽ C_{7} {(\int_{Ω} w_{n}^{4} d x)}^{\frac{1}{4}} {(\int_{Ω} {\bar{v}}^{4} d x)}^{\frac{1}{4}} \end{array}$ and $\begin{matrix} (3.16) & \int_{Ω} [exp (2 α v_{n}^{2}) - 1] {\bar{v}}^{2} d x ⩽ C_{7} {(\int_{Ω} {\bar{v}}^{4} d x)}^{\frac{1}{2}} . \end{matrix}$ Note that $\begin{array}{l} g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) \\ = g (f (w_{n} + \bar{v})) f (w_{n} + \bar{v}) - g (f (w_{n})) f (w_{n}) \\ (3.17) & = [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) + h (f (w_{n} + \bar{v})) [f^{2} (w_{n} + \bar{v}) - f^{2} (w_{n})] \end{array}$ By (3.12), (3.13), (3.14), (f2), Hölder inequality and Sobolev inequality, we have $\begin{array}{l} | \int_{B_{n}^{a}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x | \\ ⩽ \int_{B_{n}^{a}} {2 + C_{6} [exp (2 α {(w_{n} + \bar{v})}^{2}) - 1] + C_{6} [exp (2 α w_{n}^{2}) - 1]} w_{n}^{2} d x \\ ⩽ 2 {| B_{n}^{a} |}^{\frac{1}{2}} ‖ w_{n} ‖_{4}^{2} + C_{8} {| B_{n}^{a} |}^{\frac{1}{4}} ‖ w_{n} ‖_{8}^{2} \\ (3.18) & ⩽ C_{9} ({| B_{n}^{a} |}^{\frac{1}{2}} + {| B_{n}^{a} |}^{\frac{1}{4}}) . \end{array}$ Then (3.11) and (3.18) imply that for any $ε > 0$ , there exists $\hat{a} > 0$ such that $\begin{matrix} (3.19) & | \int_{B_{n}^{a}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x | ⩽ ε, \forall n \in N . \end{matrix}$ By the uniformly continuity of h on $[- \hat{a}, \hat{a}]$ , there exists $δ > 0$ such that $\begin{matrix} (3.20) & | h (t + v) - h (t) | < ε for all (t, v) \in [- \hat{a}, \hat{a}] \times [- δ, δ], \end{matrix}$ Since $| f (t + \bar{v}) - f (t) | ⩽ {max}_{ξ \in [0, 1]} f^{'} (t + ξ \bar{v}) | \bar{v} | ⩽ | \bar{v} |$ by (f1), (3.20) gives $\begin{matrix} (3.21) & | h (f (t + \bar{v})) - h (f (t)) | < ε for all (t, \bar{v}) \in [- \hat{a}, \hat{a}] \times [- δ, δ] . \end{matrix}$ Set $\begin{matrix} C^{δ} = {x \in R^{2} : | \bar{v} (x) | ⩽ δ}, D^{δ} = R^{2} ∖ C^{δ} . \end{matrix}$ Then $\begin{matrix} (3.22) & | D^{δ} | ⩽ \frac{1}{δ^{2}} \int_{D^{δ}} | \bar{v} |^{2} d x ⩽ \frac{‖ \bar{v} ‖_{2}^{2}}{δ^{2}} . \end{matrix}$ From (3.12), (3.13), (3.14), (3.21), (3.22), (f2), Hölder inequality and Sobolev inequality, we have $\begin{array}{l} | \int_{A_{n}^{\hat{a}}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x | \\ = | \int_{A_{n}^{\hat{a}} \cap C^{δ}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x \\ + \int_{A_{n}^{\hat{a}} \cap D^{δ}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x | \\ ⩽ ε \int_{A_{n}^{\hat{a}} \cap C^{δ}} w_{n}^{2} d x + \int_{A_{n}^{\hat{a}} \cap D^{δ}} {2 + C_{6} [exp (2 α {(w_{n} + \bar{v})}^{2}) - 1] + C_{6} [exp (2 α w_{n}^{2}) - 1]} w_{n}^{2} d x \\ ⩽ ε ‖ w_{n} ‖_{2}^{2} + 2 \int_{D^{δ}} | w_{n} |^{2} d x + 2 C_{6} C_{7} {(\int_{D^{δ}} w_{n}^{4} d x)}^{\frac{1}{2}} \\ ⩽ C_{4}^{2} ε + 2 \int_{D^{δ}} | w_{n} |^{2} d x + 2 C_{6} C_{7} {(\int_{D^{δ}} w_{n}^{4} d x)}^{\frac{1}{2}}, \end{array}$ which, together with $w_{n} ⇀ 0$ in $H^{1} (R^{2})$ , implies $\begin{matrix} (3.23) & | \int_{A_{n}^{\hat{a}}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x | ⩽ C_{4}^{2} ε + o (1) as n \to \infty . \end{matrix}$ By (3.12), (3.15), (3.16), (f1), (f2), Hölder inequality and Sobolev inequality, we have $\begin{array}{l} | \int_{R^{2} ∖ B_{R} (0)} h (f (w_{n} + \bar{v})) [f^{2} (w_{n} + \bar{v}) - f^{2} (w_{n})] d x | \\ ⩽ | \int_{R^{2} ∖ B_{R} (0)} 2 | h (f (w_{n} + \bar{v})) | max_{ξ \in [0, 1]} [f (w_{n} + ξ \bar{v}) f^{'} (w_{n} + ξ \bar{v})] | \bar{v} | d x | \\ ⩽ 2 \int_{R^{2} ∖ B_{R} (0)} | h (f (w_{n} + \bar{v})) | (| w_{n} | + | \bar{v} |) | \bar{v} | d x \\ ⩽ 2 \int_{R^{2} ∖ B_{R} (0)} {1 + C_{6} [exp (2 α {(w_{n} + \bar{v})}^{2}) - 1]} (| w_{n} | + | \bar{v} |) | \bar{v} | d x \\ ⩽ 2 ‖ w_{n} ‖_{2} {(\int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{2} d x)}^{\frac{1}{2}} + 2 \int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{2} d x \\ + 2 C_{6} C_{7} ‖ w_{n} ‖_{4} {(\int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{4} d x)}^{\frac{1}{4}} + 2 C_{6} C_{7} {(\int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{4} d x)}^{\frac{1}{2}} \\ ⩽ C_{10} [{(\int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{2} d x)}^{\frac{1}{2}} + \int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{2} d x + {(\int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{4} d x)}^{\frac{1}{4}} \\ (3.24) & + {(\int_{R^{2} ∖ B_{R} (0)} {\bar{v}}^{4} d x)}^{\frac{1}{2}}] . \end{array}$ From this and the decay of the integral, there exists $\bar{R} > 0$ such that $\begin{matrix} (3.25) & | \int_{R^{2} ∖ B_{\bar{R}} (0)} h (f (w_{n} + \bar{v})) [f^{2} (w_{n} + \bar{v}) - f^{2} (w_{n})] d x | ⩽ ε, \forall n \in N \end{matrix}$ and $\begin{matrix} (3.26) & | \int_{R^{2} ∖ B_{\bar{R}} (0)} g (f (\bar{v})) f (\bar{v}) d x | ⩽ ε . \end{matrix}$ Since $v_{n} \to \bar{v}$ in $L^{2} (B_{\bar{R}} (0)) \cap L^{p} (B_{\bar{R}} (0))$ , then [26, Lemma A.1] implies that there exists $w \in L^{2} (B_{\bar{R}} (0)) \cap L^{p} (B_{\bar{R}} (0))$ such that, along a subsequence, $\begin{matrix} (3.27) & | \bar{v} (x) | ⩽ w (x) and | v_{n} (x) | ⩽ w (x), a.e. x \in B_{\bar{R}} (0) . \end{matrix}$ Then by (G1), (G2) and (3.27), we have $\begin{array}{l} | g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) - g (f (\bar{v})) f (\bar{v}) | \\ ⩽ 3 (| v_{n} |^{2} + | \bar{v} |^{2}) + C_{11} {[exp (2 α v_{n}^{2}) - 1] + [exp (2 α (v_{n}^{2} + {\bar{v}}^{2})) - 1] + [exp (2 α {\bar{v}}^{2}) - 1]} \\ (3.28) & ⩽ 6 w^{2} + 3 C_{11} [exp (4 α w^{2}) - 1], a.e. x \in B_{\bar{R}} (0) . \end{array}$ Since $w^{2}, exp (4 α w^{2}) - 1 \in L^{1} (B_{\bar{R}} (0))$ and $\begin{matrix} g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v}) (f (v_{n} - \bar{v}) - g (f (\bar{v})) f (\bar{v}) \to 0, a.e. x \in B_{\bar{R}} (0), \end{matrix}$ then it follows from (3.28) and the Lebesgue dominated convergence theorem that $\begin{matrix} (3.29) & \int_{B_{\bar{R}} (0)} [g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) - g (f (\bar{v})) f (\bar{v})] d x = o (1) . \end{matrix}$ Note that $\begin{array}{l} | \int_{R^{2}} [g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) - g (f (\bar{v})) f (\bar{v})] d x | \\ ⩽ | \int_{B_{\bar{R}} (0)} [g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) - g (f (\bar{v})) f (\bar{v})] d x | \\ + | \int_{R^{2} ∖ B_{\bar{R}} (0)} g (f (\bar{v})) f (\bar{v}) d x | \\ + | \int_{R^{2} ∖ B_{\bar{R}} (0)} [g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v})] d x | \\ ⩽ | \int_{B_{\bar{R}} (0)} [g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) - g (f (\bar{v})) f (\bar{v})] d x | \\ + | \int_{A_{n}^{\hat{a}}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x | \\ + | \int_{B_{n}^{\hat{a}}} [h (f (w_{n} + \bar{v})) - h (f (w_{n}))] f^{2} (w_{n}) d x | \\ + | \int_{R^{2} ∖ B_{\bar{R}} (0)} h (f (w_{n} + \bar{v})) [f^{2} (w_{n} + \bar{v}) - f^{2} (w_{n})] d x | \\ (3.30) & + | \int_{R^{2} ∖ B_{\bar{R}} (0)} g (f (\bar{v})) f (\bar{v}) d x | . \end{array}$ Then it follows from (3.19), (3.23), (3.25), (3.26), (3.29) and (3.30) that $\begin{matrix} | \int_{R^{2}} [g (f (v_{n})) f (v_{n}) - g (f (v_{n} - \bar{v})) f (v_{n} - \bar{v}) - g (f (\bar{v})) f (\bar{v})] d x | ⩽ (C_{4}^{2} + 3) ε + o (1), \end{matrix}$ Since $ε > 0$ is arbitrary, then (3.9) follows from the above inequality. Since $G \in C^{1} (R, R)$ , the proof is standard with Trudinger–Moser inequality (1.3). □

Remark that since $V (x) \equiv V_{\infty}$ satisfies (V1) and (V2), the conclusions for I in this section are true for $I^{\infty}$ .

Lemma 3.3.

Assume that (V1), (V2) and (G1)–(G4) hold. Then $m^{\infty} ⩾ m$ .

Proof.

By Lemma 2.5, we have $M^{\infty} \neq \emptyset$ . Arguing by contradiction, we assume that $m > m^{\infty}$ . Let $ε : = m - m^{\infty}$ . Then there exists $v_{ε}^{\infty}$ such that $\begin{matrix} (3.31) & v_{ε}^{\infty} \in M^{\infty} and m^{\infty} + \frac{ε}{2} > I^{\infty} (v_{ε}^{\infty}) . \end{matrix}$ In view of Lemma 2.5, there exists $t_{ε} > 0$ such that ${(v_{ε}^{\infty})}_{t_{ε}} \in M$ . Since $V_{\infty} ⩾ V (x)$ for all $x \in R^{2}$ , it follows from (1.7), (1.14), (3.31) and Corollary 2.4 that $\begin{matrix} m^{\infty} + \frac{ε}{2} > I^{\infty} (v_{ε}^{\infty}) ⩾ I^{\infty} ({(v_{ε}^{\infty})}_{t_{ε}}) ⩾ I ({(v_{ε}^{\infty})}_{t_{ε}}) ⩾ m = m^{\infty} + ε . \end{matrix}$ This contradiction shows that $m^{\infty} ⩾ m$ . The proof is completed. □

Lemma 3.4.

Assume that (V1), (V2) and (G1)–(G4) hold. Then m is achieved.

Proof.

In view of Lemmas 2.5 and 3.1, we have $M \neq \emptyset$ and $m > 0$ . Let ${v_{n}} \subset M$ be such that $I (v_{n}) \to m$ . Since $J (v_{n}) = 0$ , then it follows from (1.7), (1.12), (2.7), (2.8), (3.2) and (3.3) that $\begin{array}{rcl} m + o (1) & = & I (v_{n}) - \frac{1}{2 + p} J (v_{n}) \\ (3.32) & ⩾ & \frac{p - 2}{2 (2 + p)} ‖ \nabla v_{n} ‖_{2}^{2} + \frac{1}{2 + p} {‖ \nabla f (v_{n}) ‖}_{2}^{2} + \frac{ρ_{1}}{2 (2 + p)} {‖ f (v_{n}) ‖}_{2}^{2} . \end{array}$ This shows that ${‖ \nabla v_{n} ‖_{2}}$ , ${‖ f (v_{n}) ‖_{2}}$ and ${‖ \nabla f (v_{n}) ‖_{2}}$ are bounded. Then ${‖ f (v_{n}) ‖}$ and ${‖ f (v_{n}) ‖_{4}}$ are bounded. By (f6), then $\begin{array}{l} \int_{R^{2}} {| f (v_{n}) |}^{4} d x ⩾ \int_{| v_{n} | ⩾ 1} {| f (v_{n}) |}^{4} d x ⩾ θ \int_{| v_{n} | ⩾ 1} v_{n}^{2} d x \\ \int_{R^{2}} {| f (v_{n}) |}^{2} d x ⩾ \int_{| v_{n} | ⩽ 1} {| f (v_{n}) |}^{2} d x ⩾ θ \int_{| v_{n} | ⩽ 1} v_{n}^{2} d x \end{array}$ which implies ${‖ v_{n} ‖_{2}}$ is bounded. Therefore ${‖ v_{n} ‖}$ is bounded in $H^{1} (R^{2})$ . Moreover, there exists a constant $C > 0$ such that $‖ v_{n} ‖ ⩽ C$ . Passing to a subsequence, we have $v_{n} ⇀ \bar{v}$ in $H^{1} (R^{2})$ , $v_{n} \to \bar{v}$ in $L_{loc}^{s} (R^{2})$ for $2 ⩽ s < \infty$ and $v_{n} \to \bar{v}$ a.e. on $R^{2}$ . There are two possible cases: 1) $\bar{v} = 0$ and 2) $\bar{v} \neq 0$ .

Case 1). $\bar{v} = 0$ , i.e. $v_{n} ⇀ 0$ in $H^{1} (R^{2})$ . Then $v_{n} \to 0$ in $L_{loc}^{s} (R^{2})$ for $2 ⩽ s < \infty$ and $v_{n} \to 0$ a.e. on $R^{2}$ . By (V1) and (V2), it is easy to show that $\begin{matrix} (3.33) & lim_{n \to \infty} \int_{R^{2}} [V_{\infty} - V (x)] f^{2} (v_{n}) d x = lim_{n \to \infty} \int_{R^{2}} (\nabla V (x) \cdot x) f^{2} (v_{n}) d x = 0 . \end{matrix}$ By (1.7), (1.12), (1.14), (1.15) and (3.33), one can get $\begin{matrix} (3.34) & I^{\infty} (v_{n}) \to m, J^{\infty} (v_{n}) \to 0 . \end{matrix}$ Set $α_{2} \in (0, π / 2 C^{2})$ . By (1.17), (1.18), (3.1), (f2), (f4), Hölder inequality, Trudinger–Moser inequality (1.3) and Sobolev inequality, we have $\begin{array}{l} \int_{R^{2}} [g (f (v_{n})) f (v_{n}) + 2 G (f (v_{n}))] d x \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v_{n}) ‖}_{2}^{2} + C_{1} \int_{R^{2}} {| f (v_{n}) |}^{3} (exp (α_{2} {| f (v_{n}) |}^{4}) - 1) d x \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v_{n}) ‖}_{2}^{2} + C_{1} \int_{R^{2}} (exp (2 α_{2} v_{n}^{2}) - 1) | v_{n} |^{3} d x \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v_{n}) ‖}_{2}^{2} + C_{1} {[\int_{R^{2}} (exp (4 α_{2} v_{n}^{2}) - 1) d x]}^{\frac{1}{2}} {(\int_{R^{2}} | v_{n} |^{6} d x)}^{\frac{1}{2}} \\ ⩽ \frac{ρ_{2}}{4} {‖ f (v_{n}) ‖}_{2}^{2} + C_{12} ‖ v_{n} ‖_{6}^{3} \\ (3.35) & ⩽ \frac{ρ_{2}}{4} {‖ f (v_{n}) ‖}_{2}^{2} + o (1), \end{array}$ where $ρ_{2} > 0$ is given by (2.8). Since $J (v_{n}) = 0$ , it follows from (1.12), (2.8) and (3.35) that $\begin{array}{l} ‖ \nabla v_{n} ‖_{2}^{2} + \frac{ρ_{2}}{2} {‖ f (v_{n}) ‖}_{2}^{2} ⩽ & ‖ \nabla v_{n} ‖_{2}^{2} + \int_{R^{2}} \frac{2 f^{2} (v_{n})}{1 + 2 f^{2} (v_{n})} | \nabla v_{n} |^{2} d x \\ + \frac{1}{2} \int_{R^{2}} [4 V (x) + \nabla V (x) \cdot x] f^{2} (v_{n}) d x \\ = & \int_{R^{2}} [g (f (v_{n})) f (v_{n}) + 2 G (f (v_{n}))] d x \\ ⩽ & \frac{ρ_{2}}{4} {‖ f (v_{n}) ‖}_{2}^{2} + o (1), \end{array}$ which implies $\begin{matrix} (3.36) & ‖ \nabla v_{n} ‖_{2}^{2} \to 0 and {‖ f (v_{n}) ‖}_{2}^{2} \to 0 . \end{matrix}$ Since $v_{n} \in M$ and $I (v_{n}) \to m$ , then (G1), (G2), (3.35), (3.36) and Lemma 3.1 give $\begin{matrix} 0 < m + o (1) = I^{\infty} (v_{n}) = \frac{1}{2} (‖ \nabla v_{n} ‖_{2}^{2} + V_{\infty} {‖ f (v_{n}) ‖}_{2}^{2}) - \int_{R^{2}} G (f (v_{n})) d x = o (1), \end{matrix}$ which is a obvious contradiction. Using Lions’ concentration compactness principle [26, Lemma 1.21], we can prove that there exist $δ > 0$ and ${y_{n}} \subset R^{2}$ such that $\int_{B_{1} (y_{n})} | v_{n} |^{2} d x > δ$ . Let ${\hat{v}}_{n} (x) = v_{n} (x + y_{n})$ . Then $‖ {\hat{v}}_{n} ‖ = ‖ v_{n} ‖$ , and by (3.34), one has $\begin{matrix} (3.37) & J^{\infty} ({\hat{v}}_{n}) = o (1), I^{\infty} ({\hat{v}}_{n}) \to m, \int_{B_{1} (0)} | {\hat{v}}_{n} |^{2} d x > δ > 0 . \end{matrix}$ Therefore, there exists $\hat{v} \in H^{1} (R^{2}) ∖ {0}$ such that, passing to a subsequence, $\begin{matrix} (3.38) & \{\begin{matrix} {\hat{v}}_{n} ⇀ \hat{v}, & in H^{1} (R^{2}); \\ {\hat{v}}_{n} \to \hat{v}, & in L_{loc}^{s} (R^{2}), \forall s \in [2, \infty); \\ {\hat{v}}_{n} \to \hat{v}, & a.e. on R^{2} . \end{matrix} \end{matrix}$ Let $w_{n} = {\hat{v}}_{n} - \hat{v}$ . Then (3.38) and Lemma 3.2 yield $\begin{matrix} (3.39) & I^{\infty} ({\hat{v}}_{n}) = I^{\infty} (\hat{v}) + I^{\infty} (w_{n}) + o (1), J^{\infty} ({\hat{v}}_{n}) = J^{\infty} (\hat{v}) + J^{\infty} (w_{n}) + o (1) . \end{matrix}$ Let $\begin{matrix} (3.40) & Ψ^{\infty} (v) : = I^{\infty} (v) - \frac{1}{p + 2} J^{\infty} (v), \forall v \in H^{1} (R^{2}) . \end{matrix}$ From (1.7), (1.12), (3.37), (3.39) and (3.40), we have $\begin{matrix} (3.41) & Ψ^{\infty} (w_{n}) = m - Ψ^{\infty} (\hat{v}) + o (1), J^{\infty} (w_{n}) = - J^{\infty} (\hat{v}) + o (1) . \end{matrix}$ If there exists a subsequence ${w_{n_{i}}}$ of ${w_{n}}$ such that $w_{n_{i}} = 0$ , then we have $\begin{matrix} (3.42) & I^{\infty} (\hat{v}) = m, J^{\infty} (\hat{v}) = 0 . \end{matrix}$ Next, we assume that $w_{n} \neq 0$ . We claim that $J^{\infty} (\hat{v}) ⩽ 0$ . Otherwise, if $J^{\infty} (\hat{v}) > 0$ , then (3.41) implies $J^{\infty} (w_{n}) < 0$ for large n. Applying Lemma 2.5 to $I^{\infty}$ , there exists $t_{n} > 0$ such that ${(w_{n})}_{t_{n}} \in M^{\infty}$ for large n. Applying Lemma 2.3 to $I^{\infty}$ , from (1.14), (1.15), (2.1), (3.40), (3.41) and Lemma 3.3, we derive $\begin{array}{l} m - Ψ^{\infty} (\hat{v}) + o (1) & = Ψ^{\infty} (w_{n}) = I^{\infty} (w_{n}) - \frac{1}{p + 2} J^{\infty} (w_{n}) \\ ⩾ I^{\infty} ({(w_{n})}_{t_{n}}) - \frac{t_{n}^{p + 2}}{p + 2} J^{\infty} (w_{n}) \\ ⩾ m^{\infty} ⩾ m, \end{array}$ which is a contradiction due to $Ψ^{\infty} (\hat{v}) > 0$ . This shows that $J^{\infty} (\hat{v}) ⩽ 0$ . Applying Lemmas 2.3 and 2.5 to $I^{\infty}$ , there exists $t_{\infty} > 0$ such that ${\hat{v}}_{t_{\infty}} \in M^{\infty}$ , moreover, it follows from (1.14), (1.15), (2.1), (2.4), (3.37), (3.40), Fatou’s lemma and Lemma 3.3 that $\begin{array}{l} m & = lim_{n \to \infty} [I^{\infty} ({\hat{v}}_{n}) - \frac{1}{p + 2} J^{\infty} ({\hat{v}}_{n})] ⩾ I^{\infty} (\hat{v}) - \frac{1}{p + 2} J^{\infty} (\hat{v}) \\ ⩾ I^{\infty} ({\hat{v}}_{t_{\infty}}) - \frac{t_{\infty}^{p + 2}}{p + 2} J^{\infty} (\hat{v}) ⩾ m^{\infty} ⩾ m, \end{array}$ which implies (3.42) holds also. In view of Lemma 2.5, there exists $\hat{t} > 0$ such that ${\hat{v}}_{\hat{t}} \in M$ . Applying Corollary 2.4 to $I^{\infty}$ , we deduce from (V1), (1.7), (1.14) and (3.42) that $\begin{matrix} m ⩽ I ({\hat{v}}_{\hat{t}}) ⩽ I^{\infty} ({\hat{v}}_{\hat{t}}) ⩽ I^{\infty} (\hat{v}) = m . \end{matrix}$ This shows that m is achieved at ${\hat{v}}_{\hat{t}} \in M$ .

Case 2). $\bar{v} \neq 0$ . In this case, the proof is similar to the previous section. □

Lemma 3.5.

Assume that (V1), (V2) and (G1)–(G4) hold. If $\bar{v} \in M$ and $I (\bar{v}) = m$ , then $\bar{v}$ is a critical point of I.

Proof.

From (G1), (G3) and (1.12), we can deduce that there exist $T_{1} \in (0, 1)$ and $T_{2} \in (1, \infty)$ such that $\begin{matrix} (3.43) & J ({\bar{v}}_{T_{1}}) > 0, J ({\bar{v}}_{T_{2}}) < 0 . \end{matrix}$ Similar to the proof of [23,24], we can prove this lemma by using $\begin{array}{rcl} (3.44) & I ({\bar{v}}_{t}) ⩽ I (\bar{v}) - \frac{1}{2} H (t, \bar{v}) = m - \frac{1}{2} H (t, \bar{v}), \forall t > 0 . \end{array}$ and $\begin{matrix} ε : = min {\frac{1}{4} H (T_{1}, \bar{v}), \frac{1}{4} H (T_{2}, \bar{v}), 1, \frac{ϱ δ}{8}} . \end{matrix}$ □

Proof of Theorem 1.1.

In view of Lemmas 2.6, 3.4 and 3.5, there exists $\bar{v} \in M$ such that $\begin{matrix} I (\bar{v}) = m = inf_{v \in E ∖ {0}} max_{t > 0} I (v_{t}), I^{'} (\bar{v}) = 0 . \end{matrix}$ This shows that v is a nontrivial solution of (1.5) such that $I (\bar{v}) = {inf}_{M} I > 0$ . □

Footnotes

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (No: 11971485, No: 12001542), Hunan Provincial Natural Science Foundation (No: 2021JJ40703), the Fundamental Research Funds for the Central Universities of Central South University (No: 2021zzts0050).

References

Ambrosetti,

Badiale and

Cingolani, Semiclassical states of nonlinears Schrödinger equations, Arch. Ration. Mech. Anal. 40 (1997), 285–300. doi:10.1007/s002050050067.

Bartsch and

Z.Q.

Wang, Existence and multiplicity results for some superlinear elliptic problems on

R^{N}

, Comm. Partial Differential Equations 20 (1995), 1725–1741. doi:10.1080/03605309508821149.

Berestycki,

Gallouët and

Kavian, Équation de champs scalaires edclidiens non linéaires dans le plan, C.R. Acad. Sci. Paris Sér. I Math. 297 (1983), 307–310.

H.S.

Brandi,

Manus,

Mainfray,

Lehner and

Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids B 5 (1993), 3539–3550. doi:10.1063/1.860828.

D.M.

Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in

R^{2}

, Comm. Partial Differential Equations 17 (1992), 407–435. doi:10.1080/03605309208820848.

S.T.

Chen,

V.D.

Rădulescu,

X.H.

Tang and

B.L.

Zhang, Ground state solutions for quasilinear Schrödinger equation with variable potential and superlinear reaction, Rev. Mat. Iberoam. 36 (2020), 1549–1570. doi:10.4171/rmi/1175.

Colin and

Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. 56 (2004), 213–226. doi:10.1016/j.na.2003.09.008.

J.M.

do Ó, N-Laplacian equations in

R^{N}

with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315. doi:10.1155/S1085337597000419.

J.M.

do Ó,

O.H.

Miyagaki and

S.H.

Soares, Soliton solutions for quasilinear Schrödinger equation: The critical exponential case, Nonlinear Anal. 67 (2007), 3357–3372. doi:10.1016/j.na.2006.10.018.

10.

J.M.

do Ó and

U.B.

Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations 38 (2010), 275–315. doi:10.1007/s00526-009-0286-6.

11.

Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan. 50 (1981), 3262–3267. doi:10.1143/JPSJ.50.3262.

12.

Laedke,

Spatschek and

Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983), 2764–2769. doi:10.1063/1.525675.

13.

A.G.

Litvak and

A.M.

Sergeev, One-dimensional collapse of plasma waves, JETP Lett. 27 (1978), 517–520.

14.

J.Q.

Liu,

Y.Q.

Wang and

Z.Q.

Wang, Soliton solutions for quasilinear Schrödinger equations, II, J. Differential Equations 187 (2003), 473–493. doi:10.1016/S0022-0396(02)00064-5.

15.

J.Q.

Liu,

Y.Q.

Wang and

Z.Q.

Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–901. doi:10.1081/PDE-120037335.

16.

J.Q.

Liu and

Z.Q.

Wang, Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc. 131 (2003), 441–448. doi:10.1090/S0002-9939-02-06783-7.

17.

V.G.

Makhankov and

V.K.

Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. 104 (1984), 1–86. doi:10.1016/0370-1573(84)90106-6.

18.

Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092. doi:10.1512/iumj.1971.20.20101.

19.

Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Japan. 42 (1977), 1824–1835. doi:10.1143/JPSJ.42.1824.

20.

Poppenberg,

Schmitt and

Z.Q.

Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329–344. doi:10.1007/s005260100105.

21.

Ruiz and

Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity 23 (2010), 1221–1233. doi:10.1088/0951-7715/23/5/011.

22.

U.B.

Severo and

G.M.

de Carvalho, Quasilinear Schrödinger equations with singular and vanishing potentials involving exponential critical growth in

R^{2}

, Nonlinear Anal. 196 (2020), 111800. doi:10.1016/j.na.2020.111800.

23.

X.H.

Tang and

S.T.

Chen, Ground state solutions of Nehari–Pohoz˘aev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations 56 (2017), 110. doi:10.1007/s00526-017-1214-9.

24.

X.H.

Tang and

S.T.

Chen, Ground state solutions of Nehari–Pohoz˘aev type for Schrödinger–Poisson problems with general potentials, Discrete Contin. Dyn. Syst. 37 (2017), 4973–5002. doi:10.3934/dcds.2017214.

25.

N.S.

Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.

26.

Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24, Birkhäuser Boston Inc., Boston, MA, 1996.

27.

Zhang,

X.H.

Tang and

Zhang, Infinitely many soutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl. 420 (2014), 1762–1775. doi:10.1016/j.jmaa.2014.06.055.