Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials

Caifeng Zhang; Maochun Zhu

doi:10.1515/ans-2023-0136

Article Open Access

Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials

Caifeng Zhang and Maochun Zhu

Published/Copyright: May 1, 2024

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From the journal Advanced Nonlinear Studies Volume 24 Issue 3

Abstract

The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint:

(0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x < + ∞ ,

where 0 < β < 2 , V ( x ) ≥ 0 and may vanish on an open set in R 2 . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 :

(0.2) − Δ u + γ u = f ( u ) | x | β ,

where the nonlinearity f has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, “Concentration-compactness principle of singular Trudinger-Moser inequalities in R n and n -Laplace equations,” Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 :

(0.3) − Δ u + V ( x ) u = f ( u ) | x | β .

Keywords: Trudinger–Moser inequalities; degenerate potential; Nehari manifold; ground states

2010 MSC: 35J10; 35J91; 46E35; 26D10

1 Introduction and main results

Let Ω be a smooth bounded domain. The classical Sobolev embedding theorems state that W 0 1 , p ( Ω ) ⊂ L q ( Ω ) for 1 ≤ q ≤ p * and p < n , where p * = n p n − p is called the Sobolev exponent. However, in the limiting case p = n , some examples show that W 0 1 , n ( Ω ) ⊈ L ∞ ( Ω ) . In this case, Trudinger’s inequality serves as an appropriate replacement. Trudinger’s inequality was first established by Trudinger [1]. More precisely, he proved when p = n there exists a constant α > 0 such that the following inequality holds (see also Pohozaev [2] and Yudovic [3]):

(1.1) sup u ∈ W 0 1 , n ( Ω ) , ‖ ∇ u ‖ n ≤ 1 1 | Ω | ∫ Ω e α | u | n n − 1 d x < ∞ .

Nevertheless, the best constant α in (1.1) is unknown. A sharp version of inequality (1.1) was given by Moser [4].

(1.2) sup u ∈ W 0 1 , n ( Ω ) , ‖ ∇ u ‖ n ≤ 1 1 | Ω | ∫ Ω e α | u | n n − 1 d x < ∞ , iff α ≤ α n ≔ n ω n − 1 1 n − 1 ,

where ω n − 1 denotes the area of the surface of the unit ball in R n . Now, the inequality (1.2) are called Trudinger–Moser inequalities. There are many extensions of Trudinger–Moser inequalities. One of the important extensions is to construct Trudinger–Moser inequalities in the whole Euclidean space. Related inequalities for the whole Euclidean space have been considered by Cao in [5] in the case n = 2 and for any dimension by do Ó [6] and Adachi and Tanaka [7] in the subcritical case, that is α < α n . When it comes to the critical case α = α n , Ruf [8] (in the case n = 2 ) and Li and Ruf [9] ( n ≥ 3 ) showed that if the Dirichlet norm is replaced by the Sobolev norm, i.e. ‖ u ‖ W 1 , n ( R n ) = ∫ R n | ∇ u | n + | u | n d x 1 n , then there holds

(1.3) sup u ∈ W 1 , n ( R n ) , ‖ u ‖ W 1 , n ( R n ) ≤ 1 ∫ R n Φ n α | u ( x ) | n n − 1 d x < ∞ iff α ≤ α n ,

where Φ n ( t ) ≔ e t − ∑ j = 0 n − 2 t j j ! .

All the proofs given above depend strictly on the Pólya-Szegö inequality and the symmetrization argument. A symmetrization-free argument was developed by Lam and Lu in [10], [11]. Using this symmetrization-free argument, they proved the following critical singular Trudinger–Moser inequality in R n :

Theorem A.

([10]) Assume n ≥ 2 , 0 ≤ β < n and γ > 0 , then

(1.4) sup ‖ u ‖ γ ≤ 1 ∫ R n Φ n α 1 − β n | u ( x ) | n n − 1 | x | β d x < ∞ iff α ≤ α n , β ≔ α n 1 − β n ,

where ‖ u ‖ γ ≔ ∫ R n | ∇ u | n + γ | u | n d x 1 n .

The singular Trudinger–Moser inequalities on the entire Euclidean space have also been considered in the paper of Adimurthi and Yang [12]. When one restricts the Trudinger–Moser functionals on a function sequence { u k } k , the Concentration-Compactness principle associated with the Trudinger–Moser inequalities makes sense which was first established by [13]. The Concentration-Compactness principle associated with Trudinger–Moser inequality (1.3) was established by do Ó, de Souza and de Medeiros in [14]. Li, Lu and Zhu [15] obtained the Concentration-Compactness principle associated with Trudinger–Moser inequalities on the entire Heisenberg group by applying the symmetrization-free argument. Later, Chen and Zhang [16] generalized the Concentration-Compactness principle associated with Trudinger–Moser inequalities on the entire Euclidean Space to the singular case:

Theorem B.

([16]) Let { u k } k be a bounded sequence in H 1 ( R 2 ) such that ‖ u k ‖ γ = 1 and u k ⇀ u 0 ≢ 0 in H 1 ( R 2 ) . If

0 < p < 1 1 − ‖ u 0 ‖ γ 2 ,

then

sup k ∫ R 2 e 4 π p 1 − β 2 u k 2 − 1 | x | β d x < + ∞ .

For more results of Trudinger–Moser inequalities and the related Concentration-Compactness principle, one can refer to [14], [17], [18], [19], [20], [21], [22], [23], [24], [25] and the references therein.

If γ is replaced by the potential function V ( x ) , we can easily deduce inequality (1.4) when V ( x ) has a positive constant lower bound. However if V ( x ) vanishes on some open set of R 2 , classical methods such as symmetrization or blow-up analysis fail. Therefore, the problem becomes fairly complicated and there are few works devoted to it. In 2021, Chen, Lu and Zhu [26] developed a method combining a new imbedding theorem involving the degenerate potential which may vanish on some open set in R 2 (Lemma 2.1, [26]) to derive the following result:

Theorem C.

([26]) Assume that the potential V ( x ) ≥ 0 satisfies V ( x ) = 0 at the ball B δ ( 0 ) centered at the origin with the radius δ and V ( x ) ≥ c 0 in R 2 \ B 2 δ ( 0 ) for some δ > 0 . Then

(1.5) sup u ∈ H 1 ( R 2 ) , ‖ u ‖ V ≤ 1 ∫ R 2 e 4 π u 2 − 1 d x < ∞ ,

where ‖ u ‖ V = ∫ R 2 | ∇ u | 2 + V ( x ) | u | 2 d x 1 2 .

The first aim of this paper is to establish the singular version of Trudinger–Moser inequality (1.5).

Theorem 1.1.

Assume that the potential V ( x ) ≥ 0 satisfies V ( x ) = 0 at the ball B δ ( 0 ) centered at the origin with the radius δ and V ( x ) ≥ V 0 in R 2 \ B 2 δ ( 0 ) for some δ > 0 . Then

(1.6) sup u ∈ H 1 ( R 2 ) , ‖ u ‖ V ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x < + ∞ .

Furthermore, we study the Concentration-Compactness principle associated with Trudinger–Moser inequality (1.6).

Theorem 1.2.

Let { u k } k ⊆ H V ( R 2 ) such that ‖ u k ‖ V = 1 and u k ⇀ u 0 ≢ 0 in H V ( R 2 ) . For any

0 < p < 1 1 − ‖ u 0 ‖ V 2 ,

one has

sup k ∫ R 2 e 4 π p 1 − β 2 u k 2 − 1 | x | β d x < + ∞ .

Trudinger–Moser inequality (1.1) and the associated Concentration-Compactness principle play an important role in studying the existence of ground states to the following equations:

(1.7) − Δ u + V ( x ) u = f ( u ) | x | β ,

where 0 ≤ β < 2 and V ( x ) ≥ 0 is a degenerate potential satisfying:

( V 1 ) V ( x ) = 0 at B δ ( 0 ) and V ( x ) ≥ V 0 in R 2 \ B 2 δ ( 0 ) for some positive V 0 and δ .

( V 2 ) There holds

sup x ∈ R 2 V ( x ) = lim | x | → ∞ V ( x ) = γ > 0 .

The nonlinear term f ( t ) is continuous and satisfies the following conditions:

There exists some β 0 > 0 such that
lim | t | → ∞ f ( t ) e α 1 − β 2 t 2 = 0 , for α > β 0 , + ∞ , for α < β 0 .
There exists μ > 2 such that 0 < μ F ( t ) = μ ∫ 0 t f ( s ) d s ≤ t f ( t ) for any t ∈ R . This is the well known (A-R) condition.
There exist positive constants t 0 and M 0 such that F ( t ) ≤ M 0 | f ( t ) | when | t | ≥ t 0 .
f ( 0 ) = 0 and f ( t ) = o ( t ) for t sufficiently close to 0.
f ( t ) is a C 1 ( R ) function and f ( t ) t is strictly increasing in ( 0 , + ∞ ) , and decreasing in ( − ∞ , 0 ) .

In order to study the existence of ground states to (1.7), we first focus on the limit equation of (1.7):

(1.8) − Δ u + γ u = f ( u ) | x | β .

Our third main result comes as

Theorem 1.3.

Assume f ( t ) satisfies (i)–(v) and

(vi) lim inf t → + ∞ t f ( t ) e − ( 1 − β 2 ) β 0 t 2 = α 0 > M , where

M = inf r > 0 ( 2 − β ) 2 1 − β 2 β 0 r 2 − β e γ ( 2 − β ) 4 r 2 .

Then equation (1.8) admits a positive ground state solution.

By applying the Concentration-Compactness principle Theorem 1.2 and Theorem 1.3, we can derive

Theorem 1.4.

Assume V ( x ) satisfies ( V 1 ) and ( V 2 ) , f ( t ) satisfies (i)–(vi), then equation (1.7) admits a positive ground state solution.

As far as we know, there are many works devoted to the existence of a ground state solution to (1.7). If β = 0 and V ( x ) is coercive, i.e.

V ( x ) ≥ C 0 > 0 and additional either 1 V ∈ L 1 ( R 2 ) or lim | x | → + ∞ V ( x ) = + ∞ ,

one can easily get E = u ∈ H 1 ( R 2 ) : ∫ R 2 V ( x ) | u | 2 d x < + ∞ can be compactly embedded into L p ( R 2 ) ( p ≥ 1 ). Then the existence of nontrivial weak solutions can be obtained by Mountain-Pass lemma, one can see [15], [27], [28] for details. In the case V ( x ) is a constant, H 1 ( R 2 ) is continuously embedded into L 2 ( R 2 ) but the embedding is not compact. Ruf and Sani [29] showed that (1.7) possesses a nontrivial ground state solution by means of the constrained minimization method on the Pohozaev manifold under the growth assumption

lim | s | → + ∞ s f ( s ) e 4 π s 2 ≥ β 0 > 0 , for some β 0 .

Later, Masmoudi and Sani [30] weakened the above growth condition by combining the Pohozaev manifold and the Trudinger–Moser inequality with exact growth in R 2 . For more results of the existence of nontrivial solutions, one can refer to [14], [15], [16], [18], [31], [32], [33] and the references therein.

When V ( x ) is the Rabinowitz type potentials:

0 < C 0 = inf R 2 V ( x ) < sup R 2 V ( x ) = lim | x | → + ∞ V ( x ) = γ < + ∞

and the nonlinearity f ( t ) is of exponential growth, the existence of semiclassical state solution was obtained by Alves and Figueiredo [34] if ε is small enough and − Δ is replaced by − ε 2 Δ . Recently, Chen, Lu and Zhu [35] removed the smallness assumption on ε and established the existence of ground state solutions to (1.7).

Recently, Chen, Lu and Zhu [26], [36] obtained the existence of a ground state solution to quasilinear equations involving the degenerate potentials ( C 0 = 0 ) and the critical exponential growth when β = 0 by using a sharp Trudinger–Moser inequality with degenerate potentials in R 2 . Our work focus on the existence of ground state solutions to (1.7) in the case β > 0 . Due to the appearance of the singular weight, it becomes difficult to get the existence result by the constrained minimization method based on the Pohozaev manifold, we can not follow the same line as [26], [29], [30], [36]. To overcome this difficulty, we first use the Moser function sequence { M n } n to establish the threshold of the least energy, and then we apply an embedding theorem in [16] to obtain the convergence of ∫ R 2 F ( u k ) | x | β d x . By managing a method combining the threshold of the least energy, the convergence of ∫ R 2 F ( u k ) | x | β d x and Nehari manifold, we can get the desired result.

The paper is organized as follows. Section 2 establishes the critical singular Trudinger–Moser inequality involving the degenerate potential and the related Concentration-Compactness principle. In Section 3, we focus on the existence of ground states of the limit equation and give the proof of Theorem 1.3. In Section 4, we prove the existence of ground states to the Schrödinger equation (1.7) by using Theorem 1.2 and Theorem 1.3.

2 Singular Trudinger–Moser inequality involving the degenerate potential and the related Concentration-Compactness principle

This section is devoted to studying the singular Trudinger–Moser inequality involving the degenerate potential and the related Concentration-Compactness principle. That is, presenting the proofs for Theorem 1.1 and Theorem 1.2. Before starting the proof, we need an important embedding lemma which was established in [26].

Lemma 2.1.

([26]) Assume u ∈ H 1 ( R 2 ) such that

∫ R 2 ( | ∇ u | 2 + V ( x ) | u | 2 ) d x < + ∞ ,

where V ( x ) satisfies ( V 1 ) and ( V 2 ) . Then there exists a positive constant c depending on δ and V 0 such that

∫ R 2 | u | 2 d x < c ∫ R 2 ( | ∇ u | 2 + V ( x ) | u | 2 ) d x .

Remark 2.2.

Lemma 2.1 implies that the standard Sobolev space H 1 ( R 2 ) and the space H V ( R 2 ) which is defined as the completion of C c ∞ ( R 2 ) under the norm  ‖ u ‖ V are equivalent.

Now, we are ready to start the Proof of Theorem 1.1.

Proof of Theorem 1.1.

Without loss of generality, we assume u ≥ 0 is smooth and compactly supported since C c ∞ ( R 2 ) is dense in the Hilbert space H V ( R 2 ) . Then we split the proof into two cases.□

Case 1. If

∫ R 2 V ( x ) | u | 2 d x = 0 .

As a consequence, one can obtain that supp u ⊆ B 2 δ ( 0 ) . The classical singular Trudinger–Moser inequality on the bounded domain (see [37]) gives that

(2.1) ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x ≤ sup v ∈ H 1 ( R 2 ) , ∫ B 2 δ ( 0 ) | ∇ v | 2 d x ≤ 1 ∫ B 2 δ ( 0 ) e 4 π 1 − β 2 v 2 − 1 | x | β d x < C δ .

Case 2. If

∫ R 2 V ( x ) | u | 2 d x > 0 .

Now, we apply the rearrangement-free method which was developed in [10], [11] and set

A ( u ) ≔ ∫ R 2 V ( x ) | u | 2 d x 1 2

and

Ω ( u ) ≔ { x ∈ R 2 | u ( x ) > A ( u ) } .

It is easy to known that A ( u ) < 1 . Now, we claim that

| Ω ( u ) | ≤ 4 π δ 2 + 1 V 0 ,

where | Ω ( u ) | denotes the measure of Ω ( u ) . With the help of (V1), one can derive that

| Ω ( u ) ∩ B 2 δ c ( 0 ) | ≤ ∫ Ω ( u ) ∩ B 2 δ c ( 0 ) u 2 A 2 ( u ) d x ≤ ∫ Ω ( u ) ∩ B 2 δ c ( 0 ) u 2 d x ∫ R 2 V ( x ) | u | 2 d x ≤ 1 V 0 .

Then it follows that

| Ω ( u ) | ≤ | Ω ( u ) ∩ B 2 δ c ( 0 ) | + | B 2 δ ( 0 ) | ≤ 1 V 0 + 4 π δ 2 .

By splitting the integral into two parts, we have

∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x = ∫ Ω ( u ) e 4 π 1 − β 2 u 2 − 1 | x | β d x + ∫ R 2 \ Ω ( u ) e 4 π 1 − β 2 u 2 − 1 | x | β d x ≕ I 1 + I 2 .

For I 2 , direct calculations give

I 2 ≤ ∫ { u ( x ) < 1 } ∑ k = 1 ∞ ( 2 π ( 2 − β ) ) k k ! u 2 k | x | β d x ≤ ∫ { u ( x ) < 1 } ∑ k = 1 ∞ ( 2 π ( 2 − β ) ) k k ! u 2 | x | β d x ≤ ∑ k = 1 ∞ ( 2 π ( 2 − β ) ) k k ! ∫ R 2 u 2 | x | β d x .

With the help of Theorem A, one can obtain that

∫ R 2 u 2 | x | β d x = ‖ u ‖ γ 2 ∫ R 2 u ‖ u ‖ γ 2 1 | x | β d x ≤ 2 2 − β ‖ u ‖ γ 2 ∫ R 2 1 | x | β exp 1 − β 2 u 2 ‖ u ‖ γ 2 − 1 d x ≤ 2 2 − β ‖ u ‖ γ 2 sup ‖ u ‖ γ ≤ 1 ∫ R 2 e 1 − β 2 u 2 − 1 | x | β d x ≤ C ‖ u ‖ γ 2 ,

where C is a positive constant independent of u . Summing up the above, one can apply Lemma 2.1 to derive that

I 2 ≤ e 4 π 1 − β 2 ∫ R 2 u 2 | x | β d x ≤ C ‖ u ‖ γ 2 ≤ C ( 1 + γ c ) ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≲ 1 .

As for I 1 , we denote two functions v and w in Ω ( u ) by

v ( x ) ≔ u ( x ) − A ( u ) and w ( x ) ≔ v ( x ) ( 1 + A 2 ( u ) ) 1 2 .

Then v , w ∈ H 0 1 ( Ω ( u ) ) and direct calculations yield that

u 2 ( x ) ≤ w 2 ( x ) + 1 + A 2 ( u ) , ∇ w ( x ) = ( 1 + A 2 ( u ) ) 1 2 ∇ v ( x ) .

Therefore, we get

(2.2) ∫ Ω ( u ) | ∇ w ( x ) | 2 d x = ( 1 + A 2 ( u ) ) ∫ Ω ( u ) | ∇ v ( x ) | 2 d x ≤ ( 1 + A 2 ( u ) ) 1 − ∫ R 2 V ( x ) u 2 d x < 1 .

Therefore, one can apply the classical singular Trudinger–Moser inequality on the bounded domain to derive that

(2.3) I 1 = ∫ Ω ( u ) e 4 π 1 − β 2 u 2 − 1 | x | β d x ≤ e 4 π 1 − β 2 ( 1 + A 2 ( u ) ) ∫ Ω ( u ) e 4 π 1 − β 2 w 2 | x | β d x < C .

Combining the estimates of I 1 and I 2 , we see that inequality (1.6) holds. Thus, we complete the Proof of Theorem 1.1.

Now, we prove the Concentration-Compactness principle associated with the Trudinger–Moser inequality (1.6).

Proof of Theorem 1.2.

By splitting the integral into two parts, we get

(2.4) ∫ R 2 e 4 π p 1 − β 2 u k 2 − 1 | x | β d x = ∫ B 1 ( 0 ) + ∫ B 1 c ( 0 ) e 4 π p 1 − β 2 u k 2 − 1 | x | β d x ≕ I 1 + I 2 .

First, we estimate I 1 . Since p < 1 1 − ‖ u 0 ‖ V 2 , there exists a ε 0 > 0 such that p ( 1 + ε 0 ) < 1 1 − ‖ u 0 ‖ V 2 . Picking q = ( 1 + ε 0 2 ) ( 2 2 − β ) , then p ( 1 − β 2 ) q < 1 1 − ‖ u 0 ‖ V 2 and 1 q ′ = β + ε 0 2 + ε 0 > β 2 . Therefore, one can apply Hölder’s inequality to derive that

(2.5) I 1 ≤ ∫ B 1 ( 0 ) e 4 π p 1 − β 2 u k 2 | x | β d x ≤ ∫ B 1 ( 0 ) e 4 π p q 1 − β 2 u k 2 d x 1 q ∫ B 1 ( 0 ) | x | − β q ′ d x 1 q ′ ≤ C ,

where the last inequality comes from the Concentration-Compactness principle of Trudinger–Moser inequality involving the degenerate potential (see [26]).

As for I 2 , still using the Concentration-Compactness principle of Trudinger–Moser inequality involving the degenerate potential, we have

I 2 ≤ ∫ B 1 c ( 0 ) e 4 π p 1 − β 2 u k 2 − 1 d x < C .

This together with inequality (2.5) gives that

sup k ∫ R 2 e 4 π p 1 − β 2 u k 2 − 1 | x | β d x < + ∞ .

Thus, we complete the proof.□

3 Existence of ground state solutions to the limit equation (1.8)

This section is devoted to the existence of ground state solutions to the limit equation (1.8). With direct calculations, we get its related functional and Nehari manifold:

J ∞ ( u ) = 1 2 ∫ R 2 ( | ∇ u | 2 + γ u 2 ) d x − ∫ R 2 F ( u ) | x | β d x

and

N ∞ = u ∈ H 1 ( R 2 ) | u ≠ 0 , N ∞ ( u ) = 0 ,

where

N ∞ ( u ) = ∫ R 2 ( | ∇ u | 2 + γ u 2 ) d x − ∫ R 2 f ( u ) u | x | β d x .

First, we show that N ∞ is not empty.

Lemma 3.1.

N ∞ is not an empty set.

Proof.

Pick u 0 ∈ H 1 ( R 2 ) be a positive and smooth function which is compactly supported. Define a new function h ( s ) by

h ( s ) ≔ N ∞ ( s u 0 ) = s 2 ∫ R 2 | ∇ u 0 | 2 + γ u 0 2 d x − ∫ R 2 s u 0 f ( s u 0 ) | x | β d x .

Then it is sufficient to show that

when s > 0 is sufficiently small, then h ( s ) > 0 .
when s > 0 is large enough, there holds h ( s ) < 0 .

For claim (i), we can deduce from assumptions (i)–(iv) on f to derive that

| f ( t ) | ≤ ε | t | + C ε t μ − 1 e β 0 1 − β 2 t 2 − 1

for any positive constant ε . Using this estimate, one can directly obtain that

(3.1) h ( s ) = s 2 ∫ R 2 | ∇ u 0 | 2 + γ u 0 2 d x − ∫ R 2 f ( s u 0 ) s u 0 | x | β d x ≥ s 2 ∫ R 2 | ∇ u 0 | 2 + γ u 0 2 d x − ε s 2 ∫ R 2 u 0 2 | x | β d x − C ε s μ ∫ R 2 u 0 μ e β 0 1 − β 2 s 2 u 0 2 − 1 | x | β d x .

Since μ > 2 , we can see that h ( s ) > 0 for s sufficiently small. Thus we complete the proof of claim (i).

As for (ii), we first give an estimate:

(3.2) t f ( t ) ≥ μ t μ F ( 1 ) − C ,

where t ∈ R and C is a constant independent of t . Through condition (ii), we have

F ( t ) ≥ t μ F ( 1 ) for t ≥ 1 .

Since F is continuous, there exists some constant C 1 such that

F ( t ) ≥ t μ F ( 1 ) − C 1 for t ≥ 0 .

As a consequence, (3.2) follows from condition (ii). Then, we can estimate h ( s ) :

(3.3) h ( s ) = s 2 ∫ R 2 | ∇ u 0 | 2 + γ u 0 2 d x − ∫ R 2 f ( s u 0 ) s u 0 | x | β d x ≤ s 2 ∫ R 2 | ∇ u 0 | 2 + γ u 0 2 d x − μ s μ F ( 1 ) ∫ Ω u 0 μ | x | β d x + C ∫ Ω u 0 | x | β d x ,

which implies that for s large enough, h ( s ) < 0 . Therefore, we finish the proof of Lemma 3.1.□

Set

(3.4) m ∞ = inf { J ∞ ( u ) | u ∈ N ∞ } .

Now, we estimate the least energy m ∞ which plays an important role in proving the existence of ground states.

Lemma 3.2.

There holds

m ∞ < 2 π β 0 .

Proof.

Let u ∈ H 1 ( R 2 ) be a positive function such that ‖ u ‖ γ = 1 . From the argument of Lemma 3.1, one can see that there exists a positive t 0 such that

N ∞ ( t 0 u ) = 0 .

From the definition of m ∞ , we have

m ∞ ≤ J ∞ ( t 0 u ) .

Construct a sequence of Moser functions { M n } n ⊂ H 1 ( R 2 ) as Yang [28], then we can apply Lemma 3.3 in [28] to derive that there exists some n 0 ∈ N such that

max t ≥ 0 J ∞ ( t M n 0 ) < 2 π β 0 ,

which implies that

m ∞ ≤ max t ≥ 0 J ∞ ( t M n 0 ) < 2 π β 0 .

Thus, we get the desired conclusion.□

Furthermore, we show that m ∞ is positive.

Lemma 3.3.

There holds m ∞ > 0 .

Proof.

Since t f ( t ) ≥ μ F ( t ) , we have m ∞ ≥ 0 . Assume on the contrary, m ∞ = 0 . Then there exists a sequence { u k } k ⊆ H 1 ( R 2 ) such that

N ∞ ( u k ) = 0 and J ∞ ( u k ) → 0 as k → + ∞ .

Therefore, direct calculations yield that

m ∞ = lim k → + ∞ 1 2 ∫ R 2 | ∇ u k | 2 + γ u k 2 d x − ∫ R 2 F ( u k ) | x | β d x = lim k → + ∞ 1 2 ∫ R 2 f ( u k ) u k | x | β d x − ∫ R 2 F ( u k ) | x | β d x ≥ 1 2 − 1 μ lim k → + ∞ ∫ R 2 f ( u k ) u k | x | β d x = lim k → + ∞ 1 2 − 1 μ ∫ R 2 | ∇ u k | 2 + γ u k 2 d x ,

which implies that

lim k → + ∞ ∫ R 2 | ∇ u k | 2 + γ u k 2 d x = 0 .

Since u k ∈ N ∞ , one can apply the conditions (i)–(v) on f to obtain that

(3.5) 1 = ∫ R 2 f ( u k ) u k | x | β 1 ‖ u k ‖ γ 2 d x ≤ ∫ R 2 1 ‖ u k ‖ γ 2 ε | u k | 2 | x | β + C ε u k μ e β 0 1 − β 2 u k 2 − 1 | x | β d x .

For k sufficiently large, we have ‖ u k ‖ γ ≪ 1 . Let v k = u k ‖ u k ‖ γ . Then singular Trudinger–Moser inequalities on unbounded domains imply that for any α ≤ 4 π ( 1 − β 2 ) ,

∫ R 2 e α v k 2 − 1 | x | β d x = ∑ i = 1 ∞ ∫ R 2 α i u k 2 i i ! | x | β ‖ u k ‖ γ 2 i d x ≤ C ,

which yields that

(3.6) ∫ R 2 u k 2 i | x | β d x ≤ C i ! ‖ u k ‖ γ 2 i α i .

Hölder’s inequality gives that

(3.7) ∫ R 2 u k p | x | β d x ≤ ∫ R 2 u k 2 i | x | β d x θ ∫ R 2 u k 2 i + 2 | x | β d x 1 − θ ≤ C ( i + 1 ) ! ‖ u k ‖ γ p α p / 2 ,

where θ ∈ [ 0,1 ] and p = 2 i θ + 2 ( i + 1 ) ( 1 − θ ) . Combining (3.5) with (3.7), we can use Hölder’s inequality and Trudinger–Moser inequality to derive that

(3.8) 1 ≤ 1 ‖ u k ‖ γ 2 ∫ R 2 ε | u k | 2 | x | β + C ε u k μ e β 0 1 − β 2 u k 2 − 1 | x | β d x ≤ C ε α + C ‖ u k ‖ γ 2 ∫ R 2 u k q ′ μ | x | β d x 1 q ′ ∫ R 2 e β 0 1 − β 2 q u k 2 − 1 | x | β d x 1 q ≤ C ε + C ‖ u k ‖ γ μ − 2 ,

where q > 1 . Since lim k → + ∞ ‖ u k ‖ γ = 0 , (3.8) can not hold. Therefore m ∞ > 0 .□

Let { u k } k ⊆ H 1 ( R 2 ) be a minimizing sequence such that

N ∞ ( u k ) = 0 and J ∞ ( u k ) → m ∞ as k → + ∞ .

Since 0 < m ∞ < 2 π β 0 , the (A-R) condition implies that { u k } k is bounded in H 1 ( R 2 ) . Without loss of generality, we assume u k ≥ 0 . Then, up to a sequence, there exists some u ∈ H 1 ( R 2 ) such that

u k ⇀ u i n H 1 ( R 2 ) , u k → u i n L loc p ( R 2 ) for any p ≥ 1 , u k → u a . e i n R 2 .

Lemma 3.4.

Let { u k } k be a bounded sequence in H 1 ( R 2 ) which converges weakly to u such that

(3.9) sup k ∫ R 2 u k f ( u k ) | x | β d x < + ∞ ,

then

lim k → + ∞ ∫ R 2 F ( u k ) | x | β d x = ∫ R 2 F ( u ) | x | β d x .

Proof.

By dividing the integral into two parts, we get

(3.10) ∫ R 2 F ( u k ) | x | β d x − ∫ R 2 F ( u ) | x | β d x = ∫ { | u k | ≤ R } F ( u k ) | x | β d x − ∫ { | u | ≤ R } F ( u ) | x | β d x + ∫ { | u k | ≥ R } F ( u k ) | x | β d x − ∫ { | u | ≥ R } F ( u ) | x | β d x ≕ I k R + J k R .

For I k R , we also split the integral into two parts.

(3.11) I k R = ∫ { | u k | ≤ R } ∩ B r ( 0 ) F ( u k ) | x | β d x − ∫ { | u | ≤ R } ∩ B r ( 0 ) F ( u ) | x | β d x + ∫ { | u k | ≤ R } ∩ B r c ( 0 ) F ( u k ) | x | β d x − ∫ { | u | ≤ R } ∩ B r c ( 0 ) F ( u ) | x | β d x ≕ I 1 r k R + I 2 r k R .

Dominated convergence theorem yields that lim k → + ∞ I 1 k R = 0 . Now, we consider the term I 2 r k R . From the conditions (i)–(v) on f , we can derive that

F ( t ) ≤ t f ( t ) ≤ ε t 2 + C ε t μ e β 0 1 − β 2 t 2 − 1 .

Since | u | < R , then

(3.12) F ( u ) ≤ ε + C ε R μ − 2 e β 0 1 − β 2 R 2 − 1 u 2 = C ( β 0 , R ) u 2 .

Combining (3.7) with (3.12), one can apply the singular Trudinger–Moser inequality to obtain that

(3.13) ∫ { | u k | ≤ R } ∩ B r c ( 0 ) F ( u k ) | x | β d x ≤ C ( β 0 , R ) ∫ { | u k | ≤ R } ∩ B r c ( 0 ) u k 2 | x | β d x ≤ C ( β 0 , R ) r β ∫ { | u k | ≤ R } ∩ B r c ( 0 ) u k 2 d x ≤ C ( β 0 , R ) r β sup k ‖ u k ‖ γ 2 .

Therefore lim r → + ∞ lim k → + ∞ I 2 r k R = 0 . As for J k R , we use condition (iii) and (3.9) to derive that

(3.14) lim R → + ∞ lim k → + ∞ ∫ { | u k | ≥ R } F ( u k ) | x | β d x ≤ lim R → + ∞ lim k → + ∞ ∫ { | u k | ≥ R } M 0 f ( u k ) | x | β d x ≤ lim R → + ∞ lim k → + ∞ M 0 R ∫ { | u k | ≥ R } u k f ( u k ) | x | β d x = 0 .

Due to the absolutely continuity of the integral ∫ R 2 F ( u ) | x | β d x , one can see that lim R → + ∞ lim k → + ∞ J k R = 0 . Combining the above estimates, we complete the proof of Lemma 3.4.□

If { u k } k ⊆ N ∞ is a minimizing sequence for m ∞ , the (A-R) condition gives that { u k } k is a bounded sequence in H 1 ( R 2 ) . Then it follows that

sup k ∫ R 2 u k f ( u k ) | x | β d x < + ∞ .

This together with Lemma 3.4 yields that

Corollary 3.5.

Let { u k } k ⊆ N ∞ be a minimizing sequence for m ∞ , then

lim k → + ∞ ∫ R 2 F ( u k ) | x | β d x = ∫ R 2 F ( u ) | x | β d x .

Lemma 3.6.

Let { u k } k ⊂ N ∞ be a minimizing sequence for m ∞ which converges weakly to some u ≢ 0 with ‖ u ‖ γ > ∫ R 2 u f ( u ) | x | β d x , then

(3.15) lim k → + ∞ ∫ R 2 u k f ( u k ) | x | β d x = ∫ R 2 u f ( u ) | x | β d x .

Proof.

Up to a sequence, we have u k → u a.e. in R 2 . The lower semicontinuity of the norm in H 1 ( R 2 ) yields that

lim k → + ∞ ‖ u k ‖ γ ≥ ‖ u ‖ γ .

Now, we divide the proof into two cases.

Case 1: If lim k → + ∞ ‖ u k ‖ γ = ‖ u ‖ γ . We apply the weak convergence of u k to obtain u k → u in H 1 ( R 2 ) . Then one can apply the result in [16] to derive that u k strongly converges to u in L p ( R 2 , | x | − β d x ) for any p ≥ 2 , 0 < β < 2 .

Now, we claim that

(3.16) sup k ∫ R 2 u k f ( u k ) | x | β p d x < + ∞ ,

where p > 1 is chosen in such a way that β < β p < 2 . Note that t f ( t ) ≤ ε t 2 + C ε t μ ( e β 0 ( 1 − β 2 ) t 2 − 1 ) . One can obtain that

(3.17) sup k ∫ R 2 u k f ( u k ) | x | β p d x < C p sup k ∫ R 2 ε p u k 2 p | x | p β + C ε u k μ p e p β 0 1 − β 2 u k 2 − 1 | x | p β d x .

Then (3.16) follows from (3.6), Hölder’s inequality and the classical singular Trudinger–Moser inequality in H 1 ( R 2 ) . By splitting the integral into three parts, we obtain that

(3.18) ∫ R 2 u k f ( u k ) | x | β d x − ∫ R 2 u f ( u ) | x | β d x = ∫ B r ( 0 ) u k f ( u k ) − u f ( u ) | x | β d x + ∫ B r c ( 0 ) ( u k − u ) f ( u k ) | x | β + ( f ( u k ) − f ( u ) ) u | x | β d x ≕ I r k + I I r k + I I I r k .

Combining the Vitali convergence theorem with (3.16), one can derive that for any r > 0 ,

(3.19) lim k → + ∞ I r k = 0 .

For I I r k , one can manage similar progress as (3.16) to get that

(3.20) sup k ∫ R 2 f ( u k ) p | x | β d x < + ∞ ,

for any p > 1 . Through Hölder’s inequality, (3.20) and u k → u in L p ( R 2 , | x | − β d x ) , we can obtain that

(3.21) lim k → + ∞ ∫ B r c ( 0 ) ( u k − u ) f ( u k ) | x | β d x ≤ sup k ∫ R 2 f ( u k ) p | x | β d x 1 p lim k → + ∞ ∫ R 2 | u k − u | p ′ | x | β d x 1 p ′ = 0 .

As for I I I r k , one can apply Hölder’s inequality again to obtain that

(3.22) ∫ B r c ( 0 ) ( f ( u k ) − f ( u ) ) u | x | β d x ≤ ∫ B r c ( 0 ) u f ( u k ) | x | β d x + ∫ B r c ( 0 ) u f ( u ) | x | β d x ≤ sup k ∫ R 2 f ( u k ) p | x | β d x 1 p ∫ B r c ( 0 ) | u | p ′ | x | β d x 1 p ′ + ∫ B r c ( 0 ) u f ( u ) | x | β d x .

By the absolute convergence of the integral, we have

(3.23) lim r → + ∞ lim k → + ∞ I I I r k = 0 .

This together with (3.19) and (3.21) yields (3.15).

Case 2: If lim k → + ∞ ‖ u k ‖ γ > ‖ u ‖ γ , define

v k ≔ u k lim k → + ∞ ‖ u k ‖ γ , v ≔ u lim k → + ∞ ‖ u k ‖ γ .

Then we show that there exists p > 1 sufficiently close to 1 such that

(3.24) p lim k → + ∞ ‖ u k ‖ γ 2 < 4 π β 0 1 − ‖ v ‖ γ 2 .

Indeed, one can use ‖ u ‖ γ > ∫ R 2 u f ( u ) | x | β d x to derive that

J ∞ ( u ) = 1 2 ‖ u ‖ γ − ∫ R 2 F ( u ) | x | β d x ≥ 1 2 ‖ u ‖ γ − 1 μ ∫ R 2 u f ( u ) | x | β d x > 0 .

This together with J ∞ ( u k ) → m ∞ and m ∞ < 2 π β 0 yields that

(3.25) lim k → + ∞ ‖ u k ‖ γ 2 1 − ‖ v ‖ γ 2 = 2 m ∞ + 2 lim k → + ∞ ∫ R 2 F ( u k ) | x | β d x − 2 J ∞ ( u ) − 2 ∫ R 2 F ( u ) | x | β d x < 4 π β 0 ,

where we have used Corollary 3.5. Now, we will show that

(3.26) sup k ∫ R 2 u k f ( u k ) | x | β p d x < + ∞ and sup k ∫ R 2 f ( u k ) p | x | β d x < + ∞ .

where p > 1 is chosen in such a way that β < β p < 2 and p lim k → + ∞ ‖ u k ‖ γ 2 < 4 π β 0 1 − ‖ v ‖ γ 2 . Notice that t f ( t ) ≤ ε t 2 + C ε t μ e β 0 t 2 − 1 . We get

sup k ∫ R 2 u k f ( u k ) | x | β p d x < C p ∫ R 2 ε p u k 2 p | x | p β + C ε u k μ p e p β 0 1 − β 2 u k 2 − 1 | x | p β d x

and

sup k ∫ R 2 f ( u k ) p | x | β d x < C p ∫ R 2 ε p u k p | x | β + C ε u k ( μ − 1 ) p e p β 0 1 − β 2 u k 2 − 1 | x | β d x .

Applying Hölder’s inequality once again, one can derive (3.26) from Theorem 1.2. Then it follows from the similar progress as Case 1 that

lim k → + ∞ ∫ R 2 u k f ( u k ) | x | β d x = ∫ R 2 u f ( u ) | x | β d x ,

which completes the proof.□

Lemma 3.7.

Let { u k } k ⊂ N ∞ be a minimizing sequence for m ∞ which converges weakly to u . Then the case u = 0 can not occur.

Proof.

Suppose u = 0 , then it follows from Corollary 3.5 that

(3.27) lim k → ∞ ‖ u k ‖ γ 2 = 2 lim k → ∞ J ∞ ( u k ) + 2 lim k → ∞ ∫ Ω F ( u k ) | x | β d x = 2 lim k → ∞ J ∞ ( u k ) = 2 m ∞ < 4 π β 0 .

Picking p > 1 sufficiently close to 1 such that p β < 2 and p lim k → ∞ ‖ u k ‖ γ 2 < 4 π β 0 , manage the similar progress as (3.16), one can use Hölder’s inequality, the Trudinger–Moser inequality in H 1 ( R 2 ) to obtain that

(3.28) sup k ∫ R 2 u k f ( u k ) | x | β p d x < + ∞ and sup k ∫ R 2 f ( u k ) p | x | β d x < + ∞ .

Now, we estimate the integral ∫ R 2 u k f ( u k ) | x | β d x , we rewrite it as the sum of I r k , I I r k and I I I r k defined as in (3.18). lim k → + ∞ I r k = 0 is a direct consequence of the Vitali convergence theorem and (3.28). lim k → + ∞ I I r k = 0 follows from Hölder’s inequality, (3.28) and u k → u in L p ( R 2 , | x | − β d x ) . Hölder’s inequality and the absolute convergence of the integral yields lim r → + ∞ lim k → + ∞ I I I r k = 0 . Hence, we derive that

(3.29) lim k → + ∞ ∫ R 2 u k f ( u k ) | x | β d x = lim k → + ∞ ∫ R 2 u k f ( u k ) − u f ( u ) | x | β d x = 0 ,

which implies that

0 < 2 m ∞ = lim k → ∞ ∫ R 2 | ∇ u k | 2 + γ | u k | 2 d x = lim k → ∞ ∫ R 2 u k f ( u k ) | x | β d x = 0 .

Thus we get a contradiction. This proves u ≠ 0 .□

Now, we are in position to prove Theorem 1.3.

Proof of Theorem 1.3.

Let { u k } k ⊆ N ∞ be a minimizing sequence for m ∞ which converges weakly to u . First, we claim that

(3.30) ‖ u ‖ γ 2 ≤ ∫ R 2 u f ( u ) | x | β d x .

Suppose on the contrary, in view of Lemma 3.6, we derive that

(3.31) lim k → ∞ ∫ R 2 u k f ( u k ) | x | β d x = ∫ R 2 u f ( u ) | x | β d x .

As a result,

‖ u ‖ γ 2 > ∫ R 2 u f ( u ) | x | β d x = lim k → ∞ ∫ R 2 u k f ( u k ) | x | β d x = lim k → ∞ ‖ u k ‖ γ 2 ,

which can not occur. Therefore, there exists s ∈ ( 0,1 ] such that s u ∈ N ∞ . Recall the definition of m ∞ , one can use the monotonicity of t f ( t ) − 2 F ( t ) and Corollary 3.5 to obtain that

m ∞ ≤ J ∞ ( s u ) = 1 2 ∫ R 2 s u f ( s u ) | x | β d x − ∫ R 2 F ( s u ) | x | β d x ≤ 1 2 ∫ R 2 u f ( u ) | x | β d x − ∫ R 2 F ( u ) | x | β d x ≤ lim k → ∞ 1 2 ∫ R 2 u k f ( u k ) | x | β d x − ∫ R 2 F ( u k ) | x | β d x = lim k → ∞ J ∞ ( u k ) = m ∞ .

This implies that s = 1 , therefore u ∈ N ∞ and J ∞ ( u ) = m ∞ . The proof is completed.□

4 Existence of ground state solutions to the equations involving the degenerate potentials

This section is devoted to the existence of ground state solutions to the following elliptic equation:

(4.1) − Δ u + V ( x ) u = f ( u ) | x | β , x ∈ R 2 ,

where V is positive satisfying (V1) and (V2), f satisfies (i)–(vi). First, we give the related functional and Nehari manifold:

J V ( u ) = 1 2 ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x − ∫ R 2 F ( u ) | x | β d x

and

N V = u ∈ H 1 ( R 2 ) | u ≠ 0 , N V ( u ) = 0 ,

where

N V ( u ) = ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x − ∫ R 2 f ( u ) | x | β d x .

Managing the similar progress as Lemma 3.1, we can get the following lemma.

Lemma 4.1.

N V is not empty.

Set m V = inf { J V ( u ) | u ∈ N V } . We can obtain the following relationship between m V and m ∞ .

Lemma 4.2.

There holds 0 < m V < m ∞ .

Proof.

First, we show m V < m ∞ . Based on the assumptions of f , one can apply Theorem 1.3 to derive that m ∞ can be attained by some u ∞ ∈ H 1 ( R 2 ) . Since 0 ≤ V ( x ) < γ , we have

∫ R 2 ( | ∇ u ∞ | 2 + V ( x ) u ∞ 2 ) d x < ∫ R 2 | ∇ u ∞ | 2 + γ u ∞ 2 d x = ∫ R 2 f ( u ∞ ) u ∞ | x | β d x .

Since f ( t ) t is increasing, there exists s ∈ ( 0,1 ) such that

∫ R 2 ( | ∇ ( s u ∞ ) | 2 + V ( x ) s 2 u ∞ 2 ) d x = ∫ R 2 f ( s u ∞ ) s u ∞ | x | β d x .

As a result, s u ∞ ∈ N V . Since t f ( t ) − 2 F ( t ) is strictly increasing, we have

(4.2) m V ≤ J V ( s u ∞ ) = 1 2 ∫ R 2 f ( s u ∞ ) s u ∞ − 2 F ( s u ∞ ) | x | β d x < J ∞ ( u ∞ ) = m ∞ .

Therefore, m V < m ∞ . With the help of Theorem 1.1, one can carry out the similar progress as Lemma 3.3 to derive that m V > 0 . Hence, there holds 0 < m V < m ∞ .□

Since 0 < m V < m ∞ , the (A-R) condition gives that { u k } is bounded in H 1 ( R 2 ) . Without loss of generality, we assume u k ≥ 0 . Then, up to a sequence, there exists some u ∈ H 1 ( R 2 ) such that

u k ⇀ u i n H 1 ( R 2 ) , u k → u i n L loc p ( R 2 ) for any p ≥ 1 , u k → u a . e i n R 2 .

Lemma 4.3.

Let { u k } k ⊆ N ∞ be a minimizing sequence for m V , then

lim k → + ∞ ∫ R 2 F ( u k ) | x | β d x = ∫ R 2 F ( u ) | x | β d x .

Proof.

Since { u k } k is bounded in H 1 ( R 2 ) , then it follows from 0 < m V < m ∞ < 2 π β 0 that

sup k ∫ R 2 u k f ( u k ) | x | β d x < + ∞ .

Therefore, one can use Theorem 1.1 instead of the singular Trudinger–Moser inequality in H 1 ( R 2 ) and follow the same line as Lemma 3.4 to get the desired conclusion.□

Lemma 4.4.

Let { u k } k ⊂ N V be a minimizing sequence for m V . Assume that { u k } k is a bounded sequence in H V ( R 2 ) which converges weakly to u ≢ 0 and ‖ u ‖ V > ∫ R 2 u f ( u ) | x | β d x , then

(4.3) lim k → + ∞ ∫ R 2 u k f ( u k ) | x | β d x = ∫ R 2 u f ( u ) | x | β d x .

Proof.

Up to a sequence, u k → u a.e. in R 2 . The lower semicontinuity of the norm in H V ( R 2 ) yields that

lim k → + ∞ ‖ u k ‖ V ≥ ‖ u ‖ V .

Then we divide the proof into two cases.

Case 1: If lim k → + ∞ ‖ u k ‖ V = ‖ u ‖ V . We apply the weak convergence of u k to obtain u k → u in H V ( R 2 ) . Since H V ( R 2 ) = H 1 ( R 2 ) , one can apply the result in [16] to derive that u k strongly converges to u in L p ( R 2 , | x | − β d x ) for any p ≥ 2 , 0 < β < 2 . Thanks to Theorem 1.1, one can manage the similar progress as Case 1 in Lemma 3.6 to derive (4.3).

Case 2: If lim k → + ∞ ‖ u k ‖ V > ‖ u ‖ V , with a slightly modification of Case 2 in Lemma 3.6, we can apply Lemma 4.3 and Theorem 1.2 to get (4.3). Therefore we get the desired result.□

Similar as Lemma 3.7, by Theorem 1.1 and the fact that

0 < m V < m ∞ < 2 π β 0 ,

we can obtain the following

Lemma 4.5.

The case u = 0 can not occur.

Remark 4.6.

Lemma 4.2 plays an important role in getting the singular Trudinger–Moser inequality and Concentration-Compactness principle for Trudinger–Moser inequality in H V ( R 2 ) which are necessary in proving Lemmas 4.4 and 4.5.

Then we are in position to show the existence of ground state solution to (4.1).

Proof of Theorem 1.4.

First, we claim that

(4.4) ‖ u ‖ V 2 ≤ ∫ R 2 u f ( u ) | x | β d x .

Suppose on the contrary, in view of Lemma 4.4, we derive that

(4.5) lim k → ∞ ∫ R 2 u k f ( u k ) | x | β d x = ∫ R 2 u f ( u ) | x | β d x .

Therefore,

‖ u ‖ V 2 > ∫ R 2 u f ( u ) | x | β d x = lim k → ∞ ∫ R 2 u k f ( u k ) | x | β d x = lim k → ∞ ‖ u k ‖ V 2 ,

which is a contrary. Hence, there exists s ∈ ( 0,1 ] such that s u ∈ N V . Recall the definition of m V , one can use the monotonicity of t f ( t ) − 2 F ( t ) to obtain that

m V ≤ J V ( s u ) = 1 2 ∫ R 2 s u f ( s u ) | x | β d x − ∫ R 2 F ( s u ) | x | β d x ≤ 1 2 ∫ R 2 u f ( u ) | x | β d x − ∫ R 2 F ( u ) | x | β d x ≤ lim k → ∞ 1 2 ∫ R 2 u k f ( u k ) | x | β d x − ∫ R 2 F ( u k ) | x | β d x = lim k → ∞ J V ( u k ) = m V .

This implies that s = 1 , therefore we have u ∈ N V and J V ( u ) = m V . Thus Theorem 1.4 is proved.□

Corresponding author: Maochun Zhu, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, P.R. China, E-mail: zhumaochun2006@126.com

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12001038

Award Identifier / Grant number: 12071185

Award Identifier / Grant number: 12061010

Research ethics: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: Authors state no conflict of interest.
Research funding: The first author was partly supported by a grant from the NNSF of China (No. 12001038). The second was supported by National Natural Science Foundation of China (Grant Nos. 12071185 and 12061010).
Data availability: Not applicable.

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Received: 2024-01-03

Accepted: 2024-04-09

Published Online: 2024-05-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2023-0136

Keywords for this article

Trudinger–Moser inequalities; degenerate potential; Nehari manifold; ground states

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