Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities

1 Introduction

In the present study, we consider the existence of solutions for nonlinear elliptic equations of the form

where V , Q , and f satisfy the following assumptions:

1. V ∈ C ( ( 0 , + ∞ ) , ( 0 , + ∞ ) ) , and there exist a 0 , a > − 2 such that

   limsup r → 0 + V ( r ) r a 0 < + ∞ , liminf r → + ∞ V ( r ) r a > 0 ;

2. Q ∈ C ( ( 0 , + ∞ ) , ( 0 , + ∞ ) ) , and there exists b 0 > − 2 such that

   limsup r → 0 + Q ( r ) r b 0 < + ∞ , limsup r → + ∞ Q ( r ) V ( r ) = 0 ;

3. f ∈ C ( R , R ) , and there exists α 0 > 0 such that

   (1.2) lim ∣ t ∣ → ∞ ∣ f ( t ) ∣ e α t 2 = 0 , for all  α > α 0

   and

   (1.3) lim ∣ t ∣ → ∞ ∣ f ( t ) ∣ e α t 2 = + ∞ , for all  α < α 0 ;

4. f ( t ) = o ( t ) as t → 0 ;

5. t f ( t ) > 0 for all t ≠ 0 , and there exist M 0 > 0 and t 0 > 0 such that

   F ( t ) ≔ ∫ 0 t f ( s ) d s ≤ M 0 ∣ f ( t ) ∣ , ∀ ∣ t ∣ ≥ t 0 .

When V ( ∣ x ∣ ) ≡ V ˆ > 0 and Q ( ∣ x ∣ ) ≡ 1 , we derive the following autonomous equation from (1.1):

There are a lot of work on the existence of solutions for (1.4) in the literature, for example, see [2–6,8,12,15–17], in which besides (F1)–(F3) and the following well-known Ambrosetti-Rabinowitz (AR)-type condition:

1. there exists θ > 2 such that

   f ( t ) t ≥ θ F ( t ) > 0 , ∀ t ≠ 0 .

1. there exists p > 2 such that f ( t ) t ≥ λ ∣ t ∣ p for all t ∈ R , where

   λ > α 0 ( p − 2 ) 4 π p p − 2 2 C p p ,

   where

   (1.5) C p ≔ inf u ∈ H 1 ( R 2 ) ‖ ∇ u ‖ 2 2 + V ˆ ‖ u ‖ 2 2 ‖ u ‖ p .

1. there holds

   (1.6) lim t → ∞ f ( t ) t e α 0 t 2 > 2 e V ˆ α 0 .

To study the existence of solutions of (1.1) via the classical Mountain-pass theorem, we will frame the variational study of (1.1) in the weighted space:

And we define

Then E is a Hilbert space on the above inner product ( ⋅ , ⋅ ) . Furthermore, for s ∈ [ 1 , + ∞ ) , we define

Ambrosetti et al. [7] and do Ó et al. [13] studied equation (1.1) by assuming that V and Q satisfy the following assumption:

1. there are A 1 , A 2 , A 3 > 0 , 0 < β 1 < 2 , and β 2 > 2 such that

   A 1 1 + ∣ x ∣ β 1 ≤ V ( x ) ≤ A 2 , 0 < Q ( x ) ≤ A 3 1 + ∣ x ∣ β 2 .

1. Q ∈ C ( ( 0 , + ∞ ) , ( 0 , + ∞ ) ) , and there exist b 0 > − 2 and b < ( a − 2 ) ∕ 2 such that

   limsup r → 0 + Q ( r ) r b 0 < + ∞ , limsup r → + ∞ Q ( r ) r b < + ∞ .

Obviously, ( F 4 ′ ) in Theorem 1.1 extends (H4); moreover, it is very crucial to overcome the obstacles caused by the critical exponential term (see [1, Lemma 3.3]). AR-type condition plays a crucial role to prove the boundedness of (Palais-Smale) sequences for the energy functional associated (1.1).

Since b < ( a − 2 ) ∕ 2 and a + 2 > 0 in (V1) and ( Q 1 ′ ), it is easy to verify that (V1) and ( Q 1 ′ ) imply (Q1). However, there are many functions V ( r ) and Q ( r ) satisfying (V1) and (Q1) but not ( Q 1 ′ ). For example,

1. Let V ( r ) ≔ r θ 1 and Q ( r ) ≔ r θ 2 . Then V and Q satisfy (V1) and (Q1) when θ 1 > θ 2 > − 2 but not ( Q 1 ′ ) when θ 2 + 1 > θ 1 ∕ 2 ;

2. Let V ( r ) ≔ log ( 1 + r ) and Q ( r ) ≔ r σ . Then V and Q satisfy (V1) and (Q1) when σ ∈ ( − 2 , 0 ] but not ( Q 1 ′ ) when σ ∈ [ − 1 , 0 ] .

To state our main results, we first introduce the following assumptions:

1. there exists p > 2 such that F ( t ) ≥ μ p ∣ t ∣ p for all t ∈ R , where

   μ > α 0 ( p − 2 ) 2 π ( 2 + b 0 ) p p − 2 2 S p p ;

2. there exist q > p > 2 , μ > 0 , λ > 0 , and ν ∈ [ 0 , 1 ] such that

   (1.8) F ( t ) ≥ μ p ∣ t ∣ p + λ q ∣ t ∣ q , ∀ t ∈ R

   and

   (1.9) ( p − 2 ) ν p ∕ ( p − 2 ) 2 p μ 2 ∕ ( p − 2 ) + ( q − 2 ) ( 1 − ν ) q ∕ ( q − 2 ) 2 q λ 2 ∕ ( q − 2 ) S 2 4 ( q − p ) ∕ ( p − 2 ) ( q − 2 ) < ( 2 + b 0 ) π α 0 S p − 2 p p − 2 ;

3. there exist q > p > 2 , μ > 0 , λ > 0 , and ν ∈ ( 0 , 1 ] such that (1.8) and the following inequality hold:

   (1.10) μ > ν p ∕ 2 1 − 1 − ν λ 2 ∕ q q q − 2 ( q − 2 ) α 0 S p 2 p p − 2 2 q ( 2 + b 0 ) π S 2 4 ( q − p ) ∕ ( p − 2 ) ( q − 2 ) p − 2 2 α 0 ( p − 2 ) 2 π ( 2 + b 0 ) p p − 2 2 S p p ;

4. t f ( t ) ≥ 2 F ( t ) for all t ∈ R .

It is easy to see that (1.10) is equivalent to (1.9). For equation (1.4), we give the following assumption:

1. there exist q > p > 2 , ν ∈ ( 0 , 1 ] , μ > 0 , and λ > 0 such that (1.8) and the following inequality hold:

   (1.11) μ > ν p ∕ 2 1 − 1 − ν λ 2 ∕ q q q − 2 ( q − 2 ) α 0 C p 2 p p − 2 4 π q V ˆ 2 ( q − p ) ∕ ( p − 2 ) ( q − 2 ) p − 2 2 α 0 ( p − 2 ) 4 π p p − 2 2 C p p .

Our main results are the following two theorems.

Since

S 2 ≥ V ˆ , so we can derive (1.10) with V ( ∣ x ∣ ) ≡ V ˆ from (1.9). Therefore, we have the following theorem by the proof [5, Corollary 1.5].

The article is organized as follows. In Section 2, we give the variational setting and some preliminary lemmas. Section 3 is devoted to the estimation of the energy functional associated with (1.1) where a fine threshold is established by a careful analysis. In Section 4, we complete the proofs of Theorems 1.2 and 1.3.

Throughout the sequel, we denote the usual Lebesgue space with norm ‖ u ‖ s = ∫ R 2 ∣ u ∣ s d x 1 s by L s ( R 2 ) , where 1 ≤ s < ∞ , B r ≔ { x ∈ R 2 : ∣ x ∣ < r } for all r > 0 , and C denotes different positive constants in different places.

2 Variational framework and preliminaries

In view of Lemmas 2.1 and 2.2, it is easy to see that the energy functional associated with problem (1.1) Φ : E → R by

is well defined under (V1), (Q1), (F1), and (F2), and by using standard arguments, we can show that Φ ∈ C 1 ( E , R ) with derivative given by

for all φ ∈ E . Now we say that u ∈ E is a weak solution to problem (1.1), if for all φ ∈ C 0 ∞ ( R 2 ) it holds that ⟨ Φ ′ ( u ) , φ ⟩ = 0 .

Finally, let

Then X is a Hilbert space when endowed with inner product

3 Minimax estimates

In this section, we give a precise estimation about the minimax level c * defined by Lemma 2.4.