On quasilinear elliptic problems with finite or infinite potential wells

Shibo Liu

doi:10.1515/math-2021-0053

Article Open Access

On quasilinear elliptic problems with finite or infinite potential wells

Shibo Liu

Published/Copyright: August 31, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

We consider quasilinear elliptic problems of the form

− div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) ,

where ϕ and f satisfy suitable conditions. The positive potential V ∈ C ( R N ) exhibits a finite or infinite potential well in the sense that V ( x ) tends to its supremum V ∞ ≤ + ∞ as ∣ x ∣ → ∞ . Nontrivial solutions are obtained by variational methods. When V ∞ = + ∞ , a compact embedding from a suitable subspace of W 1 , Φ ( R N ) into L Φ ( R N ) is established, which enables us to get infinitely many solutions for the case that f is odd. For the case that V ( x ) = λ a ( x ) + 1 exhibits a steep potential well controlled by a positive parameter λ , we get nontrivial solutions for large λ .

Keywords: Orlicz-Sobolev space; Φ-Laplacian; critical points; (PS)c sequence

MSC 2010: 35J20; 35J60; 35J70

1 Introduction

In this paper, we consider the following quasilinear elliptic problem in R N ,

(1.1) − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) .

where ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) is a C 1 -function satisfying the following assumptions:

the function t ↦ ϕ ( t ) t is increasing in ( 0 , ∞ ) ,
there exist ℓ , m ∈ ( 1 , N ) such that
(1.2) ℓ ≤ ϕ ( ∣ t ∣ ) t 2 Φ ( t ) ≤ m for all t ≠ 0 ,
where ℓ ≤ m < ℓ ∗ (note that for p ∈ ( 1 , N ) we set p ∗ = N p / ( N − p ) ),
Φ ( t ) = ∫ 0 ∣ t ∣ ϕ ( s ) s d s .

Nonlinear elliptic problems in R N like (1.1) have been extensively studied. For example, if ϕ ( t ) ≡ 1 , then problem (1.1) reduces to the following stationary Schrödinger equation:

(1.3) − Δ u + V ( x ) u = f ( u ) , u ∈ H 1 ( R N ) ,

which is a central topic in nonlinear analysis in the last decade, see [1,2,3, 4,5,6] and references therein. If ϕ ( t ) = t p − 2 , then the leading term in (1.1) is the p -Laplacian operator − Δ p and the corresponding problem has also been studied in many papers such as [7,8, 9,10]. If ϕ ( t ) = t p − 2 + t q − 2 , the leading term in (1.1) is the so-called ( p , q ) -Laplacian operator and results for the corresponding problems can be found in [11,12,13].

For general ϕ satisfying ( ϕ 1 ) and ( ϕ 2 ) , − Δ Φ u ≔ − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) is called the Φ -Laplacian of u . The Φ -Laplacian operator − Δ Φ arises in some applications such as nonlinear elasticity, plasticity and non-Newtonian fluids. Elliptic boundary value problems involving the Φ -Laplacian have been studied on a bounded domain Ω ⊂ R N in several recent papers, such as Clément et al. [14], Fukagai and Narukawa [15] and Carvalho et al. [16].

For unbounded domains such as R N , there are also some recent results on the quasilinear Φ -Laplacian problem (1.1). In Alves et al. [17], the authors studied problem (1.1) by variational methods under the following conditions on the potential V and the nonlinearity f :

V ∈ C ( R N ) , V 0 = inf R N V > 0 ;
f ∈ C ( R ) satisfies
(1.4) lim ∣ t ∣ → 0 f ( t ) ϕ ( ∣ t ∣ ) t = λ 0 , lim ∣ t ∣ → ∞ f ( t ) ϕ ∗ ( ∣ t ∣ ) t = 0 ,
where ϕ ∗ is related to Φ ∗ , the Sobolev conjugate function of Φ (see (2.4)), via
Φ ∗ ( t ) = ∫ 0 ∣ t ∣ ϕ ∗ ( s ) s d s ;
there exists θ > m such that for all t ≠ 0 ,
0 < F ( t ) ≔ ∫ 0 t f ( s ) d ≤ 1 θ t f ( t ) .

Because problem (1.1) is settled on the unbounded domain R N , to overcome the lack of compactness of the relevant Sobolev embeddings, the authors considered the cases that V is radial or Z N -periodic. Using a Strauss-type result and a Lions-type concentration lemma in Orlicz-Sobolev spaces established in the paper, they obtained nontrivial solutions for the problem via the mountain pass theorem [18].

Remark 1.1

If ϕ ( t ) = t p − 2 , then − Δ Φ is the p -Laplacian operator − Δ p and (1.1) becomes the p -Laplacian equation

− div ( ∣ ∇ u ∣ p − 2 ∇ u ) + V ( x ) ∣ u ∣ p − 2 u = f ( u ) , u ∈ W 1 , p ( R N ) ,

which has been extensively studied in the last few decades, see e.g. [8,19]. In this case, if p ∈ ( 1 , N ) , then a direct computation using (2.4) gives

Φ ∗ ( t ) = ( p ∗ ) − p ∗ p N / ( p − N ) ∣ t ∣ p ∗ , ϕ ∗ ( t ) = ( p ∗ ) 1 − p ∗ p N / ( p − N ) t p ∗ − 2 .

Consequently, the second limit in (1.4) is precisely the usual (asymptotically) subcritical growth condition

lim ∣ t ∣ → ∞ f ( t ) ∣ t ∣ p ∗ − 2 t = 0 .

While if p = N , we could not determine the functions Φ ∗ and ϕ ∗ via (2.4). For this case to apply variational methods one usually requires that f behaves like exp ( α ∣ u ∣ N / ( N − 1 ) ) , see [19] for a typical reference.

For the autonomous case that V ( x ) ≡ 0 and f ( u ) = ∣ u ∣ s − 2 − ∣ u ∣ α − 2 , nontrivial solutions for (1.1) have also been obtained in [20,21] via mountain pass theorem, thanks to the compact embeddings from the radial Orlicz-Sobolev spaces to certain Lebesgue spaces L τ ( R N ) established in these papers. The main difference of these two papers is in the assumptions on ϕ .

In [22], Chorfi and Rǎdulescu studied the following problem:

(1.5) − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + a ( x ) ∣ u ∣ α − 2 u = f ( x , u ) in R N ,

where the function ϕ is the same as in [21] and a verifies

(1.6) lim ∣ x ∣ → 0 a ( x ) = + ∞ , lim ∣ x ∣ → + ∞ a ( x ) = + ∞ .

Note that the zero order term on the left-hand side of (1.5) is a power function of u , which is different than that of (1.1). Using the strategy initiated by Rabinowitz [6], condition (1.6) enables the authors to overcome the lack of compactness and obtain a nontrivial solution for problem (1.5).

There are also some papers for the case that there is a parameter ε > 0 in (1.1), existence and multiplicity of solutions for the equation were obtained for ε small, see [23,24,25].

Our results are closely related to those of Alves et al. [17]. As mentioned before, in their paper they studied the case that the potential V is radial or Z N -periodic. In our first result, we consider the case that V satisfies the following condition due to Bartsch and Wang [26] in their study of (1.3):

for all M > 0 , μ ( V − 1 ( − ∞ , M ] ) < ∞ .

Here μ is the Lebesgue measure on R N . Note that ( V 1 ) is satisfied if V is coercive:

(1.7) lim ∣ x ∣ → ∞ V ( x ) = + ∞ .

To apply variational methods let X be a suitable subspace of the Orlicz-Sobolev space W 1 , Φ ( R N ) that will be made clear in Section 2 and consider the C 1 -functional J : X → R given by

J ( u ) = ∫ R N Φ ( ∣ ∇ u ∣ ) + ∫ R N V ( x ) Φ ( ∣ u ∣ ) − ∫ R N F ( u ) .

Then, solutions of (1.1) are critical points of J .

Theorem 1.2

Suppose ϕ satisfies ( ϕ 1 ) , ( ϕ 2 ) ; V satisfies ( V 0 ) , ( V 1 ) ; f satisfies ( f 1 ) , ( f 2 ) .

If λ 0 = 0 , then (1.1) has a nontrivial solution.
If λ 0 ≥ 0 and f is odd, then (1.1) has a sequence of solutions { u n } such that J ( u n ) → + ∞ .

Motivated by Bartsch and Wang [26], the assumption ( V 1 ) enables us to establish a compact embedding result from our working space X into subcritical Orlicz spaces, see Lemma 2.3. With this result we can regain compactness for our functional J and get critical points.

As a special case of ( V 1 ) , (1.7) can be interpreted as V has an infinite potential well. In our next result, we investigate the case that V exhibits a finite potential well:

for all x ∈ R N , V ( x ) < V ∞ ≔ lim ∣ x ∣ → ∞ V ( x ) < ∞ .

Under the assumption ( V 2 ) the above compact embedding is not valid anymore. Hence, to get critical points of J , we need the following monotonicity assumptions on ϕ and f :

for some s ≥ 2 , the function t ↦ ϕ ( t ) / t s − 2 is nonincreasing on ( 0 , ∞ ) ,
for some s ≥ 2 , the function t ↦ f ( t ) / ∣ t ∣ s − 1 is strictly increasing on ( 0 , ∞ ) and ( − ∞ , 0 ) .

Note that ( f 3 s ) implies that for all ξ ∈ R \ { 0 } , t ↦ f ( t ξ ) / t s − 1 is strictly increasing on ( 0 , ∞ ) . Our result reads as follows.

Theorem 1.3

Suppose ϕ satisfies ( ϕ 1 ) , ( ϕ 2 ) and ( ϕ 3 s ) ; V satisfies ( V 0 ) , ( V 2 ) ; f satisfies ( f 1 ) with λ 0 = 0 , ( f 2 ) and ( f 3 s ) . Then (1.1) has a nontrivial solution.

For our last result, we consider the case that the potential V ( x ) is of the form λ a ( x ) + 1 with λ > 0 and a satisfies

a ∈ C ( R N ) , a ≥ 0 and a − 1 ( 0 ) has nonempty interior;
for some M 0 > 0 we have μ ( a − 1 ( − ∞ , M 0 ] ) < ∞ .

For problem (1.3), these conditions are introduced by Bartsch and Wang [26] and characterize V as possessing a steep potential well whose height is controlled by the positive parameter λ . Our result for this case is the following theorem.

Theorem 1.4

Suppose ( ϕ 1 ) , ( ϕ 2 ) , ( a 1 ) and ( a 2 ) are satisfied, f ∈ C ( R ) satisfies ( f 1 ) with λ 0 = 0 and ( f 2 ) . Then there exists λ ∗ > 0 such that for all λ ≥ λ ∗ , the problem

(1.8) − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V λ ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) ,

has a nontrivial solution, where V λ ( x ) = λ a ( x ) + 1 .

Our Theorems 1.2 and 1.4 are generalizations of Theorem 2.1 and part of Theorem 2.4 in Bartsch and Wang [26], respectively. However, even for the semilinear case that ϕ ( t ) ≡ 1 , our Theorem 1.4 is slightly general than the corresponding result in [26], because in ( f 1 ) we only require f to be asymptotically subcritical, that is the second limit in (1.4) holds, while in [26, Theorem 2.4] the nonlinearity f is strictly subcritical, meaning that the growth of f at infinity is controlled by a subcritical power function ∣ t ∣ q − 2 t for some q ∈ ( 2 , 2 ∗ ) . See Remark 3.6 for more details. Roughly speaking, Theorem 1.3 also generalizes Rabinowitz [6, Theorem 4.27].

Both [6] and [26] are concerned on the semilinear equation (1.3). Our Φ -Laplacian equation (1.1) is much more general.

The paper is organized as follows. In Section 2, we recall some concepts and results about Orlicz spaces and prove the compact embedding lemma (Lemma 2.3) mentioned before. The existence of nontrivial solutions is proved in Section 3. Finally, in Section 4 we deal with the multiplicity result stated in Theorem 1.2(2).

2 Orlicz-Sobolev spaces

In this section, we recall some results about Orlicz spaces and Orlicz-Sobolev spaces that we will use for proving our main results. The reader is referred to [17,27] and references therein, in particular [28], for more details.

A convex, even continuous function Φ : R → [ 0 , ∞ ) is called a nice Young function, N -function for short, if Φ ( t ) = 0 is equivalent to t = 0 , and

lim t → 0 Φ ( t ) t = 0 , lim t → + ∞ Φ ( t ) t = + ∞ .

The N -function Φ satisfies the Δ 2 -condition if there is a constant K > 0 such that

Φ ( 2 t ) ≤ K Φ ( t ) for all t ≥ 0 .

Then, for an open subset Ω of R N , under the natural addition and scale multiplication,

L Φ ( Ω ) = u : Ω → R u is measurable , ∫ Ω Φ ( ∣ u ∣ ) < ∞

is a vector space. Equipped with the Luxemburg norm

∣ u ∣ Φ = inf α > 1 ∫ Ω Φ ∣ u ∣ α ≤ 1 ,

L Φ ( Ω ) is a Banach space, called Orlicz space. The Orlicz-Sobolev space W 1 , Φ ( R N ) is the completion of C 0 ∞ ( R N ) under the norm

(2.1) ∥ u ∥ 1 = ∣ ∇ u ∣ Φ + ∣ u ∣ Φ .

The complement function of Φ , denoted by Φ ˜ , is given by the Legendre transformation

Φ ˜ ( s ) = max t ≥ 0 { s t − Φ ( t ) } for s ≥ 0 .

Then,

(2.2) Φ ˜ ( Φ ′ ( t ) ) ≤ Φ ( 2 t ) for t ≥ 0 ,

and we have the Hölder inequality

(2.3) ∫ Ω ∣ u v ∣ ≤ 2 ∣ u ∣ Φ ∣ v ∣ Φ ˜

for u ∈ L Φ ( Ω ) and v ∈ L Φ ˜ ( Ω ) .

When

∫ 1 + ∞ Φ − 1 ( s ) s ( N + 1 ) / N d s = + ∞ ,

the function Φ ∗ given by

(2.4) Φ ∗ − 1 ( t ) = ∫ 0 t Φ − 1 ( s ) s ( N + 1 ) / N d s

is called the Sobolev conjugate function of Φ . It is known that similar to (1.2) we have

(2.5) ℓ ∗ ≤ ϕ ∗ ( ∣ t ∣ ) t 2 Φ ∗ ( t ) ≤ m ∗ for all t ≠ 0 .

It is also known that if Φ and Φ ˜ satisfy the Δ 2 -condition, then L Φ ( Ω ) and W 1 , Φ ( R N ) are reflexive and separable. Moreover,

(2.6) u n → u in L Φ ( Ω ) ⇔ ∫ Ω Φ ( ∣ u n − u ∣ ) → 0 ,

(2.7) u n → u in W 1 , Φ ( R N ) ⇔ ∫ R N ( Φ ( ∣ ∇ u n − ∇ u ∣ ) + Φ ( ∣ u n − u ∣ ) ) → 0 .

In addition, { u n } is bounded in L Φ ( R N ) if and only if { Φ ( ∣ u n ∣ ) } is bounded in L 1 ( R N ) . This can be seen by setting V = 1 in (2.10) below.

Let Ψ be an N -function verifying Δ 2 -condition. It is well known that if

lim ¯ t → 0 Ψ ( t ) Φ ( t ) < + ∞ , lim ¯ ∣ t ∣ → + ∞ Ψ ( t ) Φ ∗ ( t ) < + ∞ ,

then we have a continuous embedding W 1 , Φ ( R N ) ↪ L Ψ ( R N ) . Moreover, if

(2.8) lim ∣ t ∣ → 0 Ψ ( t ) Φ ( t ) < + ∞ , lim ∣ t ∣ → ∞ Ψ ( t ) Φ ∗ ( t ) = 0 ,

then the embedding W 1 , Φ ( R N ) ↪ L loc Ψ ( R N ) is compact, such Ψ is called subcritical.

For the study of problem (1.1), we introduce the following subspace of W 1 , Φ ( R N ) . Assuming ( V 0 ) , ( ϕ 1 ) , and ( ϕ 2 ) on the linear subspace

X = u ∈ W 1 , Φ ( R N ) ∫ R N V ( x ) Φ ( ∣ u ∣ ) < ∞ ,

we equip the norm ∥ u ∥ = ∣ ∇ u ∣ Φ + ∣ u ∣ Φ , V , where

∣ u ∣ Φ , V = inf α > 0 ∫ Ω V ( x ) Φ ∣ u ∣ α ≤ 1 .

Then ( X , ∥ ⋅ ∥ ) is a separable reflexive Banach space, which will be simply denoted by X . If V is bounded, then X is precisely the original Orlicz-Sobolev space W 1 , Φ ( R N ) , the norm ∥ ⋅ ∥ is equivalent to the one given in (2.1).

Lemma 2.1

Assume that ( V 0 ) , ( ϕ 1 ) and ( ϕ 2 ) hold and for t ≥ 0 let

(2.9) ξ 0 ( t ) = min { t ℓ , t m } , ξ 1 ( t ) = max { t ℓ , t m } .

Then for all u ∈ X we have

(2.10) ξ 0 ( ∣ u ∣ Φ , V ) ≤ ∫ R N V ( x ) Φ ( ∣ u ∣ ) ≤ ξ 1 ( ∣ u ∣ Φ , V ) .

Proof

According to [27, Lemma 2.1], we have

(2.11) ξ 0 ( ρ ) Φ ( t ) ≤ Φ ( ρ t ) ≤ ξ 1 ( ρ ) Φ ( t ) for ρ , t ≥ 0 .

Taking ρ = ∣ u ∣ Φ , V and t = ∣ u ( x ) ∣ / ∣ u ∣ Φ , V , we get

∫ R N V ( x ) Φ ( ∣ u ∣ ) = ∫ R N V ( x ) Φ ∣ u ∣ Φ , V ∣ u ∣ ∣ u ∣ Φ , V ≤ ξ 1 ( ∣ u ∣ Φ , V ) ∫ R N V ( x ) Φ ∣ u ∣ ∣ u ∣ Φ , V ≤ ξ 1 ( ∣ u ∣ Φ , V )

because by the definition of ∣ ⋅ ∣ Φ , V , the integral in the last line is not greater than 1. The first inequality in (2.10) can be proved similarly.□

Remark 2.2

Similar to (2.11), because of (2.5), for

ξ 0 ∗ ( t ) = min { t ℓ ∗ , t m ∗ } , ξ 1 ∗ ( t ) = max { t ℓ ∗ , t m ∗ } ,

we have

ξ 0 ∗ ( ρ ) Φ ∗ ( t ) ≤ Φ ∗ ( ρ t ) ≤ ξ 1 ∗ ( ρ ) Φ ∗ ( t ) for ρ , t ≥ 0 .

Because m < ℓ ∗ , using this and (2.11) we have

0 < Φ ( t ) Φ ∗ ( t ) ≤ Φ ( 1 ) ξ 1 ( t ) Φ ∗ ( 1 ) ξ 0 ∗ ( t ) ≤ Φ ( 1 ) Φ ∗ ( 1 ) t m t ℓ ∗ → 0 as ∣ t ∣ → ∞ .

Therefore, Φ is an N -function verifying Δ 2 -condition and (2.8).

Lemma 2.3

Suppose ϕ satisfies ( ϕ 1 ) , ( ϕ 2 ) ; V satisfies ( V 0 ) , ( V 1 ) . Then for any N -function Ψ verifying Δ 2 -condition and (2.8), the embedding X ↪ L Ψ ( R N ) is compact. In particular, X ↪ L Φ ( R N ) is compact.

Proof

Assume that { u n } is a sequence in X such that u n ⇀ 0 in X , we want to show that u n → 0 in L Ψ ( R N ) . First, we have ∥ u n ∥ ≤ C 1 for some C 1 > 0 .

For any ε > 0 , by (2.8) there is k > 0 such that

Ψ ( t ) = Ψ ( ∣ t ∣ ) ≤ ε Φ ∗ ( ∣ t ∣ ) , for ∣ t ∣ > k .

Since the embedding X ↪ L Φ ∗ ( R N ) is continuous and ∥ u n ∥ ≤ C 1 , we deduce that { u n } in bounded in L Φ ∗ ( R N ) . Hence,

(2.12) ∫ ∣ u n ∣ > k Ψ ( ∣ u n ∣ ) ≤ ε ∫ R N Φ ∗ ( ∣ u n ∣ ) ≤ ε ξ 1 ∗ ( ∣ u n ∣ Φ ∗ ) ≤ C 2 ε ,

for some C 2 > 0 , where we have used an inequality for Φ ∗ similar to (2.10).

Given M > 0 and R > 0 , set

A R = { x ∈ R N ∣ ∣ x ∣ ≥ R , V ( x ) ≥ M } , B R = { x ∈ R N ∣ ∣ x ∣ ≥ R , V ( x ) < M } .

By the first limit in (2.8), there is κ > 0 such that for t ∈ [ 0 , k ] we have Ψ ( t ) ≤ κ Φ ( t ) . Because { ∣ u n ∣ Φ , V } is bounded, we can take M > 0 large enough such that

κ M ξ 1 ( ∣ u n ∣ Φ , V ) < ε .

It follows from (2.8) that Ψ is bounded in [ 0 , k ] . By ( V 1 ) , we have μ ( B R ) → 0 as R → ∞ , thus we can choose R > 0 such that

μ ( B R ) ⋅ sup [ 0 , k ] Ψ < ε .

Using the above inequalities and Lemma 2.1 we have

∫ A R ∩ { ∣ u n ∣ ≤ k } Ψ ( ∣ u n ∣ ) ≤ κ ∫ A R ∩ { ∣ u n ∣ ≤ k } Φ ( ∣ u n ∣ ) ≤ κ M ∫ R N V ( x ) Φ ( ∣ u n ∣ ) ≤ κ M ξ 1 ( ∣ u n ∣ Φ , V ) ≤ ε , ∫ B R ∩ { ∣ u n ∣ ≤ k } Ψ ( ∣ u n ∣ ) ≤ μ ( B R ) ⋅ sup [ 0 , k ] Ψ < ε .

Consequently,

∫ ∣ u n ∣ ≤ k Ψ ( ∣ u n ∣ ) ≤ ∫ ∣ x ∣ ≤ R + ∫ A R ∩ { ∣ u n ∣ ≤ k } + ∫ B R ∩ { ∣ u n ∣ ≤ k } Ψ ( ∣ u n ∣ ) ≤ ∫ ∣ x ∣ ≤ R Ψ ( ∣ u n ∣ ) + 2 ε .

Now using (2.12) we get

(2.13) ∫ R N Ψ ( ∣ u n ∣ ) = ∫ ∣ u n ∣ > k Ψ ( ∣ u n ∣ ) + ∫ ∣ u n ∣ ≤ k Ψ ( ∣ u n ∣ ) ≤ ( C 2 + 2 ) ε + ∫ ∣ x ∣ ≤ R Ψ ( ∣ u n ∣ ) .

Since the embedding X ↪ L loc Ψ ( R N ) is compact, from u n ⇀ 0 in X and (2.13) we have

lim ¯ n → ∞ ∫ R N Ψ ( ∣ u n ∣ ) ≤ ( C 2 + 2 ) ε .

Because ε is arbitrary, this implies

∫ R N Ψ ( ∣ u n ∣ ) → 0

and u n → 0 in L Ψ ( R N ) .□

3 Nontrivial solutions

From now on, we assume the conditions ( ϕ 1 ) , ( ϕ 2 ) , ( V 0 ) and ( f 1 ) . Then, the functional J : X → R given by

(3.1) J ( u ) = ∫ R N Φ ( ∣ ∇ u ∣ ) + ∫ R N V ( x ) Φ ( ∣ u ∣ ) − ∫ R N F ( u )

is of class C 1 . The derivative of J is given by

⟨ J ′ ( u ) , v ⟩ = ∫ R N ϕ ( ∣ ∇ u ∣ ) ∇ u ⋅ ∇ v + ∫ R N V ( x ) ϕ ( ∣ u ∣ ) u v − ∫ R N f ( u ) v u , v ∈ X .

Thus, critical points of J are precisely weak solutions of our problem (1.1).

Under the assumptions ( ϕ 1 ) , ( ϕ 2 ) , ( V 0 ) , ( f 1 ) and ( f 2 ) , it has also been proved in [17, Lemma 4.1] that J satisfies the mountain pass geometry: for some ρ > 0 and φ ∈ C 0 ∞ ( R N ) \ { 0 } ,

(3.2) inf ∥ u ∥ = ρ J ( u ) = η > 0 , lim t → + ∞ J ( t φ ) = − ∞ .

Remark 3.1

In [17], J ( t φ ) → − ∞ as t → + ∞ is only verified for φ ∈ C 0 ∞ ( R N ) \ { 0 } . But we can prove that J is anti-coercive on any finite dimensional subspace, see the verification of condition (2) of Proposition 4.3 in the proof of Theorem 1.2(2). Therefore, the limit in (3.2) is in fact valid for any φ ∈ X .

Denote I = [ 0 , 1 ] and set

(3.3) c = inf γ ∈ Γ max t ∈ [ 0 , 1 ] J ( γ ( t ) )

being Γ = { γ ∈ C ( I , X ) ∣ γ ( 0 ) = 0 , J ( γ ( 1 ) ) < 0 } . Note that c ≥ η > 0 .

According to the mountain pass theorem [18,29], there is a sequence { u n } ⊂ X such that

(3.4) J ( u n ) → c , J ′ ( u n ) → 0 .

Such sequence is called a ( P S ) c sequence (named after R. Palais and S. Smale). Under the assumptions ( ϕ 1 ) , ( ϕ 2 ) , ( V 0 ) , ( f 1 ) and ( f 2 ) , it has been shown in [17, Lemma 4.2] that the ( P S ) c sequence { u n } we just obtained is bounded in X .

The following result has been established in [17, Lemma 4.3].

Lemma 3.2

Suppose ( ϕ 1 ) , ( ϕ 2 ) , ( V 0 ) and ( f 1 ) hold. Let { u n } be a ( P S ) c sequence of J . If u n ⇀ u in X , then ∇ u n → ∇ u a.e. in R N and J ′ ( u ) = 0 .

3.1 Proof of Theorem 1.2(1)

By the above arguments, we know that J has a bounded ( P S ) c sequence { u n } . Since X is reflexive, we may assume that u n ⇀ u in X . By Lemma 3.2, u is a critical point of J . We need to show that u ≠ 0 . Thanks to the compact embedding established in Lemma 2.3, this can be achieved as in [17, p. 454]. For the reader’s convenience, we include the argument below.

Assume that u = 0 . By Lemma 2.3, the embedding X ↪ L Φ ( R N ) is compact. Thus, u n → 0 in L Φ ( R N ) and we get

(3.5) ∫ R N Φ ( ∣ u n ∣ ) → 0 .

By ( f 1 ) , for any ε > 0 , there exists C ε > 0 such that

∣ f ( t ) t ∣ ≤ ε Φ ∗ ( ∣ t ∣ ) + C ε Φ ( ∣ t ∣ ) .

Using this inequality and (3.5), and the boundedness of { u n } in L Φ ∗ ( R N ) , we deduce

∫ R N f ( u n ) u n → 0 .

Now, because ⟨ J ′ ( u n ) , u n ⟩ → 0 , we obtain

∫ R N ϕ ( ∣ ∇ u n ∣ ) ∣ ∇ u n ∣ 2 + ∫ R N V ( x ) ϕ ( ∣ u n ∣ ) u n 2 → 0 .

From this and ( ϕ 2 ) we get

∫ R N Φ ( ∣ ∇ u n ∣ ) + ∫ R N V ( x ) Φ ( ∣ u n ∣ ) → 0 .

That is u n → 0 in X . But J ( u n ) → c > 0 , this is a contradiction.

Remark 3.3

In Lemma 4.1, we will show that J satisfies the ( P S ) condition. Hence, the ( P S ) c sequence { u n } has a subsequence that converges to a nonzero critical point u at the level c > 0 . We include the above argument for its simplicity.

3.2 Proof of Theorem 1.3

For convenience, in this subsection we assume all the conditions on ϕ , V and f required in Theorem 1.3. As before, J has a bounded ( P S ) c sequence { u n } , u n ⇀ u in X = W 1 , Φ ( R N ) and u is a critical point of J . To show that u ≠ 0 , we need to consider the limiting functional J ∞ : X → R ,

J ∞ ( u ) = ∫ R N Φ ( ∣ ∇ u ∣ ) + ∫ R N V ∞ Φ ( ∣ u ∣ ) − ∫ R N F ( u ) .

Lemma 3.4

If u = 0 , then { u n } is also a ( P S ) c sequence of J ∞ .

Proof

By condition ( V 2 ) , for any ε > 0 , there is R > 0 such that

∣ V ( x ) − V ∞ ∣ < ε for ∣ x ∣ ≥ R .

If u = 0 , then u n ⇀ 0 in X . By the compactness of the embedding X ↪ L loc Φ ( R N ) , we have

∫ ∣ x ∣ < R Φ ( ∣ u n ∣ ) → 0 .

Consequently,

∣ J ∞ ( u n ) − J ( u n ) ∣ = ∫ R N ( V ∞ − V ( x ) ) Φ ( ∣ u n ∣ ) = ∫ ∣ x ∣ < R + ∫ ∣ x ∣ ≥ R ( V ∞ − V ( x ) ) Φ ( ∣ u n ∣ ) ≤ V ∞ ∫ ∣ x ∣ < R Φ ( ∣ u n ∣ ) + ε ∫ ∣ x ∣ ≥ R Φ ( ∣ u n ∣ ) ≤ V ∞ ∫ ∣ x ∣ < R Φ ( ∣ u n ∣ ) + C ε ,

because { u n } is bounded in L Φ ( R N ) . It follows that

lim ¯ n → ∞ ∣ J ∞ ( u n ) − J ( u n ) ∣ ≤ C ε ,

which implies J ∞ ( u n ) − J ( u n ) → 0 .

In a similar manner, we can prove

∥ J ∞ ′ ( u n ) − J ′ ( u n ) ∥ = sup h ∈ X , ∥ h ∥ = 1 ∫ R N ( V ∞ − V ( x ) ) u n h → 0 .

Thus, J ∞ ( u n ) → c and J ∞ ′ ( u n ) → 0 .□

Considering the constant potential V ∞ as a Z N -periodic function, it has been shown in the proof of [17, Theorem 1.8(b)] that for { u n } , the bounded ( P S ) c sequence of J ∞ obtained in Lemma 3.4, there exists a sequence { y n } ⊂ R N such that setting v n ( x ) = u n ( x − y n ) for x ∈ R N , then v n ⇀ v in X , and v is a nonzero critical point of J ∞ .

We claim that J ∞ ( v ) ≤ c . In addition to the obvious fact that v n → v a.e. in R N , by applying Lemma 3.2 to J ∞ we also have ∇ v n → ∇ v a.e. in R N . By the assumptions ( ϕ 2 ) , ( f 2 ) , and θ > m , we have

(3.6) Φ ( ∣ t ∣ ) − 1 θ ϕ ( ∣ t ∣ ) t 2 ≥ 1 − m θ Φ ( ∣ t ∣ ) ≥ 0 , 1 θ f ( t ) t − F ( t ) ≥ 0

for t ≥ 0 . Hence, we may apply the Fatou lemma to get

(3.7) c = lim n → ∞ J ∞ ( v n ) − 1 θ ⟨ J ∞ ′ ( v n ) , v n ⟩ = lim ̲ n → ∞ ∫ R N Φ ( ∣ ∇ v n ∣ ) − 1 θ ϕ ( ∣ ∇ v n ∣ ) ∣ ∇ v n ∣ 2 + ∫ R N V ∞ Φ ( ∣ v n ∣ ) − 1 θ ϕ ( ∣ v n ∣ ) ∣ v n ∣ 2 + ∫ R N 1 θ f ( v n ) v n − F ( v n ) ≥ ∫ R N Φ ( ∣ ∇ v ∣ ) − 1 θ ϕ ( ∣ ∇ v ∣ ) ∣ ∇ v ∣ 2 + ∫ R N V ∞ Φ ( ∣ v ∣ ) − 1 θ ϕ ( ∣ v ∣ ) ∣ v ∣ 2 + ∫ R N 1 θ f ( v ) v − F ( v ) = J ∞ ( v ) − 1 θ ⟨ J ∞ ′ ( v ) , v ⟩ = J ∞ ( v ) .

Define a C 1 -function ϱ : [ 0 , ∞ ) → R by

ϱ ( t ) = J ∞ ( t v ) = ∫ R N Φ ( t ∣ ∇ v ∣ ) + ∫ R N V ∞ Φ ( t ∣ v ∣ ) − ∫ R N F ( t v ) .

Then for t > 0 ,

ϱ ′ ( t ) = ⟨ J ∞ ′ ( t v ) , v ⟩ = t ∫ R N ( ϕ ( t ∣ ∇ v ∣ ) ∣ ∇ v ∣ 2 + V ∞ ϕ ( t ∣ v ∣ ) v 2 ) − ∫ R N f ( t v ) v .

Hence, for the s ≥ 2 in assumptions ( ϕ 3 s ) and ( f 3 s ) we have

ϱ ′ ( t ) > 0 ⇔ 1 t s − 2 ∫ R N ( ϕ ( t ∣ ∇ u ∣ ) ∣ ∇ u ∣ 2 + V ∞ ϕ ( t ∣ v ∣ ) v 2 ) > ∫ R N f ( t v ) v t s − 1 , ϱ ′ ( t ) = 0 ⇔ 1 t s − 2 ∫ R N ( ϕ ( t ∣ ∇ u ∣ ) ∣ ∇ u ∣ 2 + V ∞ ϕ ( t ∣ v ∣ ) v 2 ) = ∫ R N f ( t v ) v t s − 1 , ϱ ′ ( t ) < 0 ⇔ 1 t s − 2 ∫ R N ( ϕ ( t ∣ ∇ u ∣ ) ∣ ∇ u ∣ 2 + V ∞ ϕ ( t ∣ v ∣ ) v 2 ) < ∫ R N f ( t v ) v t s − 1 .

Since ϱ ′ ( 1 ) = ⟨ J ∞ ′ ( v ) , v ⟩ = 0 , by ( ϕ 3 s ) and ( f 3 s ) and a monotonicity argument we see that

ϱ ′ ( t ) > 0 for t ∈ ( 0 , 1 ) , ϱ ′ ( t ) < 0 for t ∈ ( 1 , ∞ ) .

Hence,

(3.8) J ∞ ( v ) = ϱ ( 1 ) = max t ≥ 0 ϱ ( t ) = max t ≥ 0 J ∞ ( t v ) .

Now, we are ready to conclude the proof of Theorem 1.3. For the bounded ( P S ) c sequence { u n } given in (3.4), we know that the weak limit u of a subsequence is a critical point of J . If u = 0 , by Lemma 3.4, this { u n } is also a ( P S ) c sequence of the limiting functional J ∞ , which will produce a nonzero critical point v of J ∞ satisfying J ∞ ( v ) ≤ c , see (3.7).

We also know that J ∞ ( t v ) → − ∞ as t → + ∞ , see Remark 3.1. Choose T > 0 such that J ∞ ( T v ) < 0 and define γ : [ 0 , 1 ] → X , γ ( t ) = t T v . Then γ ( 0 ) = 0 ,

J ( γ ( 1 ) ) < J ∞ ( γ ( 1 ) ) = J ∞ ( T v ) < 0

because V ( x ) < V ∞ for all x ∈ R N , hence γ ∈ Γ .

By assumption ( V 2 ) , (3.7) and (3.8), we see that for t ∈ ( 0 , 1 ] ,

(3.9) J ( γ ( t ) ) < J ∞ ( γ ( t ) ) ≤ J ∞ ( v ) ≤ c .

Because γ ( 0 ) = 0 , J ( 0 ) = 0 and c > 0 , (3.9) is also true at t = 0 . Hence,

max t ∈ [ 0 , 1 ] J ( γ ( t ) ) = J ( γ ( t 0 ) ) < J ∞ ( v ) ≤ c

for some t 0 ∈ [ 0 , 1 ] , contradicting the definition of c given in (3.3). Therefore, u ≠ 0 and it is a nonzero critical point of J .

3.3 Proof of Theorem 1.4

On the subspace

E = u ∈ W 1 , Φ ( R N ) ∫ R N a ( x ) Φ ( ∣ u ∣ ) < ∞

of W 1 , Φ ( R N ) , we equip the norm

∥ u ∥ = ∣ ∇ u ∣ Φ + ∣ u ∣ Φ , ( a + 1 ) .

Then E becomes a Banach space. To prove Theorem 1.4, we only need to find nonzero critical points of J λ : E → R ,

J λ ( u ) = ∫ R N Φ ( ∣ ∇ u ∣ ) + ∫ R N ( λ a ( x ) + 1 ) Φ ( ∣ u ∣ ) − ∫ R N F ( u ) .

As before, J λ verifies the assumptions of the mountain pass theorem thus has a ( P S ) c λ sequence { u n λ } n = 1 ∞ satisfying

(3.10) J λ ( u n λ ) → c λ > 0 , J λ ′ ( u n λ ) → 0 ,

where

(3.11) c λ = inf γ ∈ Γ λ max t ∈ [ 0 , 1 ] J λ ( γ ( t ) )

being Γ λ = { γ ∈ C ( I , E ) ∣ γ ( 0 ) = 0 , J λ ( γ ( 1 ) ) < 0 } .

Moreover, the sequence { u n λ } n = 1 ∞ is bounded in E . Going to a subsequence if necessary, u n λ ⇀ u λ in E , and u λ is a critical point of J λ due to Lemma 3.2. We need to show that u λ ≠ 0 . Although the basic idea can be traced back to [26, §5], we need to create the required estimates more carefully because our differential operator − Δ Φ is much more complicated than in [26].

Take an N -function satisfying (2.8). From ( f 1 ) with λ 0 = 0 , for any ε > 0 , there exists C ε > 0 such that

(3.12) 1 ℓ f ( t ) t − F ( t ) ≤ ε ( Φ ( ∣ t ∣ ) + Φ ∗ ( ∣ t ∣ ) ) + C ε Ψ ( ∣ t ∣ ) .

By assumption ( a 1 ) , we can take v ∈ E \ { 0 } , such that supp v is contained in the interior of a − 1 ( 0 ) . Then, by the mountain pass characterization of c λ in (3.11), we have

(3.13) c λ ≤ max t ≥ 0 J λ ( t v ) = max t ≥ 0 ∫ Φ ( t ∣ ∇ v ∣ ) + ∫ Φ ( t ∣ v ∣ ) − ∫ F ( t v ) = c ˜ < + ∞ ,

see the proof of [17, Lemma 4.1(b)] for more details.

Lemma 3.5

There exists α > 0 such that for all λ ≥ 1 ,

lim ̲ n → ∞ ∫ R N Ψ ( ∣ u n λ ∣ ) ≥ α .

Proof

It has been shown in [17, Lemma 4.1(a)] that u = 0 is a strict local minimizer of J 0 . Since for λ > 0 we have J λ ≥ J 0 , it follows that Γ λ ⊂ Γ 0 . By (3.11), it is easy to see that c λ ≥ c 0 > 0 .

For simplicity of notation, in the proof of Lemmas 3.5 and 3.7 we drop the superscript λ and write u n for u n λ . From (3.10), using (3.6), we have (note that λ ≥ 1 )

(3.14) c λ = lim n → ∞ J λ ( u n ) − 1 θ ⟨ J λ ′ ( u n ) , u n ⟩ ≥ lim ¯ n → ∞ ∫ R N Φ ( ∣ ∇ u n ∣ ) − 1 θ ϕ ( ∣ ∇ u n ∣ ) ∣ ∇ u n ∣ 2 + ∫ R N ( λ a ( x ) + 1 ) Φ ( ∣ u n ∣ ) − 1 θ ϕ ( ∣ u n ∣ ) ∣ u n ∣ 2 ≥ 1 − m θ lim ¯ n → ∞ ∫ R N Φ ( ∣ ∇ u n ∣ ) + ∫ R N ( λ a ( x ) + 1 ) Φ ( ∣ u n ∣ ) .

Since the first integral in the last line is nonnegative, it follows that

(3.15) lim ¯ n → ∞ ∫ R N ( λ a ( x ) + 1 ) Φ ( ∣ u n ∣ ) ≤ θ m θ − m c λ .

Moreover, as indicated in (3.13), { c λ } λ ≥ 1 is bounded above by c ˜ , it follows from (3.14) that { u n } is bounded in E by a positive constant, which is independent of λ . Therefore, by the continuous embedding E ↪ L Φ ∗ , there exists d > 0 such that

(3.16) ∫ R N Φ ∗ ( ∣ u n ∣ ) ≤ d .

Using ( ϕ 2 ) we have Φ ( t ) ≤ ℓ − 1 ϕ ( t ) t 2 for t ≥ 0 , then using (3.12), (3.16) and (3.15) we get

(3.17) c λ = lim n → ∞ J λ ( u n ) − 1 ℓ ⟨ J λ ′ ( u n ) , u n ⟩ = lim n → ∞ ∫ R N Φ ( ∣ ∇ u n ∣ ) − 1 ℓ ϕ ( ∣ ∇ u n ∣ ) ∣ ∇ u n ∣ 2 + ∫ R N ( λ a ( x ) + 1 ) Φ ( ∣ u n ∣ ) − 1 ℓ ϕ ( ∣ u n ∣ ) u n 2 + ∫ R N 1 ℓ f ( u n ) u n − F ( u n ) ≤ lim ̲ n → ∞ ∫ R N 1 ℓ f ( u n ) u n − F ( u n )

≤ lim ̲ n → ∞ ε ∫ R N Φ ( ∣ u n ∣ ) + ∫ R N Φ ∗ ( ∣ u n ∣ ) + C ε ∫ R N Ψ ( ∣ u n ∣ ) ≤ ε lim ¯ n → ∞ d + ∫ R N ( λ a ( x ) + 1 ) Φ ( ∣ u n ∣ ) + C ε lim ̲ n → ∞ ∫ R N Ψ ( ∣ u n ∣ ) ≤ ε d + θ m θ − m c λ ε + C ε lim ̲ n → ∞ ∫ R N Ψ ( ∣ u n ∣ ) .

Noting that c λ ≥ c 0 > 0 , choosing ε small enough at the very beginning, the desired conclusion follows from (3.17).□

Remark 3.6

Comparing with the argument in [26] for the case ϕ ( t ) ≡ 1 , because instead of being strictly subcritical (meaning that f ( t ) is controlled by some subcritical N -function Ψ ), our nonlinearity f ( t ) is only asymptotically subcritical, in the estimate (3.17) the term involving Φ ∗ presents. Hence, we need the uniform bound (3.16), which in turn is ensured by the upper bound of { c λ } given in (3.13).

Lemma 3.7

For any ε > 0 , there exists λ ∗ ≥ 1 and R > 0 , such that for λ ≥ λ ∗ we have

lim ¯ n → ∞ ∫ ∣ x ∣ ≥ R Ψ ( ∣ u n λ ∣ ) < ε .

Proof

For R > 0 , we set

A R = { x ∈ R N ∣ ∣ x ∣ ≥ R , a ( x ) ≥ M 0 } , B R = { x ∈ R N ∣ ∣ x ∣ ≥ R , a ( x ) < M 0 } .

As in (2.12), because of (2.8) there exists k > 0 such that

(3.18) ∫ ∣ u n ∣ > k Ψ ( ∣ u n ∣ ) ≤ ε 4 .

Using assumption ( a 2 ) , as R → ∞ we have μ ( B R ) → 0 ; therefore, we can fix R > 0 such that

(3.19) ∫ B R ∩ { ∣ u n ∣ ≤ k } Ψ ( ∣ u n ∣ ) ≤ μ ( B R ) ⋅ sup t ∈ [ 0 , k ] Ψ ( t ) < ε 4 .

By the first limit from (2.8), there is κ > 0 such that Ψ ( t ) ≤ κ Φ ( t ) for t ∈ [ 0 , k ] . Thus, using (3.15) and (3.13) we see that if λ is large enough,

(3.20) ∫ A R ∩ { ∣ u n ∣ ≤ k } Ψ ( ∣ u n ∣ ) ≤ κ ∫ A R Φ ( ∣ u n ∣ ) ≤ κ λ M 0 + 1 ∫ A R ( λ a ( x ) + 1 ) Φ ( ∣ u n ∣ ) ≤ κ λ M 0 + 1 2 θ m θ − m c λ ≤ κ λ M 0 + 1 2 θ m θ − m c ˜ < ε 4 .

For such large λ , combining (3.18), (3.19) and (3.20), we see that for the chosen R > 0 ,

lim ¯ n → ∞ ∫ ∣ x ∣ ≥ R Ψ ( ∣ u n ∣ ) ≤ lim ¯ n → ∞ ∫ ∣ u n ∣ > k + ∫ B R ∩ { ∣ u n ∣ ≤ k } + ∫ A R ∩ { ∣ u n ∣ ≤ k } Ψ ( ∣ u n ∣ ) < ε . □

Having proven Lemmas 3.5 and 3.7, we are ready to complete the proof of Theorem 1.4. Set ε = α / 2 in Lemma 3.7 and fix λ ∗ > 0 and R > 0 as in the lemma. If λ ≥ λ ∗ , for { u n λ } , the ( P S ) c λ sequence of J λ , we have u n λ ⇀ u λ and u λ is a critical point of J λ . Since the embedding E ↪ L Ψ ( B R ) is compact,

∫ ∣ x ∣ < R Ψ ( ∣ u λ ∣ ) = lim n → ∞ ∫ ∣ x ∣ < R Ψ ( ∣ u n λ ∣ ) ≥ lim ̲ n → ∞ ∫ R N Ψ ( ∣ u n λ ∣ ) − lim ¯ n → ∞ ∫ ∣ x ∣ ≥ R Ψ ( ∣ u n λ ∣ ) ≥ ε 2 .

Therefore, u λ is a nonzero critical point of J λ .

4 Multiple solutions

Lemma 4.1

Under the assumptions of Theorem 1.2, J satisfies the ( P S ) condition, that is, for any c ∈ R , all ( P S ) c sequences of J have convergent subsequence.

Proof

Let { u n } be a ( P S ) c sequence of J . Then { u n } is bounded and we may assume that u n ⇀ u in X . First, we show that up to a subsequence

(4.1) ∫ R N f ( u n ) ( u n − u ) → 0 .

By assumption ( f 1 ) , for any ε > 0 , there is C ε > 0 such that

(4.2) ∣ f ( t ) ∣ ≤ ε ϕ ∗ ( ∣ t ∣ ) ∣ t ∣ + C ε ϕ ( ∣ t ∣ ) ∣ t ∣ = ε Φ ∗ ′ ( ∣ t ∣ ) + C ε Φ ′ ( ∣ t ∣ ) .

For u ∈ X , by Hölder inequality (2.3) we have

∫ R N Φ ′ ( ∣ u ∣ ) ∣ v ∣ ≤ 2 ∣ Φ ′ ( ∣ u ∣ ) ∣ Φ ˜ ∣ v ∣ Φ .

Note that from (2.2), (2.11) and Lemma 2.1 with V ≡ 1 , we have

∫ R N Φ ˜ ( ∣ Φ ′ ( ∣ u ∣ ) ∣ ) ≤ ∫ R N Φ ( 2 ∣ u ∣ ) ≤ ξ 1 ( 2 ) ∫ R N Φ ( ∣ u ∣ ) ≤ 2 m ξ 1 ( ∣ u ∣ Φ ) .

Therefore, since { u n } is bounded in L Φ ( R N ) , { Φ ′ ( ∣ u n ∣ ) } is also bounded in L Φ ˜ ( R N ) . Similarly, { Φ ∗ ′ ( ∣ u n ∣ ) } is bounded in L Φ ∗ ˜ ( R N ) . (We remind the reader that instead of the Sobolev conjugate function of Φ ˜ , here Φ ∗ ˜ is the complement function of Φ ∗ .) Therefore,

(4.3) M ≔ 2 sup n ∣ Φ ∗ ′ ( ∣ u n ∣ ) ∣ Φ ∗ ˜ ∣ u n − u ∣ Φ ∗ < + ∞ .

Because u n ⇀ u in X , by Lemma 2.3 we have u n → u in L Φ ( R N ) . Now, using (4.2) and Hölder inequality we get

∫ R N f ( u n ) ( u n − u ) ≤ ε ∫ R N Φ ∗ ′ ( ∣ u n ∣ ) ∣ u n − u ∣ + C ε ∫ R N Φ ′ ( ∣ u n ∣ ) ∣ u n − u ∣ ≤ 2 ε ∣ Φ ∗ ′ ( ∣ u n ∣ ) ∣ Φ ∗ ˜ ∣ u n − u ∣ Φ ∗ + 2 C ε ∣ Φ ′ ( ∣ u n ∣ ) ∣ Φ ˜ ∣ u n − u ∣ Φ .

Since u n → u in L Φ ( R N ) , using (4.3) and the boundedness of { Φ ′ ( ∣ u n ∣ ) } in L Φ ˜ ( R N ) , it follows that

lim ¯ n → ∞ ∫ R N f ( u n ) ( u n − u ) ≤ M ε ,

which implies (4.1).

To prove that u n → u in X , we adapt the argument of [16, Appendix A], where for V ( x ) ≡ 0 , a Φ -Laplacian problem on a bounded domain is considered. Let A : X → X ∗ be defined by

⟨ A ( u ) , v ⟩ = ∫ R N ϕ ( ∣ ∇ u ∣ ) ∇ u ⋅ ∇ v + ∫ R N V ( x ) ϕ ( ∣ u ∣ ) u v .

Then it is well known that

A is hemicontinuous, i.e., for all u , v , w ∈ X , the function
t ↦ ⟨ A ( u + t v ) , w ⟩
is continuous on [ 0 , 1 ] .
A is strictly monotone: ⟨ A ( u ) − A ( v ) , u − v ⟩ > 0 for u , v ∈ X with u ≠ v .

By [30, Lemma 2.98], we know that A is pseudomonotone, i.e., for { u n } ⊂ X ,

(4.4) u n ⇀ u in X , lim ¯ n → ∞ ⟨ A ( u n ) , u n − u ⟩ ≤ 0

together imply A ( u n ) ⇀ A ( u ) in X ∗ and ⟨ A ( u n ) , u n ⟩ → ⟨ A u , u ⟩ .

For our bounded ( P S ) c sequence { u n } , (4.1) implies that (4.4) holds up to a subsequence. Therefore,

(4.5) ⟨ A ( u n ) , u n ⟩ → ⟨ A u , u ⟩ .

According to Lemma 3.2, in addition to the well known u n → u a.e. in R N we also have ∇ u n → ∇ u a.e. in R N . By the continuity of Φ we get

(4.6) f n ≔ Φ ( ∣ ∇ u n − ∇ u ∣ ) + V ( x ) Φ ( ∣ u n − u ∣ ) → 0 a.e. in R N .

Let g n : R N → R be defined by

g n = 2 m − 1 ℓ { ϕ ( ∣ ∇ u n ∣ ) ∣ ∇ u n ∣ 2 + V ( x ) ϕ ( ∣ u n ∣ ) ∣ u n ∣ 2 + Φ ( ∣ ∇ u ∣ ) + V ( x ) Φ ( ∣ u ∣ ) } .

Then by the monotonicity and convexity of Φ , using (2.11) and ( ϕ 2 ) we get

∣ f n ∣ ≤ Φ 2 ∣ ∇ u n ∣ + 2 ∣ ∇ u ∣ 2 + V ( x ) Φ 2 ∣ u n ∣ + 2 ∣ u ∣ 2 ≤ Φ ( 2 ∣ ∇ u n ∣ ) + Φ ( 2 ∣ ∇ u ∣ ) 2 + V ( x ) Φ ( 2 ∣ u n ∣ ) + Φ ( 2 ∣ u ∣ ) 2 ≤ 2 m − 1 { [ Φ ( ∣ ∇ u n ∣ ) + Φ ( ∣ ∇ u ∣ ) ] + V ( x ) [ Φ ( ∣ u n ∣ ) + Φ ( ∣ u ∣ ) ] } ≤ 2 m − 1 ℓ { ϕ ( ∣ ∇ u n ∣ ) ∣ ∇ u n ∣ 2 + V ( x ) ϕ ( ∣ u n ∣ ) ∣ u n ∣ 2 + Φ ( ∣ ∇ u ∣ ) + V ( x ) Φ ( ∣ u ∣ ) } = g n .

We have

(4.7) g n → g ≔ 2 m − 1 ℓ { ϕ ( ∣ ∇ u ∣ ) ∣ ∇ u ∣ 2 + V ( x ) ϕ ( ∣ u ∣ ) ∣ u ∣ 2 + Φ ( ∣ ∇ u ∣ ) + V ( x ) Φ ( ∣ u ∣ ) }

a.e. in R N , and g ∈ L 1 ( R N ) . Moreover, using (4.5) we get

∫ R N ( ϕ ( ∣ ∇ u n ∣ ) ∣ ∇ u n ∣ 2 + V ( x ) ϕ ( ∣ u n ∣ ) ∣ u n ∣ 2 ) = ⟨ A ( u n ) , u n ⟩ → ⟨ A ( u ) , u ⟩ = ∫ R N ( ϕ ( ∣ ∇ u ∣ ) ∣ ∇ u ∣ 2 + V ( x ) ϕ ( ∣ u ∣ ) ∣ u ∣ 2 ) ,

which implies

(4.8) ∫ R N g n → ∫ R N g .

Now, (4.6), (4.7), (4.8) and the generalized Lebesgue dominated theorem give

∫ R N [ Φ ( ∣ ∇ u n − ∇ u ∣ ) + V ( x ) Φ ( ∣ u n − u ∣ ) ] = ∫ R N f n → 0 ,

that is u n → u in X .□

Lemma 4.2

The functional ℱ : X → R defined by

ℱ ( u ) = ∫ R N F ( u )

is weakly-strongly continuous, that is, if u n ⇀ u in X , then ℱ ( u n ) → ℱ ( u ) .

Proof

Suppose { u n } ⊂ X satisfies u n ⇀ u in X . Then u n → u in L Φ ( R N ) , by Lemma 2.3. Thus, Φ ( ∣ u n − u ∣ ) → 0 in L 1 ( R N ) , which also implies

Φ ( 2 ∣ u n − u ∣ ) → 0 in L 1 ( R N ) .

By [31, Theorem 4.9], there exists k ∈ L 1 ( R N ) such that up to a subsequence,

Φ ( 2 ∣ u n − u ∣ ) → 0 a.e. in R N , Φ ( 2 ∣ u n − u ∣ ) ≤ k a.e. in R N .

By the monotonicity and convexity of Φ ,

Φ ( ∣ u n ∣ ) ≤ Φ ( ∣ u n − u ∣ + ∣ u ∣ ) ≤ 1 2 Φ ( 2 ∣ u n − u ∣ ) + 1 2 Φ ( 2 ∣ u ∣ ) ≤ 1 2 ( k + Φ ( 2 ∣ u ∣ ) ) .

Since k + Φ ( 2 ∣ u ∣ ) ∈ L 1 ( R N ) , and Φ ( ∣ u n ∣ ) → Φ ( ∣ u ∣ ) a.e. in R N , we deduce

(4.9) ∫ R N Φ ( ∣ u n ∣ ) → ∫ R N Φ ( ∣ u ∣ ) .

For any ε > 0 , choose C ε > 0 such that

∣ F ( t ) ∣ ≤ ε Φ ∗ ( ∣ t ∣ ) + C ε Φ ( ∣ t ∣ ) .

Then we have

∣ ℱ ( u n ) − ℱ ( u ) ∣ = ∫ R N F ( u n ) − ∫ R N F ( u ) ≤ ε ∫ R N Φ ∗ ( ∣ u n ∣ ) + ∫ R N Φ ∗ ( ∣ u ∣ ) + C ε ∫ R N Φ ( ∣ u n ∣ ) − ∫ R N Φ ( ∣ u ∣ ) .

Using (4.9) we get

(4.10) lim ¯ n → ∞ ∣ ℱ ( u n ) − ℱ ( u ) ∣ ≤ ε ∫ R N Φ ∗ ( ∣ u n ∣ ) + ∫ R N Φ ∗ ( ∣ u ∣ ) .

Since { u n } in bounded in X , by the continuous embedding X ↪ L Φ ∗ ( R N ) we see that { u n } is bounded in L Φ ∗ ( R N ) , letting ε → 0 in (4.10) we deduce ℱ ( u n ) → ℱ ( u ) .□

Now we are ready to prove the second part of Theorem 1.2. We need the following symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [18].

Proposition 4.3

[32, Theorem 9.12] Let X = Y ⊕ Z be an infinite dimensional Banach space with dim Y < ∞ . Suppose J ∈ C 1 ( X ) is even, satisfies ( P S ) , J ( 0 ) = 0 and

for some ρ > 0 , inf ∂ B ρ ∩ Z J > 0 , where B ρ = { u ∈ X ∣ ∥ u ∥ < ρ } ;
for any finite dimensional subspace W ⊂ X , there is an R = R ( W ) such that J ≤ 0 on W \ B R ( W ) ,

then J has a sequence of critical values c j → + ∞ .

4.1 Proof of Theorem 1.2(2)

We know that the C 1 -functional J given in (3.1) is even, satisfies ( P S ) and J ( 0 ) = 0 . To get an unbounded sequence of critical values of J , it suffices to verify the two assumptions in Proposition 4.3.

Verification of (1). Since X is separable reflexive Banach space, there exist { e i } 1 ∞ ⊂ X and { f i } 1 ∞ ⊂ X ∗ such that ⟨ f i , e j ⟩ = δ j i and

X = span ¯ { e i ∣ i ≥ 1 } , X ∗ = span ¯ w ∗ { f i ∣ i ≥ 1 } .

Let

Y k = span ¯ { e i ∣ i < k } , Z k = span ¯ { e i ∣ i ≥ k } .

In Lemma 4.2, we have proved that the functional ℱ is weakly-strongly continuous. Therefore, by [33, Lemma 3.3] we see that

(4.11) β k = sup u ∈ ∂ B 1 ∩ Z k ∣ ℱ ( u ) ∣ → 0 .

For u ∈ ∂ B 1 , we have ∣ ∇ u ∣ Φ ≤ 1 and ∣ u ∣ Φ , V ≤ 1 . Hence, there exists a constant c > 0 such that

∫ R N Φ ( ∣ ∇ u ∣ ) + ∫ R N V ( x ) Φ ( ∣ u ∣ ) ≥ ∣ ∇ u ∣ Φ ℓ + ∣ u ∣ Φ , V ℓ ≥ c .

Using (4.11), we can choose k such that β k < c . Set Z = Z k and Y = Y k . Then dim Y < ∞ , X = Y ⊕ Z , for u ∈ ∂ B 1 ∩ Z we have

J ( u ) = ∫ R N Φ ( ∣ ∇ u ∣ ) + ∫ R N V ( x ) Φ ( ∣ u ∣ ) − ℱ ( u ) ≥ c − β k > 0 .

This verifies condition (1) of Proposition 4.3.

Verification of (2). Because θ > m , condition ( f 2 ) implies

lim ∣ t ∣ → ∞ F ( t ) ∣ t ∣ m = + ∞ .

Let W be any given finite dimensional subspace of X and { u n } be a sequence in W such that ∥ u n ∥ → ∞ . Then

v n = u n ∥ u n ∥ → v

for some v ∈ W ∩ ∂ B 1 . For x ∈ { v ≠ 0 } , we have

∣ u n ( x ) ∣ = ∥ u n ∥ ∣ v n ( x ) ∣ → + ∞ .

Applying the Fatou lemma and noting F ≥ 0 , we deduce

1 ∥ u n ∥ m ∫ R N F ( u n ) ≥ 1 ∥ u n ∥ m ∫ v ≠ 0 F ( u n ) = ∫ v ≠ 0 F ( u n ) ∣ u n ∣ ∣ v n ∣ → + ∞ .

Consequently, applying Lemma 2.1 we get

J ( u n ) = ∫ R N Φ ( ∣ ∇ u n ∣ ) + ∫ R N V ( x ) Φ ( ∣ u n ∣ ) − ∫ R N F ( u n ) ≤ ξ 1 ( ∣ ∇ u n ∣ Φ ) + ξ 1 ( ∣ u ∣ Φ , V ) − ∫ R N F ( u n ) ≤ ∣ ∇ u n ∣ Φ m + ∣ ∇ u n ∣ Φ ℓ + ∣ u n ∣ Φ , V m + ∣ u n ∣ Φ , V ℓ − ∫ R N F ( u n ) ≤ 2 ( ∥ u n ∥ m + ∥ u n ∥ ℓ ) − ∫ R N F ( u n ) = 2 ∥ u n ∥ m 1 + ∥ u n ∥ ℓ − m − 1 ∥ u n ∥ m ∫ R N F ( u n ) → − ∞

because ℓ ≤ m . Hence, condition (2) of Proposition 4.3 is verified, and the proof of Theorem 1.2(2) is completed.

Acknowledgements

This work was completed while the author was visiting the Abdus Salam International Centre for Theoretical Physics (ICTP). Therefore, the author would like to thank ICTP for the hospitality. The author would also like to thank the reviewers for their careful reading of the manuscript and useful suggestions.

Funding information: This work was supported by NSFC (12071387, 11971436).
Conflict of interest: The author states no conflict of interest.

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Received: 2020-08-28

Revised: 2021-02-01

Accepted: 2021-06-03

Published Online: 2021-08-31

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

Articles in the same Issue

https://doi.org/10.1515/math-2021-0053

Keywords for this article

Orlicz-Sobolev space; Φ-Laplacian; critical points; (PS)c sequence

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