On quasilinear elliptic problems with finite or infinite potential wells

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(\phi(|\nabla u|)\nabla u\big)+V(x)\phi (|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$ satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,\Phi }(\mathbb{R}^{N})$ into $L^{\Phi}(\mathbb{R}^{N})$ is established, which enables us to get infinitely many solutions for the case that $f$ is odd. For the case that $V(x)=\lambda a(x) + 1$ exhibits a steep potential well controlled by a positive parameter $\lambda$, we get nontrivial solutions for large $\lambda$.

Nonlinear elliptic problems in R N like (1.1) have been extensively studied. For example, if φ(t) ≡ 1, then the problem (1.1) reduces to the following stationary Schrödinger equation 3) which is a central topic in nonlinear analysis in the last decads, see [19,23,26,28,30,32] and the reference therein. If φ(t) = t p−2 , then the leading term in (1.1) is the p-Laplacian operator −∆ p and the corresponding problem has also been studied in many papers such as [8,9,25,27]. If φ(t) = t p−2 + t q−2 , the leading term in (1.1) is the so-called This work was supported by NSFC (11671331, 11971436) and a special fund from Ministry of Education of China (20180707). It was completed while the author was visiting the Abdus Salam International Centre for Theoretical Physics (ICTP), he would like to thank ICTP for the hospitality.
(p, q)-Laplacian operator and results for the corresponding problems can be found in [10,16,24].
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 [5], the authors studied the problem (1.1) by variational methods under the following conditions on the potential V and the nonlinearity f .
where φ * is related to Φ * , the Sobolev conjugate function of Φ (see (2.4)), via Because the 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 [6]. 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 [7,31] 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 on the assumptions on φ.
In [17], Chorfi and Rǎdulescu studied the following problem where the function φ is the same as in [7] and a verifies lim |x|→0 a(x) = +∞, lim |x|→+∞ a(x) = +∞. (1.6) Note that the zero order term on the left hand side of (1.5) is a power function of u, which is different to that of (1.1). Using the strategy initiated by Rabinowitz [30], condition (1.6) enables the authors to overcome the lack of compactness and obtain a nontrivial solution for the 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 [2][3][4].
Our results are closely related to those of Alves et al [5]. 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 [11] in their study of (1.3). ( (1.7) 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 Then, solutions of (1.1) are critical points of J.
Motivated by Bartsch and Wang [11], 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 inverstigate the case that V exhibits a finite potential well: 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 : (φ s 3 ) for some s ≥ 2, the function t → φ(t)/t s−2 is nonincreasing on (0, ∞), ( f s 3 ) for some s ≥ 2, the function t → f (t)/ |t| s−1 is strictly increasing on (0, ∞) and (−∞, 0). Note that ( f s 3 ) implies that for all ξ ∈ R\{0}, t → f (tξ)/t s−1 is strictly increasing on (0, ∞). Our result reads as follows.
Our Theorems 1.1 and 1.3 are generalizations of Theorem 2.1 and part of Theorem 2.4 in Bartsch and Wang [11], respectively. However, even for the semilinear case that φ(t) ≡ 1, our Theorem 1.3 is slightly general than the corresponding result in [11], because in ( f 1 ) we only require f to be asymptotically subcritical, that is the second limit in (1.4) holds, while in [11,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 1.2 also generalizes Rabinowitz [30,Theorem 4.27].
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 1.1 (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 refereed to [5,21] and the references therein, in particular [1], for more details.
A convex, even continuous function Φ : Then, for an open subset Ω of R N , under the natural addition and scale multiplication, is a vector space. Equipped with the Luxemburg norm The complement function of Φ, denoted byΦ, is given by the Legendre transformatioñ 2) and we have the Hölder inequalityˆΩ is called the Sobolev conjugate function of Φ. It is known that similar to (1.2) we have It is also known that, if Φ andΦ satisfy the ∆ 2 -condition, then L Φ (Ω) and W 1,Φ (R N ) are reflexive and separable. Moreover, This can be seem by setting V = 1 in (2.10) below. Let Ψ be an N-function verifying ∆ 2 -condition. It is well known that if is compact, such Ψ is call 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 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 Then for all u ∈ X we have Proof. According to [21, Lemma 2.1], we have 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.
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 ). Firstly, we have u n ≤ C 1 for some C 1 > 0. For any ε > 0, by (2.8) there is k > 0 such that 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 ). Hencê for some C 2 > 0, where we have used an inequality for Φ * similar to (2.10).
Given M > 0 and R > 0, set Using the above inequalities and Lemma 2.1 we havê Consequently,ˆ| Because ε is arbitrary, this impliesˆR

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 is of class C 1 . The derivative of J is given by 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 [5, Lemma 4.1] that J satisfies the mountain pass geometry: for some ρ > 0 and ϕ ∈ Remark 3.1. In [5], J(tϕ) → −∞ 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 verifivation of condition (2) of Proposition 4.3 in the proof of Theorem 1.1 (2) below. Therefore, the limit in (3.2) is in fact valid for any ϕ ∈ X.
The following result has been established in [5,Lemma 4.3]. 3.1. Proof of Theorem 1.1 (1). By the above arguments, we know that J has a bounded (PS ) 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 [5, 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 getˆR N Φ(|u n |) → 0. (3.5) Using this inequality and (3.5), and the boundedness of {u n } in L Φ * (R N ), we deducê From this and (φ 2 ) we getˆR 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 (PS ) condition. Hence the (PS ) c sequence {u n } has a subsequence 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 1.2. For convenience, in this subsection we assume all the conditions on φ, V and f required in Theorem 1.2. As before, J has a bounded (PS ) 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, Proof. By condition (V 2 ), for any ε > 0, there is R > 0 such that If u = 0, then u n ⇀ 0 in X. By the compactness of the embedding X ֒→ L Φ loc (R N ) we haveˆ| x|<R Φ(|u n |) → 0.

Consequently,
In a similar manner we can prove 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 [5, Theorem 1.8 (b)] that for {u n }, the bounded (PS ) 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 for t ≥ 0. Hence, we may apply the Fatou lemma to get Define a C 1 -function ̺ : [0, ∞) → R by Then for t > 0, Hence, for the s ≥ 2 in assumptions (φ s 3 ) and ( f s 3 ) we have Since ̺ ′ (1) = J ′ ∞ (v), v = 0, by (φ s 3 ) and ( f s 3 ) and a monotonicity argument we see that ̺ ′ (t) > 0 for t ∈ (0, 1), ̺ ′ (t) < 0 for t ∈ (1, ∞). Hence, Now, we are ready to conclude the proof of Theorem 1.2. For the bounded (PS ) c sequence {u n } given in (3.4), we known 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 (PS ) c sequence of the limiting functional J ∞ , which will produce a nonzero critical point v of J ∞ satisfying J ∞ (v) ≤ c, see (3.7).

Proof of Theorem 1.3. On the subspace
Then E becomes a Banach space. To prove Theorem 1.3 we only need to find nonzero critical points of J λ : E → R, As before, J λ verifies the assumptions of the mountain pass theorem thus has a (PS ) c λ sequence u λ being Γ λ = { γ ∈ C(I, E)| γ(0) = 0, J λ (γ(1)) < 0}. Moreover, the sequence u λ n ∞ n=1 is bounded in E. Going to a subsequence if necessary, u λ n ⇀ 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 [11, §5], we need to create the required estimates more carefully because our differential operator −∆ Φ is much more complicated than in [11].
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 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 Proof. It has been shown in [5, 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 λ n . From (3.10), using (3.6) we have (note that λ ≥ 1) (3.14) Since the first integral in the last line is nonnegative, it follows that Moreover, as indicated in (3.13), {c λ } λ≥1 is bounded above byc, 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 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 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 [11] for the case φ(t) ≡ 1, because instead of being strictly subcritical (meaning that f (t) is controlled by some subcritical Nfunction 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).
Having proven Lemmas 3.5 and 3.7, we are ready to complete the proof of Theorem 1.3. Set ε = α/2 in Lemma 3.7 and fix λ * > 0 and R > 0 as in the lemma. If λ ≥ λ * , for u λ n , the (PS ) c λ sequence of J λ , we have u λ n ⇀ u λ and u λ is a critical point of J λ . Since the embedding E ֒→ L Ψ (B R ) is compact, Therefore, u λ is a nonzero critical point of J λ . By assumption ( f 1 ), for any ε > 0, there is C ε > 0 such that

Multiple solutions
For u ∈ X, by Hölder inequality (2.3) we havê Note that from (2.2), (2.11) and Lemma 2.1 with V ≡ 1, we havê Therefore, since {u n } is bounded in L Φ (R N ), {Φ ′ (|u n |)} is also bounded in LΦ(R N ). Similarly, Φ ′ * (|u n |) is bounded in L Φ * (R N ). (We remind the reader that instead of the Sobolev conjugate function ofΦ, here Φ * is the complement function of Φ * .) Therefore 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 ˆR To prove that u n → u in X, we adapt the argument of [15,Appendix A], where for V(x) ≡ 0, a Φ-Laplacian problem on a bounded domain is considered. Let A : X → X * be defined by Then it is well known that • A is hemicontinuous, i.e., for all u, v, w ∈ X, the function By [14,Lemma 2.98], we know that A is pseudomonotone, i.e., for {u n } ⊂ X, together imply A(u n ) ⇀ A(u) in X * and A(u n ), u n → Au, u . For our bounded (PS ) c sequence {u n }, (4.1) implies that (4.4) holds up to a subsequence. Therefore A(u n ), u n → Au, u . (4.5) 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 Let g n : R N → R be defined by Then by the monotonicity and convexity of Φ, using (2.11) and (φ 2 ) we get We have a.e. in R N , and g ∈ L 1 (R N ). Moreover, using (4.5) we get Now, (4.6), (4.7), (4.8) and the generalized Lebesgue dominated theorem giveŝ is weakly-strongly continuous, that is, if u n ⇀ u in X, then F (u n ) → F (u).
: 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 1.1 (2). We know that the C 1 -functional J given in (3.1) is even, satisfies (PS ) 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 i j and X = span {e i | i ≥ 1} , X * = span w * f i i ≥ 1 .
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