Existence of weak solutions to a class of Schr

https://doi.org/10.4134/JKMS.j180785 pISSN: 0304-9914 / eISSN: 2234-3008

EXISTENCE OF WEAK SOLUTIONS TO A CLASS OF

SCHR ¨ODINGER TYPE EQUATIONS INVOLVING THE

FRACTIONAL p-LAPLACIAN IN R^N

Jae-Myoung Kim, Yun-Ho Kim, and Jongrak Lee

Abstract. We are concerned with the following elliptic equations:

(−∆)^s_pu + V (x)|u|^p−2u = λg(x, u) in R^N,

where (−∆)^s_pis the fractional p-Laplacian operator with 0 < s < 1 < p <

+∞, sp < N , the potential function V : R^N → (0, ∞) is a continuous potential function, and g : R^N×R → R satisfies a Carath´eodory condition.

We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter λ for which the problem admits at least one nontrivial weak solution when the nonlinearity g has the subcritical growth condition.

1. Introduction

A great attention has been drawn to the study of nonlocal type operators in view of the mathematical theory to concrete some phenomena: social sciences, quantum mechanics, materials science, continuum mechanics, phase transition phenomena, image process and Levy process [6, 8, 14, 21, 24, 32, 33] and the references therein. In particular, the fractional Schr¨odinger equation which is initially introduced by Laskin [24] has received considerable attention in recent years (see e.g. [20, 40, 47]).

Motivated by huge interest in the current literature, exploiting variational methods, we investigate the existence of nontrivial weak solutions for the frac- tional p-Laplacian problems. To be more precise, we consider the existence results of a nontrivial weak solution for the following nonlinear elliptic equa- tions of the fractional p-Laplace type:

(Pλ) (−∆)^s_pu + V (x)|u|^p−2u = λg(x, u) in R^N,

where λ is a real parameter, 0 < s < 1 < p < +∞, sp < N , the potential function V : R^N → (0, ∞) is continuous, g : R^N × R → R is a Carath´eodory

Received November 20, 2018; Accepted January 24, 2019.

2010 Mathematics Subject Classification. 35R11, 35A15, 35J60, 49R05.

Key words and phrases. fractional p-Laplacian, variational methods, critical point theory.

c

2019 Korean Mathematical Society 1529

function, and (−∆)^s_p is the fractional p-Laplacian operator defined as (−∆)^s_pu(x) = 2 lim

ε&0

Z

R^N\B^N_ε(x)

|u(x) − u(y)|^p−2(u(x) − u(y))

|x − y|^{N +ps} dy for x ∈ R^N, where B^N_ε (x) := {y ∈ R^N : |y − x| ≤ ε}. Many researchers have been extensively studied the fractional p-Laplacian type problems in various way; see [4, 5, 8, 13, 14, 19, 22, 26, 36, 38, 44, 47] and the references therein.

Since the seminal work of Ambrosetti and Rabinowitz in [3], critical point theory has become one of the most effective analytic tools for establishing the existence of solutions to elliptic equations of variational type. Afterward, many results for the existence of nontrivial solutions to nonlinear elliptic problems involving the fractional p-Laplacian type have been obtained; see for example [4, 5, 7, 15, 20, 22, 26, 38, 41, 42, 46]. Especially, the existence and multiplicity results for the fractional p-Laplacian type problems have been studied by Iannizzotto et al. [22] and Servadei [38] for a bounded domain in R^N. The key ingredient for obtaining these results is the Ambrosetti and Rabinowitz condition ((AR)- condition for short) in [3];

(AR) There exist positive constants C0 and η such that η > p and 0 < ηG(x, t) ≤ g(x, t)t for x ∈ Ω and |t| ≥ C0, where G(x, t) =Rt

0g(x, s) ds, and Ω is a bounded domain in R^N. It is well known that the (AR)-condition is essential to verify the compact- ness condition of the Euler-Lagrange functional which plays a central role in applying critical point theory. However this condition is very restrictive and eliminates many nonlinearities. Miyagaki and Souto [34] established the exis- tence of a nontrivial solution for the superlinear problems without assuming the (AR)-condition. Inspired by this paper, the existence of at least one solu- tion and infinitely many solutions for the p-Laplacian problem in a bounded domain Ω ⊂ R^N was presented by Liu-Li [31] under the following assumption:

(LL) There exists C∗> 0 such that

G(x, t) ≤ G(x, τ ) + C_∗

for each x ∈ Ω, 0 < t < τ or τ < t < 0, where G(x, t) = g(x, t)t − pG(x, t).

See also [50] for p = 2. In this direction, Wei-Su in [42] showed that the fractional Laplacian problem possesses infinitely many weak solutions. On the other hand, the existence and multiplicity of weak solutions for the p-Laplacian equation in case of the whole space R^N were obtained by Liu [30] under the following assumption:

(Je) There exists η ≥ 1 such that

ηG(x, t) ≥ G(x, τ t)

for all (x, t) ∈ R^N× R and τ ∈ [0, 1], where G(x, t) = g(x, t)t − pG(x, t) and G(x, t) =Rt

0g(x, s)ds.

Recently, under this condition, Torres in [41] obtained the existence result for the fractional p-Laplacian problem by using the mountain pass theorem. In fact, the condition above is originally due to Jeanjean [23] in the case of p = 2.

Following in [31], the condition (Je) is weaker than the condition that for each x ∈ R^N,

g(x, t)

|t|^p−1 is an increasing function of t ∈ R^N\ {0}.

In the last few decades, there were extensive studies dealing with the p-Laplac- ian problems by the assumption (Je); see [27–29] for the p-Laplacian and [2,39, 45] for the p(x)-Laplacian. In this respect, authors in [7, 20, 46] extended the existence of infinitely many weak solutions to the fractional Laplacian problems.

The purpose of this study is twofold. First, by using the mountain pass theorem under the Cerami condition that is slightly weaker than the well known Palais-Smale condition, we present the existence of a nontrivial weak solution for our problem when the condition on g has mild and different assumptions from the condition (Je) based on the arguments in [28, 44, 48]. Second, we concretely provides an estimate of the positive interval of the parameters λ for which the problem (Pλ) admits at least one nontrivial weak solution when the nonlinearity g has the subcritical growth condition (but may not always be p- superlinear). To do this, we give an abstract result that is based on the work of Bonanno [9]. It is worth noticing that we obtain the existence of the nontrivial weak solution for our problem without the facts that the energy functional associated with (P_λ) satisfies the Cerami condition and the mountain pass geometry that is crucial to apply the mountain pass theorem.

This paper is structured as follows. In Section 2, we recall briefly some basic results for the fractional Sobolev spaces. And under various conditions on g, we obtain several existence results of nontrivial weak solutions for problem (Pλ) by utilizing the variational principle. Also we obtain the existence of at least one nontrivial weak solution whenever the parameter λ belongs to a positive interval.

2. Preliminaries and existence of a nontrivial weak solution In this section, we briefly recall some definitions and basic properties of the fractional Sobolev spaces. We refer the reader to [1,21,35] for further references.

Let s ∈ (0, 1) and p ∈ (1, +∞). We define the fractional Sobolev space W^s,p(R^N) as follows:

W^s,p(R^N) :=



u ∈ L^p(R^N) : Z

R^N

Z

R^N

|u(x) − u(y)|^p

|x − y|^{N +ps} dxdy < +∞

 ,

endowed with the norm

||u||_Ws,p(R^N):=



||u||^p_Lp(R^N)+ |u|^p_Ws,p(R^N)

¹_p , where

||u||^p_Lp(R^N):=

Z

R^N

|u|^pdx and |u|^p_Ws,p(R^N):=

Z

R^N

Z

R^N

|u(x) − u(y)|^p

|x − y|^{N +ps} dxdy.

Let s ∈ (0, 1) and 1 < p < +∞. Then W^s,p(R^N) is a separable and re- flexive Banach space. Also, the space C₀^∞(R^N) is dense in W^s,p(R^N), that is W₀^s,p(R^N) = W^s,p(R^N) (see e.g. [1, 35]).

Lemma 2.1 ([17]). Let Ω ⊂ R^N a bounded open set with Lipschitz boundary, s ∈ (0, 1) and p ∈ (1, +∞). Then we have the following continuous embeddings:

W^s,p(Ω) ,→ L^q(Ω) for all q ∈ [1, p^∗_s], if sp < N ; W^s,p(Ω) ,→ L^q(Ω) for every q ∈ [1, ∞), if sp = N ; W^s,p(Ω) ,→ C_b^0,λ(Ω) for all λ < s − N/p, if sp > N, where p^∗_s is the fractional critical Sobolev exponent, that is

p^∗_s:=

( _{N p}

N −sp if sp < N, +∞ if sp ≥ N.

In particular, the space W^s,p(Ω) is compactly embedded in L^q(Ω) for any q ∈ [p, p^∗_s).

Lemma 2.2 ([35, 37]). Let 0 < s < 1 < p < +∞ with ps < N . Then there exists a positive constant C = C(N, p, s) such that for all u ∈ W^s,p(R^N),

||u||_Lp∗s(R^N)≤ C |u|_Ws,p(R^N).

Consequently, the space W^s,p(R^N) is continuously embedded in L^q(R^N) for any q ∈ [p, p^∗_s]. In particular, we denote the best constant S_s,p in the fractional Sobolev inequality by

S_s,p= inf

u∈L^p∗s(R^N)\{0},|u|

W s,p (RN )<∞

|u|^p_Ws,p(R^N)

||u||^p

L^p∗s(R^N)

. For our analysis, we assume that

(V) V ∈ C(R^N), inf_x∈RNV (x) > 0, measx ∈ R^N : V (x) ≤ M < +∞ for all M ∈ R.

When V satisfies (V), the basic space

X_s(R^N) :=u ∈ W^s,p(R^N) : V |u|^p∈ L¹(R^N) denote the completion of C₀^∞(R^N) with respect to the norm

||u||_Xs(RN):=



|u|^p_Ws,p(R^N)+ ||V ^1pu||^p_Lp(R^N)

¹_p .

Lemma 2.3 ([41]). Let 0 < s < 1 < p < +∞ with ps < N and suppose that the assumption (V) holds. Then there is a compact embedding Xs(R^N) ,→ L^q(R^N) for q ∈ [p, p^∗_s).

2.1. Existence of a weak solution via the mountain pass theorem In this subsection, we deal with the existence of a nontrivial weak solution for the problem (P_λ) under suitable assumptions.

Definition 2.4. Let 0 < s < 1 < p < +∞. We say that u ∈ Xs(R^N) is a weak solution of the problem (Pλ) if

Z

R^N

Z

R^N

|u(x) − u(y)|^p−2(u(x) − u(y))(v(x) − v(y))

|x − y|^{N +ps} dxdy

+ Z

R^N

V (x) |u(x)|^p−2uv dx = λ Z

R^N

g(x, u)v dx for all v ∈ X_s(R^N).

Let us define a functional Φs,p: Xs(R^N) → R by Φ_s,p(u) = 1

p Z

R^N

Z

R^N

|u(x) − u(y)|^p

|x − y|^{N +ps} dxdy +1 p

Z

R^N

V (x) |u(x)|^p dx.

So then from Lemma 3.2 of [41], the functional Φs,pis well defined on Xs(R^N), Φ_s,p ∈ C¹(X_s(R^N), R) and its Fr´echet derivative is given by for any v ∈ Xs(R^N),

hΦ⁰_s,p(u), vi = Z

R^N

Z

R^N

|u(x) − u(y)|^p−2(u(x) − u(y))(v(x) − v(y))

|x − y|^{N +ps} dxdy

+ Z

R^N

V (x) |u(x)|^p−2uv dx.

Lemma 2.5 ([36]). Let 0 < s < 1 < p < +∞ and let the assumption (V) hold. Then the functional Φ⁰_s,p is of type (S+), i.e., if un * u in Xs(R^N) and lim sup_n→∞Φ⁰_s,p(un) − Φ⁰_s,p(u), un− u ≤ 0, then un→ u in Xs(R^N) as n → ∞.

Denoting G(x, t) =Rt

0g(x, s) ds and we suppose that for 1 < p < q < p^∗_s and x ∈ R^N,

(G1) g : R^N × R → R satisfies the Carath´eodory condition.

(G2) There exist nonnegative functions a ∈ L^q0(R^N) ∩ L^∞(R^N) and b ∈ L^∞(R^N) such that

|g(x, t)| ≤ a(x) + b(x) |t|^q−1 for all (x, t) ∈ R^N × R, where 1/q + 1/q⁰= 1.

(G3) There exists δ > 0 such that

G(x, t) ≤ 0 for x ∈ R^N, |t| < δ.

(G4) lim_|t|→∞ ^G(x,t)_|t|p = ∞ uniformly for almost all x ∈ R^N. (G5) There exist c0≥ 0, r0≥ 0, and τ > ^N_ps such that

|G(x, t)|^τ ≤ c0|t|^{τ p}G(x, t)

for all (x, t) ∈ R^N × R and |t| ≥ r0, where G(x, t) = (1/p)g(x, t)t − G(x, t) ≥ 0.

(G6) There exist µ > p and % > 0 such that µG(x, t) ≤ tg(x, t) + %t^p for all (x, t) ∈ R^N × R.

Under the assumptions (G1) and (G2), we define the functional Ψ : Xs(R^N)

→ R by

Ψ(u) = Z

R^N

G(x, u) dx.

Then it follows from the same arguments as those of Proposition 1.12 in [43]

that Ψ ∈ C¹(Xs(R^N), R) and its Fr´echet derivative is hΨ⁰(u), vi =

Z

R^N

g(x, u)v dx

for any u, v ∈ X_s(R^N). Next we define a functional I_λ: X_s(R^N) → R by Iλ(u) = Φs,p(u) − λΨ(u).

Then we know that the functional Iλ∈ C¹(Xs(R^N), R) and its Fr´echet deriv- ative is

hI_λ⁰(u), vi = Z

R^N

Z

R^N

|u(x) − u(y)|^p−2(u(x) − u(y))(v(x) − v(y))

|x − y|^{N +ps} dxdy

+ Z

R^N

V (x) |u(x)|^p−2uv dx − λ Z

R^N

g(x, u)v dx for any u, v ∈ Xs(R^N).

Remark 2.6. As seen before, there were many existence results of solution for elliptic problems under the assumption (Je): see [27–29, 39, 45]. The authors in [28], however gave some examples which satisfy the assumptions (G5), (G6) not (Je). In that sense, our analysis is motivated by this counterexample. For example,

g(x, t) = a(x) |t|^p−2t(4 |t|³+ 2t sin t − 4 cos t), where 0 < inf_RNa ≤ sup_RNa < ∞.

First of all, in this setting, we need the following lemma.

Lemma 2.7. Let 0 < s < 1 < p < +∞ with ps < N . Assume that (V) and (G1)–(G2) hold. Then Ψ and Ψ⁰ are weakly strongly continuous in Xs(R^N).

Proof. Let {u_n} be a sequence in X_s(R^N) such that u_n * u in X_s(R^N) as n → ∞. Then {un} is bounded in Xs(R^N) and Lemma 2.3 guarantees that the embeddings Xs(R^N) ,→ L^p(R^N) and Xs(R^N) ,→ L^q(R^N) are compact for p < q < p^∗_s. So we know that

(2.1) un → u in L^p(R^N) and un→ u in L^q(R^N) as n → ∞.

First we prove that Ψ is weakly strongly continuous in Xs(R^N). Let un → u in L^p(R^N) ∩ L^q(R^N) as n → ∞. By the convergence principle, there exist a subsequence {un_k} and a function v ∈ L^p(R^N) ∩ L^q(R^N) such that un_k(x) → u(x) as k → ∞ for almost all x ∈ R^N and |un_k(x)| ≤ v(x) for all k ∈ N and for almost all x ∈ R^N. Therefore taking (G2) into account, we deduce

Z

R^N

|G(x, unk) − G(x, u)| dx

≤ Z

R^N

|G(x, unk)| + |G(x, u)| dx

≤ Z

R^N

a(x)|u_nk(x)| + b(x)|u_nk(x)|^q+ a(x)|u(x)| + b(x)|u(x)|^qdx

≤ Z

R^N

a(x)|v(x)| + b(x)|v(x)|^q+ a(x)|u(x)| + b(x)|u(x)|^qdx

and thus the integral at the left-hand side is dominated by an integrable func- tion. Since g is the Carath´eodory function, we have that G(x, un_k) → G(x, u) as k → ∞ for almost all x ∈ R^N by (G1). Therefore, the Lebesgue convergence theorem tells us that

Z

R^N

G(x, un_k) dx → Z

R^N

G(x, u) dx

as k → ∞. This implies that Ψ is weakly strongly continuous in Xs(R^N).

Next, we show that Ψ⁰is weakly strongly continuous in Xs(R^N). Using (G2) and H¨older’s inequality, we assert that

Z

R^N

|g(x, un) − g(x, u)|^q

0

dx (2.2)

≤ C1

Z

R^N

|g(x, un)|^q0+ |g(x, u)|^q0 dx

≤ C2

Z

R^N

|a(x)|^q0+ ||b||^q_L⁰_∞

(R^N)(|u_n(x)|^q+ |u(x)|^q) dx

for some positive constants C₁, C₂. Since u_n→ u as n → ∞ in L^p(R^N)∩L^q(R^N) and almost all in R^N, it follows from (2.2) and the convergence principle that

|g(x, un) − g(x, u)|^q0 ≤ h(x) for almost all x ∈ R^N and for some h ∈ L¹(R^N),

and thus g(x, u_n) → g(x, u) as n → ∞ for almost all x ∈ R^N. This together with the Lebesgue convergence Theorem yields that

||Ψ⁰(un) − Ψ⁰(u)||_X^∗

s(R^N)= sup

||ϕ||

Xs(RN )≤1

|hΨ⁰(un) − Ψ⁰(u), ϕi|

= sup

||ϕ||_{Xs(RN )}≤1

Z

R^N

|g(x, un) − g(x, u)| |ϕ(x)| dx

≤

Z

R^N

|g(x, u_n) − g(x, u)|^q0 dx

_q0¹

→ 0 as n → ∞.

This completes the proof. 

The following assertion is to show that the energy functional Iλsatisfies the mountain pass geometry.

Lemma 2.8. Let 0 < s < 1 < p < +∞ with ps < N . Assume that (V) and (G1)–(G4) hold. Then the geometric conditions in the mountain pass theorem are satisfied, i.e.,

(1) u = 0 is a strict local minimum for Iλ, (2) Iλ is unbounded from below in Xs(R^N).

Proof. By the condition (G3), it is trivial that u = 0 is a strict local minimum for Iλ. Next we prove the condition (2). By the condition (G4), for any positive constant M, we can choose a constant δ > 0 such that

(2.3) G(x, t) ≥ M |t|^p

for |t| > δ and for almost all x ∈ R^N. Take v ∈ Xs(R^N) \ {0}. Then the relation (2.3) implies that

Iλ(tv) = Φs,p(tv) − λΨ(tv) ≤ t^p 1 p||v||^p_X

s(R^N)− λM Z

R^N

|v(x)|^pdx



for sufficiently large t > 1. If M is large enough, then we conclude that Iλ(tv) → −∞ as t → ∞ and therefore the functional I_λ is unbounded from

below. This completes the proof. 

With the help of Lemmas 2.5 and 2.7, we show that the energy functional I_λ satisfies the Cerami condition ((C)_c-condition for brevity), i.e., for c ∈ R, any sequence {u_n} ⊂ X_s(R^N) such that

Iλ(un) → c and ||I_λ⁰(un)||_X^∗

s(R^N)(1 + ||un||_Xs(RN)) → 0 as n → ∞ has a convergent subsequence. The basic idea of the proofs for the following Lemmas 2.9 and 2.11 comes from the paper [28]. These play a decisive role in showing the existence of a nontrivial weak solution for problem (Pλ).

Lemma 2.9. Let 0 < s < 1 < p < +∞ with ps < N . Assume that (V) and (G1)–(G5) hold. For any λ > 0, the functional Iλ satisfies the (C)c-condition.

Proof. For c ∈ R, let {un} be a (C)_c-sequence in X_s(R^N), that is, (2.4) Iλ(un) → c and ||I_λ⁰(un)||_X∗

s(R^N)(1 + ||un||_Xs(RN)) → 0 as n → ∞, which show that

(2.5) c = I_λ(u_n) + o(1) and hI_λ⁰(u_n), u_ni = o(1),

where o(1) → 0 as n → ∞. It follows from Lemmas 2.5 and 2.7 that Φ⁰_s,p and Ψ⁰are mappings of type (S+). Since I_λ⁰ is of type (S+) and Xs(R^N) is reflexive, it suffices to verify that the sequence {u_n} is bounded in X_s(R^N). Suppose to the contrary that the sequence {u_n} is unbounded in X_s(R^N). So then we may suppose that

||un||_Xs(RN)> 1 and ||un||_Xs(RN)→ ∞ as n → ∞.

Define a sequence {wn} by wn = un/||un||_Xs(RN). Then it is obvious that {wn} ⊂ Xs(R^N) and ||wn||_Xs(RN)= 1. Hence, up to a subsequence, still denoted by {w_n}, we obtain w_n* w in X_s(R^N) as n → ∞ and by Lemma 2.3, we have (2.6) wn→ w a.e. in R^N and wn→ w in L^p0(R^N) as n → ∞ for p ≤ p₀< p^∗_s. Set Ω =x ∈ R^N : w(x) 6= 0 . Due to the condition (2.5), we have that

c = Iλ(un) + o(1) = Φs,p(un) − λΨ(un) + o(1) (2.7)

= 1 p||un||^p_X

s(R^N)− λ Z

R^N

G(x, u_n) dx + o(1).

Since ||un||_Xs(RN)→ ∞ as n → ∞, we assert that (2.8)

Z

R^N

G(x, un) dx = 1 λp||un||^p_X

s(R^N)−c λ+o(1)

λ → ∞ as n → ∞.

From the assumptions (G1) and (G2), we have that there exists a positive constant M such that |G(x, t)| ≤ M for all (x, t) ∈ R^N×[−t0, t0]. This together with (G4) yields that there is a real number M0such that G(x, t) ≥ M0for all (x, t) ∈ R^N× R, and thus

(2.9) G(x, un) − M0

||u_n||^p_X

s(R^N)

≥ 0

for all x ∈ R^N and for all n ∈ N. By the convergence (2.6), we know that

|un| = |wn| ||un||_Xs(RN) → ∞ as n → ∞ for all x ∈ Ω. Furthermore, owing to the condition (G4), we have

(2.10) lim

n→∞

G(x, un)

||un||^p_X

s(R^N)

= lim

n→∞

G(x, un)

|un|^p |wn|^p= ∞

for all x ∈ Ω. Hence we get that meas(Ω) = 0. Indeed, if meas(Ω) 6= 0, then according to (2.7)–(2.10), and the Fatou lemma, we deduce that

1

λ = lim inf

n→∞

R

R^NG(x, un) dx λR

R^NG(x, u_n) dx + c − o(1)

= lim inf

n→∞

Z

R^N

G(x, un)

1 p||un||^p_X

s(R^N)

dx

≥ lim inf

n→∞

Z

Ω

pG(x, un)

||un||^p_X

s(R^N)

dx − lim sup

n→∞

Z

Ω

pM0

||un||^p_X

s(R^N)

dx

= lim inf

n→∞

Z

Ω

p(G(x, u_n) − M₀)

||un||^p_X

s(R^N)

dx (2.11)

≥ Z

Ω

lim inf

n→∞

p(G(x, un) − M0)

||un||^p_X

s(R^N)

dx

= Z

Ω

lim inf

n→∞

pG(x, u_n)

|un|^p_X

s(R^N)

|wn|^p_X

s(R^N)dx − Z

Ω

lim sup

n→∞

pM₀

||un||^p_X

s(R^N)

dx

= ∞,

which is a contradiction. Thus w(x) = 0 for almost all x ∈ R^N. Observe that

c + 1 ≥ Iλ(un) −1

phI_λ⁰(un), uni

= 1 p||un||^p_X

s(R^N)− λ Z

R^N

G(x, un) dx −1 p||un||^p_X

s(R^N)

(2.12)

+λ p Z

R^N

g(x, un)undx

≥ λ Z

R^N

G(x, un) dx

for n large enough and G is defined in (G5). Let us define Ω_n(a, b) := {x ∈ R^N : a ≤ |u_n(x)| < b} for a ≥ 0. By the convergence (2.6), we note that (2.13) wn→ 0 in L^p0(R^N) and wn → 0 a.e. in R^N as n → ∞ for p ≤ p₀< p^∗_s. Hence from the relation (2.8) we get

(2.14) 0 < 1

λp ≤ lim sup

n→∞

Z

R^N

|G(x, un)|

||un||^p_X

s(R^N)

dx.

On the other hand, we can choose a positive constant K_p0such that ||w_n||_Lp0(R^N)

≤ Kp₀||wn||_Xs(RN)for p ≤ p0< p^∗_s, because ||wn||_Xs(RN)= 1. From the assump- tion (G2) and (2.13), we have

Z

Ω_n(0,d)

G(x, un)

||un||^p_X

s(R^N)

dx ≤ Z

Ω_n(0,d)

a(x) |un(x)| +^b(x)_q |un(x)|^q

||un||^p_X

s(R^N)

dx

≤ 1

||u_n||^p_X

s(R^N)

||a||_Lq0(R^N)||un||_Lq(R^N)

+||b||L^∞(R^N)

q Z

Ωn(0,d)

|un(x)|^q−p|wn(x)|^p dx

≤ C₃||a||_Lq0(R^N)

||un||^p−1_X

s(R^N)

||wn||_Lq(R^N)

+||b||_L∞(R^N)

q d^q−p Z

R^N

|wn(x)|^p dx

≤ KqC4

||un||^p−1_X

s(R^N)

+||b||L^∞(R^N)

q d^q−p Z

R^N

|wn(x)|^p dx → 0 (2.15)

as n → ∞,

where C₃ and C₄ are positive constants. Set τ⁰ = τ /(τ − 1). Since τ > N/ps, we see that p < τ⁰p < p^∗_s. Hence, it follows from (G5), (2.12), and (2.13) that

Z

Ωn(d,∞)

|G(x, un)|

||un||^p_X

s(R^N)

dx

= Z

Ω_n(d,∞)

|G(x, u_n)|

|un(x)|^p |w_n(x)|^pdx

≤ (Z

Ωn(d,∞)

 |G(x, un)|

|u_n(x)|^p

^τ dx

)¹_τ( Z

Ωn(d,∞)

|wn(x)|^τ0p )_{τ 0}¹

≤ c₀^τ1 (Z

Ω_n(d,∞)

G(x, un) dx )¹_τ(

Z

R^N

|wn(x)|^τ0p )_{τ 0}¹

≤ c₀^τ1 c + 1 λ

¹_τ ( Z

R^N

|wn(x)|^τ

0p

)_{τ 0}¹

→ 0 as n → ∞.

(2.16)

Combining (2.15) with (2.16), we have Z

R^N

|G(x, un)|

||un||^p_X

s(R^N)

dx = Z

Ωn(0,d)

|G(x, un)|

||un||^p_X

s(R^N)

dx + Z

Ωn(d,∞)

|G(x, un)|

||un||^p_X

s(R^N)

dx → 0 as n → ∞, which contradicts (2.14). This completes the proof.  Theorem 2.10. Let 0 < s < 1 < p < +∞ with ps < N . Assume that (V) and (G1)–(G5) hold. Then the problem (Pλ) has a nontrivial weak solution for all λ > 0.

Proof. Note that Iλ(0) = 0. By Lemmas 2.8 and 2.9, we verify that all con- ditions of the mountain pass theorem are satisfied. Consequently, the problem (Pλ) has a nontrivial weak solution for all λ > 0. 

Lemma 2.11. Let 0 < s < 1 < p < +∞ with ps < N . Assume that (V), (G1)–(G4), and (G6) hold. Then the functional Iλ satisfies the (C)c-condition for any λ > 0.

Proof. Let {u_n} be a (C)_c-sequence in X_s(R^N) satisfying (2.4). Then the relation (2.5) is fulfilled. As in the proof of Lemma 2.9, we only show that {u_n} is bounded in X_s(R^N). To this end, arguing by contradiction, suppose that

||un||_Xs(RN) → ∞ as n → ∞. Let vn = un/||un||_Xs(RN). Then ||vn||_Xs(RN)= 1 and for p ≤ p₀ < p^∗_s, ||v_n||Lp0(R^N) ≤ Kp0||vn||X_s(R^N) = K_p0, where K_p0 is the Sobolev constant. Passing to a subsequence, we may assume that v_n * v in X_s(R^N) as n → ∞, then by Lemma 2.3,

v_n→ v in L^p0(R^N) and v_n(x) → v(x) almost all x ∈ R^N as n → ∞.

By the assumption (G6), one has c + 1 ≥ Iλ(un) − 1

µhI_λ⁰(un), uni

= 1 p||u_n||^p_X

s(R^N)− λ Z

R^N

G(x, u_n) dx −1 µ||u_n||^p_X

s(R^N)

+λ µ Z

R^N

g(x, u_n)u_ndx

≥  1 p−1

µ



||un||^p_X

s(R^N)−λ%

µ Z

R^N

|un(x)|^pdx which implies

(2.17) 1 ≤ λ%p

µ − plim sup

n→∞

||vn||^p_Lp(R^N)= λ%p

µ − p||v||^p_Lp(R^N).

Hence, it follows from (2.17) that v 6= 0. If we follow the same argument as in Lemma 2.9, we can check the relations (2.8), (2.9), and (2.10) and hence yield the relation (2.11). Therefore we can conclude a contradiction. Thus, {u_n} is

bounded in X_s(R^N). 

Remark 2.12. Although we replace (G5) with (G6) in the assumption of Theo- rem 2.10, we assert that the problem (Pλ) possesses a nontrivial weak solution for all λ > 0 via Lemma 2.11.

2.2. Another approach for the existence of a nontrivial weak solution In this subsection, we also give the existence of a nontrivial solution for our problem without the assumptions (G5) and (G6) which play a decisive role in obtaining the fact that the energy functional Iλ satisfies the (C)c-condition.

To show this result, for the time being, we need the following additional as- sumptions for g:

(H) m(x) > 0 for all x ∈ R^N and m ∈ L^γ−qγ (R^N) with some γ satisfying q < γ < p^∗_s.

(G7) There exist nonnegative functions a1∈ L^p0(R^N) ∩ L^r0(R^N), b1∈ L

p∗s

p∗s −r(R^N) ∩ L^∞(R^N) with 1 < r < p^∗_s such that

|g(x, t)| ≤ a1(x) + b1(x) |t|^r−1 holds for all (x, t) ∈ R^N × R.

(G8) lim sup_s→0 ^|g(x,s)|

m(x)|s|^ξ−1 < +∞ uniformly for almost all x ∈ R^N with q < ξ < p^∗_s.

(G9) lim sup_|s|→∞

ess sup_x∈RN |g(x,s)|

m(x)|s|^q−1



< +∞, where p < q < p^∗_s. We recall the functional Iλ: Xs(R^N) → R by

(2.18) Iλ(u) = Φs,p(u) − λΨ(u).

Here, Ψ can be defined as in Section 2.1 under the assumptions (G1) and (G7).

We observe that the growth of g allowed by both (G2) and (G7) is subcritical.

The assumptions (G1) and (G8) imply that g(x, 0) = 0 for almost all x ∈ R^N. Furthermore, lim sup_s→0m(x)|s|^|G(x,s)|_ξ < +∞ uniformly almost everywhere in R^N, by L’Hˆopital’s rule. From the analogous argument as in [16], define the crucial value

(2.19) Cg= ess sup

s6=0,x∈R^N

|g(x, s)|

m(x) |s|^q−1,

where q < ξ. So then Cg is a positive constant. The assumptions (G8) and (G9) yield Cg< ∞, and furthermore the following relation

ess sup

s6=0,x∈R^N

|G(x, s)|

m(x) |s|^q =Cg

q holds.

Under the circumstance for the functional Iλin (2.18), we need the following lemma. However, since the growth condition for g in (G7) is different from that in (G2), we cannot follow the lines of the proof as in Lemma 2.7 to get the following assertion directly. Hence we give the proof.

Lemma 2.13. Let 0 < s < 1 < p < +∞ with ps < N . Assume that (V), (G1), and (G7) hold. Then Ψ and Ψ⁰ are weakly strongly continuous in Xs(R^N).

Proof. In the same way to that of Proposition 1.12 in [43], it is obvious that the functional Ψ is Gˆateaux differentiable in Xs(R^N). So, we only need to prove that the functionals Ψ and Ψ⁰ are weakly strongly continuous in Xs(R^N). Let us assume that u_n* u in X_s(R^N) as n → ∞. First we prove that Ψ is weakly strongly continuous on X_s(R^N). Observe that

|Ψ(u_n) − Ψ(u)| ≤ Z

R^N

|G(x, u_n) − G(x, u)| dx

≤ Z

R^N\B_K^N(0)

|G(x, un) − G(x, u)| dx

+ Z

B^N_K(0)

|G(x, un) − G(x, u)| dx, (2.20)

where B_K^N(0) := {x ∈ R^N : |x| ≤ K} for K ∈ N. For the first term in the right side of the inequality (2.20), the assumption (G7) and the H¨older inequality imply that

Z

R^N\B_K^N(0)

|G(x, un) − G(x, u)| dx

≤ Z

R^N\B_K^N(0)

|G(x, u_n)| + |G(x, u)| dx

≤ Z

R^N\B_K^N(0)

a1(x)(|un(x)| + |u(x)|) + b1(x)

r (|un(x)|^r+ |u(x)|^r) dx

≤ C5||a1||_Lp0(R^N\B_K^N(0))||un+ u||_Lp(R^N\B_K^N(0))

+ C6||b1||

L

p∗s

p∗s −r(R^N\B_K^N(0))

||un+ u||^r_Lp∗s(R^N\B^N_K(0))

for some positive constants C₅ and C₆. Note that a₁ ∈ L^p0(R^N) and b₁ ∈ L

p∗s

p∗s −r(R^N). Then, for any ε > 0, there exists N (K) ∈ N such that K > N (K) implies

Z

R^N\B_K^N(0)

|a1(x)|^p0dx < ε and Z

R^N\B^N_K(0)

|b1(x)|

p∗s

p∗s −r dx < ε.

Since Xs(R^N) is reflexive, {un} is bounded in Xs(R^N) and {un + u} is also bounded in X_s(R^N). By Lemma 2.2, {u_n} is bounded in L^p∗s(R^N) and so {un+ u} is bounded in L^p∗s(R^N). Consequently, we assert that

(2.21)

Z

R^N\B_K^N(0)

|G(x, un) − G(x, u)| dx < 2C7ε for some positive constant C7.

Next, we consider the second term in the right side of the inequality (2.20).

Since u_n * u in X_s(R^N) as n → ∞, it is easy to check that u_n * u in X_s(B_K^N(0)) and u_n → u in L^r(B^N_K(0)) by Lemma 2.1. By the convergence principle, there exist a subsequence, still denoted by {un}, in Xs(B_K^N(0)) and a function v ∈ L^p(B_K^N(0)) ∩ L^p∗rs(B^N_K(0)) such that u_n(x) → u(x) for almost all x ∈ B_K^N(0) as n → ∞ and |un(x)| ≤ v(x) for all n ∈ N and for almost all x ∈ B^N_K(0). Therefore from (G7), we get

Z

B^N_K(0)

|G(x, un)| dx ≤ Z

B_K^N(0)

a1(x)|un(x)| + b1(x)|un(x)|^rdx

≤ Z

B_K^N(0)

a1(x)|v(x)| + b1(x)|v(x)|^rdx

and thus {G(x, un)} is uniformly integrable on B_K^N(0). By the assumptions (G1) and (G7), we deduce that G(x, un) → G(x, u) as n → ∞ and

|G(x, u)| ≤ ||a1||L^∞(R^N)|u(x)| + ||b1||L^∞(R^N)|u(x)|^r< ∞

for almost all x ∈ B_K^N(0), respectively. By the Vitali convergence theorem, Z

B_K^N(0)

G(x, un) dx → Z

B^N_K(0)

G(x, u) dx

as n → ∞. Hence for above ε > 0, there exists N (K) ∈ N such that (2.22)

Z

B^N_K(0)

|G(x, un) − G(x, u)| dx < ε for K > N (K). From (2.21) and (2.22), we have

Z

R^N

G(x, un) dx → Z

R^N

G(x, u) dx

as n → ∞. This implies that Ψ is weakly strongly continuous on Xs(R^N).

Next we prove that Ψ⁰is weakly strongly continuous on Xs(R^N). Note that sup

||ϕ||_{Xs(RN )}≤1

hΨ⁰(un) − Ψ⁰(u), ϕi 

= sup

||ϕ||

Xs(RN )≤1

   Z

R^N

(g(x, un) − g(x, u))ϕ dx   

≤ sup

||ϕ||_{Xs(RN )}≤1

   Z

B_K^N(0)

(g(x, u_n) − g(x, u))ϕ dx  (2.23) 

+ sup

||ϕ||_{Xs(RN )}≤1

   Z

R^N\B^N_K(0)

(g(x, un) − g(x, u))ϕ dx   

for some constant K and for any ϕ ∈ Xs(R^N). Since 1 < p < p^∗_s, the compact embedding

W^s,p(B^N_K(0)) ,→ L^p(B^N_K(0)) implies un→ u in L^p(B_K^N(0)) as n → ∞.

This together with the continuity of the Nemytskii operator with g and acting from L^p(B_K^N(0)) into L^r0(B_K^N(0)) yields that it is easy to see that the first term in the right side of the inequality (2.23) tends to 0 as n → ∞. For the second term in (2.23), we have

   Z

R^N\B_K^N(0)

(g(x, u_n) − g(x, u))ϕdx  

≤ Z

R^N\B^N_K(0)

(a1(x) + b1(x) |un(x)|^r−1) |ϕ| dx

+ Z

R^N\B^N_K(0)

(a1(x) + b1(x) |u(x)|^r−1) |ϕ| dx

≤ 2||a1||_Lp0(R^N\B^N_K(0))||ϕ||_Lp(R^N)

+ ||b1||

L

p∗s

p∗s −r(R^N\B_K^N(0))

||un||^r−1

L^p∗s(R^N)+ ||u||^r−1

L^p∗s(R^N)
||ϕ||_Lp∗s(R^N). According to the assumption (G7), for above ε > 0, there exists N (K) ∈ N such that

||a₁||_Lp0(R^N\B_K^N(0))< ε and ||b₁||

L

p∗s

p∗s −r(R^N\B_K^N(0))

< ε

for K > N (K). As the sequence {u_n} is bounded in X_s(R^N), invoking Lemma 2.2, {un} is bounded in L^p(R^N) and also in L^p∗s(R^N). Thus, it is immediate that

(2.24)

    Z

R^N\B_K^N(0)

(g(x, un) − g(x, u))ϕ dx    

≤ C8ε for a positive constant C8. Due to (2.24), we can deduce that

Z

R^N

(g(x, u_n) − g(x, u))ϕ dx → 0 as n → ∞.

This implies that Ψ⁰ is weakly strongly continuous in Xs(R^N). Therefore, the

proof is completed. 

As mentioned earlier, we give an abstract critical point theory based on [9] to get our main result of this subsection. To do this, we briefly introduce the following definitions and some properties for locally Lipschitz continuous functionals.

Let X be a real Banach space. A functional I : X → R is called locally Lipschitz continuous when, for every u ∈ X, there exist a neighborhood U of u and a constant L ≥ 0 such that

|I(v) − I(w)| ≤ L||v − w||X for all v, w ∈ U.

Let u, v ∈ X. The symbol I^◦(u; v) indicates the generalized directional deriva- tive of I at point u along direction v, namely

I^◦(u; v) := lim sup

w→u,t→0⁺

I(w + tv) − I(w)

t .

The generalized gradient of the function I at u, denoted by ∂I(u), is the set

∂I(u) :=u^∗∈ X : hu^∗, vi ≤ I^◦(u; v) for all v ∈ X .

Definition 2.14. A functional I : X → R is called Gˆateaux differentiable at u ∈ X if there is ψ ∈ X^∗(denoted by I⁰(u)) such that

lim

t→0⁺

I(u + tv) − I(u)

t = I⁰(u)(v)

for all v ∈ X. It is called continuously Gˆateaux differentiable if it is Gˆateaux differentiable for any u ∈ X and the function u → I⁰(u) is a continuous map from X to its dual X^∗.

We recall that if I is continuously Gˆateaux differentiable, then it is locally Lipschitz continuous and one has I^◦(u; v) = I⁰(u)(v) for all u, v ∈ X.

For a real Banach space (X, || · ||_X), we say that a functional I(u) = Φ(u) − Ψ(u) satisfies the (C)^[µ]-condition cut off upper at µ if any Cerami sequence {xn} ⊂ X for I with 0 < Φ(xn) < µ has a convergent subsequence of {xn}.

We recall the key lemma in [49] (see also [25]) to prove the abstract critical points result under the (C)^[µ]-condition (Theorem 2.16).

Lemma 2.15. Let X be a real Banach space and let I : X → R be a locally Lipschitz continuous function with bounded from below. Then, for all mini- mizing sequence of I, {un}_n∈N ⊆ X, there exists a minimizing sequence of I, {vn}_n∈N⊆ X, such that for any n ∈ N,

I(vn) ≤ I(un) and I^◦(vn; h) ≥ −εn||h||X

1 + ||v_n||_X for all h ∈ X, and n ∈ N, where εⁿ→ 0⁺.

Let us introduce two functions ϕ1(µ) = inf

v∈Φ⁻¹((0,µ))

sup_u∈Φ−1((0,µ))Ψ(u) − Ψ(v) µ − Φ(v)

and

ϕ2(µ) = sup

v∈Φ⁻¹((0,µ))

Ψ(v) − sup_u∈Φ−1((−∞,0])Ψ(u) Φ(v)

for all µ ∈ R.

Using Lemma 2.15, we obtain the following result; see [12] for the case of the Palais-Smale condition. The proof of this theorem proceeds in the analogous way to those of Theorems 2.3 and 2.4 in [12]. For the sake of convenience, we give the proof.

Theorem 2.16. Let X be a real Banach space and let Φ, Ψ : X → R be two locally Lipschitz continuous functionals. Suppose that

there exists µ ∈ R such that ϕ1(µ) < ϕ₂(µ).

Moreover, assume that for each λ ∈ Λ :=

1 ϕ2(µ),_ϕ¹

1(µ)



the functional Iλ :=

Φ − λΨ satisfies (C)^[µ]-condition. Then, for each λ ∈ Λ, the functional Iλ has a nontrivial point u0,λ in Φ⁻¹((0, µ)) such that Iλ(u0,λ) ≤ Iλ(u) for all u in Φ⁻¹((0, µ)) with u0,λ being a critical point of Iλ.