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Existence of infinitely many solutions for p -Laplacian equations in RN
Nonlinear Analysis 92 (2013) 72–81
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Existence of infinitely many solutions for p-Laplacian equations in RN ✩ Xiaoyan Lin a,∗ , X.H. Tang b a
Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, PR China
b
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China
highlights • We give a direct sum decomposition of the Sobolev space corresponding to the quasilinear elliptic problem. • We obtain the existence of infinitely many solutions of the quasilinear elliptic problem. • We establish new theorems under weaker assumptions on the nonlinearity.
article
abstract
info
Article history: Received 27 January 2013 Accepted 22 June 2013 Communicated by Enzo Mitidieri
Based on the cohomological linking method for cones and a new direct sum decomposition of W 1,p (RN ), we study the existence and multiplicity of solutions of a p-Laplacian equation in RN with sign-changing potential and subcritical p-superlinear nonlinearity. Our assumptions are mild and different from those studied previously. © 2013 Elsevier Ltd. All rights reserved.
MSC: 35J20 35J60 Keywords: p-Laplacian equation p-superlinear Sign-changing potential
1. Introduction Consider the following p-Laplacian equation
− △p u + V (x)|u|p−2 u = f (x, u), u ∈ W 1,p (RN ),
x ∈ RN ,
(1.1)
where △p u := div(|∇ u|p−2 ∇ u) is the p-Laplacian operator with p > 1, V : RN → R and f : RN × R → R. For p = 2, (1.1) turns into the following semilinear Schrödinger equation
−△u + V (x)u = f (x, u), u ∈ H 1 (RN ),
x ∈ RN ,
(1.2)
✩ This work is partially supported by the NNSF (No: 11171351) of China and supported by Scientific Research Fund of Hunan Provincial Education Department (08A053) and supported by Hunan Provincial Natural Science Foundation of China (No. 11JJ2005). ∗ Corresponding author. Tel.: +86 13466651234. E-mail addresses: [email protected] (X. Lin), [email protected] (X.H. Tang).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.06.011
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
73
which has been studied extensively; see for example [1–13] and the references quoted in them. Most of them deal with the problem (1.2) with a constant sign potential V (x), and the others treat the case where V (x) is sign-changing. For the second case, the classical proof is based on the fact that the spectrum σ (−△ + V ) of self-adjoint operator −△ + V induces a suitable direct sum decomposition of H 1 (RN ). For general case p > 1, many results for (1.2) can be extended to (1.1) when the potential V (x) is of constant sign; see for example [14–17] and the references therein. However, if V (x) is sign-changing, the quasilinear problem (1.1) is far more difficult as − △p is no longer a self-adjoint operator and so a complete description of its spectrum is not available. For these reasons, few papers have treated this case so far. Using the cohomological index of Fadell and Rabinowitz [18], Degiovanni and Lancelotti [19] developed a new linking theorem over cones by constructing an unbounded sequence of minimax eigenvalues; then, making use of the linking theorem, they have proved that the quasilinear elliptic boundary value problem admits a nontrivial solution on a bounded domain. Some related results can be found in [20–22] for a similar problem on a bounded domain. In a recent paper [23], Liu and Zheng obtained a theorem on the existence of a nontrivial solution of (1.1) by applying a linking theorem over cones which was mainly developed in [19], where the following assumptions on V and f are introduced: (V1) V ∈ C (RN , R) and inf RN V (x) > −∞; (V2) for any M > 0, meas x ∈ RN : V (x) ≤ M < ∞, where meas(·) denotes the Lebesgue measure in RN ; (S0) f ∈ C (RN +1 , R), and there exist constants c1 , c2 > 0 and q ∈ (p, p∗ ) such that
|f (x, t )| ≤ c1 |t |p−1 + c2 |t |q−1 ,
∀ (x, t ) ∈ RN × R,
(1.3)
where p = ∞ if N ≤ p and p = pN /(N − p) if N > p; (S1) f (x, t ) = o(|t |p−1 ), as |t | → 0, uniformly in x ∈ RN ; t F (x,t ) (S2) F (x, t ) := 0 f (x, s)ds ≥ 0, ∀ (x, t ) ∈ RN × R, lim|t |→∞ |t |p = ∞, uniformly in x ∈ RN ; (S3) there exists θ ≥ 1 such that ∗
∗
θ F (x, t ) ≥ F (x, st ), where F (x, t ) =
1 tf p
∀ (x, t ) ∈ RN × R, s ∈ [0, 1],
(x, t ) − F (x, t ).
Now, we are able to state a main result in [23]. Theorem 1.1 ([23, Theorem 1.1]). Assume that V and f satisfy (V1), (V2), (S0), (S1), (S2) and (S3). Then problem (1.1) possesses a nontrivial solution. Condition (V2) was introduced in [2], and then was used by many authors, for example, [13]. Condition (S3) is due to Jeanjean [24]. In [6], this condition is also used together with a Cerami type argument in singularly perturbed elliptic problems in RN with autonomous nonlinearity. As shown in [25], condition (S3) is somewhat weaker than the condition that f (x, t )/|t |p−1 is nondecreasing in t ∈ (−∞, 0) ∪ (0, +∞) for all x ∈ RN . Moreover, there are many functions (e.g. f (x, t ) = a|t |p−2 t ln(1 + |t |), a > 0) which satisfy (S3), but do not satisfy the following classical condition which was introduced by Ambrosetti and Rabinowitz in [26]: (AR) there exists µ > p such that 0 < µF (x, t ) ≤ tf (x, t ),
t ̸= 0.
However, (AR)-condition does not imply condition (S3); see the example in [11]. Therefore, (S2) and (S3) are complementary super-quadratic conditions to (AR). In this paper, we are interested in the existence of a nontrivial solution or infinitely many nontrivial solutions of (1.1) with sign-changing potential and subcritical p-superlinear nonlinearity. By using a linking theorem over cones in [19], we will establish theorems on the existence of a nontrivial solution of (1.1) with mild assumptions deeply different from those studied in previous related works. For any R > 0, the Sobolev embedding theorem implies W 1,p (BR ) ↩→ L2 (BR ), where BR = {x ∈ RN : |x| < R}. Based on the above fact, we can construct a direct sum decomposition of W 1,p (RN ). Then a Z2 -version of the Mountain Pass Theorem is applied to obtain the existence theorem of infinitely many nontrivial solutions of (1.1). As far as we are aware there were no such multiplicity results in this situation. To state our results, we first introduce the following assumptions:
(V2′ ) there exists a constant d0 > 0 such that lim meas x ∈ RN : |x − y| ≤ d0 , V (x) ≤ M = 0, |y|→+∞
∀M > 0,
where meas(·) denotes the Lebesgue measure in RN ; (S2 ) lim|t |→∞ |F |(tx|,pt )| = ∞, a.e. x ∈ RN , and there exists r0 ≥ 0 such that ′
F ( x, t ) ≥ 0 ,
∀ (x, t ) ∈ RN × R,
(S4) f (x, −t ) = −f (x, t ), ∀ (x, t ) ∈ R × R; N
|t | ≥ r0 ;
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X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
(S5) F (x, t ) :=
1 tf p
(x, t ) − F (x, t ) ≥ 0, and there exist c0 > 0 and κ > max{1, N /p} such that
|F (x, t )|κ ≤ c0 |t |pκ F (x, t ),
∀ (x, t ) ∈ RN × R,
|t | ≥ r0 ;
(S6) there exist µ > p and ϱ > 0 such that
µF (x, t ) ≤ tf (x, t ) + ϱt p ,
∀ (x, t ) ∈ RN × R;
(S7) there exist µ > p and r1 > 0 such that
µF (x, t ) ≤ tf (x, t ),
∀ (x, t ) ∈ RN × R,
|t | ≥ r1 .
Now, we are ready to state the main results of this paper. Theorem 1.2. Assume that V and f satisfy (V1), (V2′ ), (S0), (S1), (S2) and (S5). Then problem (1.1) possesses a nontrivial solution. Theorem 1.3. Assume that V and f satisfy (V1), (V2′ ), (S0), (S1), (S2) and (S6). Then problem (1.1) possesses a nontrivial solution. It is easy to check that (S0) and (S7) imply (S6). Thus, we have the following corollary. Corollary 1.4. Assume that V and f satisfy (V1), (V2′ ), (S0), (S1), (S2) and (S7). Then problem (1.1) possesses a nontrivial solution. Theorem 1.5. Assume that V and f satisfy (V1), (V2′ ), (S0), (S2′ ), (S4) and (S5). Then problem (1.1) possesses infinitely many nontrivial solutions. Theorem 1.6. Assume that V and f satisfy (V1), (V2′ ), (S0), (S2′ ), (S4) and (S6). Then problem (1.1) possesses infinitely many nontrivial solutions. Corollary 1.7. Assume that V and f satisfy (V1), (V2′ ), (S0), (S2′ ), (S4) and (S7). Then problem (1.1) possesses infinitely many nontrivial solutions. Remark 1.8. Obviously, (V2′ ), (S2′ ) and (S5) are weaker than (V2), (S2) and (AR), respectively. In particular, we remove the usual condition (S1), and F (x, t ) is allowed to be sign-changing in Theorems 1.5, 1.6 and Corollary 1.7. It is easy to check that function f (x, t ) = a|t |p−2 t ln (1 + |t |) with a > 0 satisfies (S0), (S1), (S2), (S3) and (S5). However, the following function f (x, t ) = 3|t |
2
t
|s|1+sin s sds + |t |4+sin t t ,
(1.4)
0
satisfies (S5) not (S3) for p = 2; see [11, Section 3]. In addition, the following functions f (x, t ) = a(x)|t |p−1 t (p + 3)t 2 − 2(p + 2)t + p + 1 ,
(1.5)
f (x, t ) = a(x)|t |p−2 t 4|t |3 + 2t sin t − 4 cos t
(1.6)
and f (x, t ) = a(x)
m
bi |t |βi t ,
(1.7)
i=1
satisfy (S0), (S2′ ), (S4) and (S7), where b1 > 0, bi ∈ R, i = 2, 3, . . . , m; β1 > β2 > · · · > βm ≥ p − 2, a ∈ C (RN , R), and 0 < infRN a ≤ supRN a < ∞. One can see that they satisfy neither (AR) nor (S3). 2. Existence of a nontrivial solution By (V1), V (x) is bounded from below and so there is an a0 > 0 such that V (x) + a0 ≥ 1,
∀ x ∈ RN .
Define
E=
u ∈ W 1,p (RN ) :
RN
|∇ u|p + [V (x) + a0 ]|u|p dx < +∞ .
(2.1)
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
75
Then E is a reflexive and separable Banach space with the norm
∥ u∥ = RN
|∇ u|p + [V (x) + a0 ]|u|p dx
1/p
,
u ∈ E.
Evidently, E is continuously embedded into W 1,p (RN ) and hence continuously embedded into Ls (RN ) for p ≤ s ≤ p∗ , i.e., there exists γs > 0 such that
∥u∥s ≤ γs ∥u∥,
∀ u ∈ E,
(2.2)
where ∥u∥s denotes the usual norm in L (R ) for all p ≤ s ≤ p . In fact we further have the following lemma due to [2]. s
∗
N
Lemma 2.1 ([2, Lemma 3.1]). Suppose that (V1) and (V2′ ) hold. Then the embedding from E into Ls (RN ) is compact for p ≤ s < p∗ . Now we define a functional Φ on E by
Φ ( u) =
1 p
RN
|∇ u|p + V (x)|u|p dx −
RN
F (x, u)dx,
∀ u ∈ E.
(2.3)
Under assumptions (V1) and (S0), the functional Φ defined by (2.3) is of class C 1 (E , R). Moreover,
Φ (u) =
1 p
p
∥ u∥ −
a0 p
p p
∥ u∥ −
⟨Φ ′ (u), v⟩ = ⟨H ′ (u), v⟩ −
RN
F (x, u)dx,
∀ u ∈ E,
a0 |u|p−2 u + f (x, u) v dx,
RN
(2.4)
∀ u, v ∈ E ,
(2.5)
where H ( u) =
1
p
RN
|∇ u|p + (V (x) + a0 )|u|p dx,
∀ u ∈ E.
(2.6)
We say that ϕ ∈ C 1 (X , R) satisfies (C)c -condition if any sequence {un } such that
ϕ(un ) → c ,
∥ϕ ′ (un )∥(1 + ∥un ∥) → 0
(2.7)
has a convergent subsequence. Lemma 2.2 ([19, Corollary 2.9], [23, Lemma 2.2]). Let X be a real Banach space and C− and C+ two symmetric closed cones in X , C− ∩ C+ = {0} and Index(C− \ {0}) = Index(X \ C+ ) < ∞,
(2.8)
where and in the sequel, Index is the Z2 -cohomological index of [18]. Let r− > r+ > 0 and e ∈ X \ C− with ∥e∥ = 1. If ϕ ∈ C 1 (X , R) satisfies (C)c -condition for all c > 0, and inf ϕ(S+ ) > sup ϕ(∂ Q ),
sup ϕ(Q ) < ∞,
where S+ = {u ∈ C+ : ∥u∥ = r+ }, Q = {u + te : u ∈ C− , t ≥ 0, ∥u + te∥ ≤ r− }, and
∂ Q = {u ∈ C− : ∥u∥ ≤ r− } ∪ {u + te : u ∈ C− , t ≥ 0, ∥u + te∥ = r− }, then ϕ has a critical point with value c ≥ infS+ ϕ > 0. Set M = {u ∈ E : ∥u∥p = 1} and
λn = inf sup ∥u∥ : A ⊆ M, A is symmetric and Index(A) ≥ n , p
n ∈ Z.
(2.9)
u∈A
Lemma 2.3 ([23, Theorem 3.8]). Suppose that (V1) and (V2′ ) are satisfied. Then 0 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · and λn → +∞ as n → ∞.
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X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
By Lemma 2.3, we can choose an integer m ≥ 1 such that max{λm , a0 } < λm+1 . Set C− = {u ∈ E : ∥u∥p ≤ λm ∥u∥pp },
C+ = {u ∈ E : ∥u∥p ≥ λm+1 ∥u∥pp }.
(2.10)
Lemma 2.4 ([23, Theorem 3.10]). Suppose that (V1) and (V2′ ) are satisfied. Then C− and C+ are two symmetric closed cones in E , C− ∩ C+ = {0} and Index(C− \ {0}) = Index(E \ C+ ) = m.
(2.11)
Lemma 2.5. Suppose that (V1), (V2′ ), (S0) and (S1) are satisfied. Then there exist constants r+ , α > 0 such that Φ (u) ≥ α for u ∈ C+ and ∥u∥ = r+ . Proof. Choose ε ∈ (0, (λm+1 − a0 )/2), in view of conditions (S0) and (S1), there exists a constant Cε > 0 such that
|F (x, u)| ≤
ε p
|u|p + Cε |u|q ,
∀ (x, u) ∈ RN × R.
(2.12)
From (2.4), (2.10) and (2.12), one has
Φ (u) =
≥
1 p 1 p
∥ u∥ p − ∥ u∥ p −
a0
∥u∥pp −
p a0 + ε p
RN
F (x, u)dx
∥u∥pp − Cε ∥u∥qq
λm+1 − (a0 + ε) ∥u∥p − Cε γqq ∥u∥q pλm+1 λm+1 − a0 ∥u∥p − Cε γqq ∥u∥q . ≥ 2pλm+1
≥
Since p < q, the assertion follows.
Lemma 2.6. Suppose that (V1), (V2′ ), (S0) and (S2) are satisfied. Let e ∈ E \ C− with ∥e∥ = 1. Then there exists a constant r− > r+ such that sup Φ (Q ) < ∞ and sup Φ (∂ Q ) ≤ 0, where Q = {u + te : w ∈ C− , t ≥ 0, ∥u + te∥ ≤ r− } .
(2.13)
Proof. It is sufficient to show that Φ (u) → −∞ as u ∈ C− + R+ e and ∥u∥ → ∞. Arguing indirectly, assume that for some sequence {un + tn e} ⊂ C− + Re with ∥un + tn e∥ → ∞, there is M > 0 such that Φ (un + tn e) ≥ −M for all n ∈ N. Set + tn e = wn + sn e, then ∥vn ∥ = 1. According to [19, Proposition 2.12], there exists a constant β ≥ 1 such that vn = ∥uun + t e∥ n
n
∥u∥ + ∥e∥ ≤ β∥u + e∥,
∀ u ∈ C− .
(2.14)
It follows from (2.14) and ∥wn + sn e∥ = 1 that
∥wn ∥ + sn = ∥wn ∥ + |sn |∥e∥ ≤ β∥wn + sn e∥ = β.
(2.15)
Hence, passing to a subsequence, we may assume that sn → s0 ≥ 0, wn ⇀ w in E, wn → w a.e. on R , and so, by Lemma 2.1, N
wn → w in Lp (RN ). So, we have
1/p 1 = ∥wn + sn e∥ ≤ ∥wn ∥ + sn ≤ λ1m/p ∥wn ∥p + sn → λm ∥w∥p + s0 .
This shows that w + s0 e ̸= 0. Let A := {x ∈ RN : w(x) + s0 e(x) ̸= 0}, then meas(A) > 0. For a.e. x ∈ A, we have limn→∞ |un (x)| = ∞. Hence, it follows from (2.4), (S2) and Fatou’s lemma that
−M Φ (un + tn e) ≤ lim p n →∞ ∥ un + t n e ∥ ∥un + tn e∥p 1 a0 F (x, un + tn e) p lim − ∥vn ∥pp − |v | dx n n→∞ p p |un + tn e|p RN 1 F (x, un + tn e) − lim inf |wn + sn e|p dx n →∞ p |un + tn e|p RN 1 F (x, un + tn e) − lim inf |wn + sn e|p dx p n →∞ N p | u + t e | n n R −∞,
0 = lim
n→∞
= ≤ ≤ =
which is a contradiction.
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
77
Lemma 2.7. Suppose that (V1), (V2′ ), (S0), (S2′ ) and (S5) are satisfied. Then any sequence {un } ⊂ E satisfying
Φ (un ) → c > 0,
⟨Φ ′ (un ), un ⟩ → 0
(2.16)
is bounded in E. Proof. To prove the boundedness of {un }, arguing by contradiction, suppose that ∥un ∥ → ∞. Let vn = un /∥un ∥. Then ∥vn ∥ = 1 and ∥vn ∥s ≤ γs ∥vn ∥ = γs for p ≤ s ≤ p∗ . Observe that for n large 1 F (x, un )dx. (2.17) c + 1 ≥ Φ (un ) − ⟨Φ ′ (un ), un ⟩ = p
RN
For 0 ≤ a < b, let
Ωn (a, b) = x ∈ RN : a ≤ |un (x)| < b .
(2.18)
Passing to a subsequence, we may assume that vn ⇀ v in E, then by Lemma 2.1, vn → v in Ls (RN ), p ≤ s < p∗ , and vn → v a.e. on RN . If v = 0, then vn → 0 in Ls (RN ), p ≤ s < p∗ , vn → 0 a.e. on RN . Hence, it follows from (2.4) and (2.16) that 1 p
|F (x, un )| dx. ∥ un ∥ p
≤ lim sup n→∞
RN
(2.19)
On the other hand, by virtue of (S0), one has
Ωn (0,r0 )
|F (x, un )| |vn |p dx ≤ | un | p
p
≤
c1
c1 p
+ +
c2 q c2 q
q −2 r0
q −2
Ωn (0,r0 )
r0
|vn |p dx
|vn |p dx → 0.
RN
(2.20)
Set κ ′ = κ/(κ − 1). Since κ > max{1, N /p}, one sees that pκ ′ ∈ (p, p∗ ). Hence, from (S5) and (2.17), one has
Ωn (r0 ,∞)
|F (x, un )| |vn |p dx ≤ | un | p
Ωn (r0 ,∞)
1/κ
≤ c0
|F (x, un )| |un |p
1/κ
κ
Ωn (r0 ,∞)
1/κ
Ωn (r0 ,∞)
≤ [c0 (c + 1)]1/κ
F (x, un )dx
′
|vn |pκ dx ′
1/κ ′
1/κ ′
RN
|vn |pκ dx RN
|vn |pκ dx ′
dx
1/κ ′
→ 0.
(2.21)
Combining (2.20) with (2.21), we have
RN
|F (x, un )| dx = ∥ un ∥ p
Ωn (0,r0 )
|F (x, un )| |vn |p dx + |un |p
Ωn (r0 ,∞)
|F (x, un )| |vn |p dx → 0, |un |p
which contradicts (2.19). Set A := {x ∈ RN : v(x) ̸= 0}, then meas(A) > 0. For a.e. x ∈ A, we have limn→∞ |un (x)| = ∞. Hence A ⊂ Ωn (r0 , ∞) for large n ∈ N, it follows from (S0), (S2′ ), (2.4) and Fatou’s lemma that
Φ (un ) ∥un ∥ ∥un ∥p 1 a0 F ( x , un ) p = lim − ∥vn ∥pp − |v | dx n n→∞ p p | un | p RN 1 F (x, un ) F (x, un ) p p ≤ lim − |v | dx − |v | dx n n p p n→∞ p Ωn (0,r0 ) |un | Ωn (r0 ,∞) |un | 1 c1 c2 F (x, un ) p ≤ lim sup + + r0q−2 |vn |p dx − |v | dx n p p p q n→∞ RN Ωn (r0 ,∞) |un | 1 c1 c2 F ( x , un ) ≤ + + r0q−2 γpp − lim inf |vn |p dx p n→∞ p p q Ωn (r0 ,∞) |un | 1 c1 c2 |F (x, un )| + r0q−2 γpp − lim inf [χΩn (r0 ,∞) (x)]|vn |p dx = + n →∞ p p q |un |p RN
0 = lim
n→∞
c + o(1) p
= lim
n→∞
78
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
≤
1 p
+
c1 p
+
c2 q
q −2
r0
γpp −
lim inf RN
F (x, un )
|un |p
n→∞
[χΩn (r0 ,∞) (x)]|vn |p dx
= −∞,
(2.22)
which is a contradiction. Thus {un } is bounded in E.
Lemma 2.8. Suppose that (V1), (V2′ ), (S0) and (S2′ ) are satisfied. Then any sequence {un } ⊂ E satisfying (2.16) has a convergent subsequence in E. Proof. Lemma 2.7 implies that {un } is bounded in E. Going if necessary to a subsequence, we can assume that un ⇀ u in E. Since E is a reflexive Banach space, it is isometrically isomorphic to a locally uniformly convex space. So the weak convergence and norm convergence imply strong convergence. Therefore we only need to show that ∥un ∥ → ∥u∥. Using Lemma 2.1, un → u in Ls (RN ) for p ≤ s < p∗ , which, together with [11, Lemma 2.3], one has
RN
|un |p−2 un − |u|p−2 u |un − u|dx → 0
(2.23)
|f (x, un ) − f (x, u)| |un − u|dx → 0.
(2.24)
and
RN
Observe that
⟨H ′ (un ) − H ′ (u), un − u⟩ = ⟨Φ ′ (un ) − Φ ′ (u), un − u⟩ + a0 + RN
RN
p−2 |un | un − |u|p−2 u (un − u)dx
[f (x, un ) − f (x, u)](un − u)dx.
(2.25)
It is clear that
⟨Φ ′ (un ) − Φ ′ (u), un − u⟩ → 0.
(2.26)
From (2.23)–(2.26), we have
⟨H ′ (un ) − H ′ (u), un − u⟩ → 0.
(2.27)
In virtue of [23, Lemma 3.1], one has
⟨H ′ (un ) − H ′ (u), un − u⟩ ≥ (∥un ∥p − ∥u∥p )(∥un ∥ − ∥u∥) ≥ 0. Hence ∥un ∥ → ∥u∥ as n → ∞ and the assertion follows.
(2.28)
Lemma 2.9. Suppose that (V1), (V2′ ), (S0), (S2′ ) and (S6) are satisfied. Then any sequence {un } ⊂ E satisfying (2.16) has a convergent subsequence in E. Proof. Employing Lemma 2.8, we only prove that {un } is bounded in E. To this end, arguing by contradiction, suppose that ∥un ∥ → ∞. Let vn = un /∥un ∥. Then ∥vn ∥ = 1 and ∥vn ∥s ≤ γs ∥vn ∥ = γs for p ≤ s ≤ p∗ . Passing to a subsequence, we may assume that vn ⇀ v in E, then by Lemma 2.1, vn → v in Ls (RN ), p ≤ s < p∗ , and vn → v a.e. on RN . By (2.4), (2.5), (2.16) and (S6), one has c + 1 ≥ Φ ( un ) −
= ≥
1
µ
⟨Φ ′ (un ), un ⟩
µ−p (µ − p)a0 ∥ un ∥ p − ∥un ∥pp + pµ pµ
RN
µ−p (µ − p)a0 + pϱ ∥ un ∥ p − ∥un ∥pp , pµ pµ
1
µ
f (x, un )un − F (x, un ) dx
for large n ∈ N,
(2.29)
which implies 1≤
(µ − p)a0 + pϱ (µ − p)a0 + pϱ lim sup ∥vn ∥pp = lim sup ∥v∥pp . µ−p µ−p n→∞ n→∞
(2.30)
Hence, it follows from (2.30) that v ̸= 0. By a similar fashion as (2.22), we can conclude a contradiction. Thus, {un } is bounded in E. Proof of Theorem 1.2. Let X = E , C− and C+ be defined by (2.10). Then Lemmas 2.4–2.6 and 2.8 imply that Φ satisfies all conditions of Lemma 2.2. Thus, problem (1.1) possesses a nontrivial solution.
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
79
Proof of Theorem 1.3. Let X = E , C− and C+ be defined by (2.10). Then Lemmas 2.4–2.6 and 2.9 imply that Φ satisfies all conditions of Lemma 2.2. Thus, problem (1.1) possesses a nontrivial solution. 3. Existence of infinity many solutions In this section, we are concerned with the existence of infinity many solutions for (1.1). By the Sobolev embedding theorem, we have the following lemma.
√
Lemma 3.1. Let Q0 := (x1 , x2 , . . . , xN ) ∈ RN : 0 ≤ xi < d0 / N , i = 1, 2, . . . , N . Then there exists a constant C0 > 0 such that
p/p′
p′
≤ C0
|u| dx
|∇ u|p + |u|p dx,
(3.1)
Q0
Q0
where p′ = 2p if p ≥ N and p′ = (N − 1)p/(N − p) if p < N. Lemma 3.2. Suppose that (V1) and (V2′ ) are satisfied. Then for any r > 0 and M > 0
p
|u| dx ≤ |x|>r
1
+ C0 [εr (M )]
M
(p′ −p)/p′
∥u∥p ,
∀ u ∈ E,
(3.2)
where
εr (M ) := sup meas x ∈ RN : |x − y| ≤ d0 , V (x) ≤ M . |y|≥r
Proof. Let Q0 be defined as in Lemma 3.1. Then Q0 ⊂ Bd0 . Choose a sequence {z (i) }i∈N ⊂ RN such that
RN =
∞
z (i) + Q0 ,
z (i) + Q0 ∩ z (j) + Q0 = ∅,
∀ i ̸= j.
(3.3)
i =1
Let Θi = z (i) + Q0 . For any r > 0 and M > 0, let A(r , M ) := {x ∈ RN : |x| > r , V (x) > M } and B(r , M ) := {x ∈ RN : |x| > r , V (x) ≤ M }. Then
p
|u| dx ≤ A(r ,M )
1
M
1
V (x)|u| dx ≤ p
A(r ,M )
M
RN
V (x)|u|p dx ≤
1 M
∥ u∥ p .
(3.4)
On the other hand, from (3.1), (3.3) and the Hölder inequality, one has
|u|p dx = B(r ,M )
∞ i =1
|u|p dx B(r ,M )∩Θi
∞ (p′ −p)/p′ ≤ [meas(B(r , M ) ∩ Θi )] i =1
≤ [εr (M )]
(p′ −p)/p′
∞
′
′
′
′
p/p′
∞ |∇ u|p + |u|p dx
Θi
|∇ u|p + |u|p dx RN
≤ C0 [εr (M )](p −p)/p ∥u∥p . ′
|u| dx B(r ,M )∩Θi
|u| dx
i=1
≤ C0 [εr (M )](p −p)/p
p/p′
Θi
i =1
≤ C0 [εr (M )](p −p)/p
p′
p′
′
Both (3.4) and (3.5) imply that (3.2) holds.
(3.5)
80
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
Lemma 3.3 ([27]). Let X be an infinite dimensional Banach space, X = Y ⊕ Z , where Y is finite dimensional. If I ∈ C 1 (X , R) satisfies (C)c -condition for all c > 0, and (I1) I (0) = 0, I (−u) = I (u) for all u ∈ X ; (I2) there exist constants ρ, α > 0 such that I |∂ Bρ ∩Z ≥ α ;
(I3) for any finite dimensional subspace X˜ ⊂ X , there is R = R(X˜ ) > 0 such that I (u) ≤ 0 on X˜ \ BR ; then I possesses an unbounded sequence of critical values. Lemma 3.4. Suppose that (V1), (V2′ ), (S0) and (S2′ ) are satisfied. Then for any finite dimensional subspace E˜ ⊂ E, there holds
Φ (u) → −∞,
u ∈ E˜ .
∥u∥ → ∞,
(3.6)
Proof. Arguing indirectly, assume that for some sequence {un } ⊂ E˜ with ∥un ∥ → ∞, there is M > 0 such that Φ (un ) ≥ −M for all n ∈ N. Set vn = un /∥un ∥, then ∥vn ∥ = 1. Passing to a subsequence, we may assume that vn ⇀ v in E. Since E˜ is finite dimensional, then vn → v ∈ E˜ in E, vn → v a.e. on RN , and so ∥v∥ = 1. Hence, we can deduce a contradiction in a similar way as (2.22). Corollary 3.5. Suppose that (V1), (V2′ ), (S0) and (S2′ ) are satisfied. Then for any finite dimensional subspace E˜ ⊂ E, there is R = R(E˜ ) > 0 such that
Φ (u) ≤ 0,
∀ u ∈ E˜ ,
∥u∥ ≥ R.
(3.7)
Since εr (M ) → 0 as r → ∞, in view of Lemma 3.2, we can choose R0 > 0 such that
|u|p dx ≤ |x|>R0
1 2(2a0 + pc1 )
∥ u∥ p ,
∀ u ∈ E.
(3.8)
Let {ej } is a total orthonormal basis of L2 (BR0 ) and define Xj = Rej , j ∈ N, Yk = ⊕kj=1 Xj ,
Zk = ⊕∞ j=k+1 Xj ,
k ∈ N.
(3.9)
Lemma 3.6. Suppose that (V1) and (V2′ ) are satisfied. Then for p ≤ s < p∗
βk :=
sup
u∈Zk , ∥u∥ 1,p =1 W (BR )
∥u∥Ls (BR0 ) → 0,
k → ∞.
(3.10)
0
Proof. Note that W 1,p (BR0 ) ↩→ Ls (BR0 ) for 1 ≤ s ≤ p∗ . Thus 0 < βk+1 ≤ βk < ∞, and so that βk → β ≥ 0, k → ∞. For k ≥ 0, there exists uk ∈ Zk such that ∥uk ∥W 1,p (BR ) = 1 and ∥uk ∥Ls (BR ) > βk /2. By definition of Zk , uk ⇀ 0 in L2 (BR0 ), and so 0
0
uk ⇀ 0 in W 1,p (BR0 ). Hence Lemma 2.1 implies that uk → 0 in Ls (BR0 ). Thus β = 0.
By Lemma 3.6, we can choose an integer m ≥ 1 such that
|u| dx ≤ |x|≤R0
1
p
2(2a0 + pc1 ) |x|≤R0
∀ u ∈ Zm ∩ W 1,p (BR0 ).
|∇ u|p + |u|p dx,
(3.11)
Let η(x) = 0 if |x| ≤ R0 and η(x) = 1 if |x| > R0 . Define Y = {(1 − η)u : u ∈ E , (1 − η)u ∈ Ym }
(3.12)
Z = {(1 − η)u : u ∈ E , (1 − η)u ∈ Zm } + {ηv : v ∈ E }.
(3.13)
and
Then Y and Z are subspaces of E, and E = Y ⊕ Z . Lemma 3.7. Suppose that (V1), (V2′ ) and (S0) are satisfied. Then there exist constants ρ, α > 0 such that Φ |∂ Bρ ∩Z ≥ α . Proof. By (3.8), (3.11) and (3.13), we have
p p
p
∥ u∥ = |x|≤R0
≤
|u| dx +
1
|u|p dx |x|>R0
|∇ u|p + |u|p dx +
2(2a0 + pc1 ) |x|≤R0 1 ≤ ∥u∥p , ∀u ∈ Z . 2a0 + pc1
1 2(2a0 + pc1 )
∥u∥p
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
81
Hence it follows from (S0), (2.2), (2.4) and the above inequality that
Φ (u) =
≥ ≥ ≥
1 p 1 p 1 p
∥ u∥ p − ∥ u∥ p − ∥ u∥ p −
1 2p
a0
p 2a0 + pc1 2p 1 2p
∥u∥pp −
RN
F (x, u)dx
∥u∥pp −
c2 q
∥u∥qq
q
∥ u∥ p −
c2 γq q
∥ u∥ q
q
∥ u∥ p −
c2 γq q
∥ u∥ q .
Since p < q, the assertion follows.
Proof of Theorem 1.5. Let X = E , Y and Z be defined by (3.12) and (3.13), respectively. Lemmas 2.7, 2.8 and 3.7 and Corollary 3.5 imply that Φ satisfies all conditions of Lemma 3.3. Thus, problem (1.1) possesses infinitely many nontrivial solutions. By a similar fashion, instead of Lemmas 2.7 and 2.8, by Lemma 2.9, we can prove Theorem 1.6. References [1] C.O. Alves, G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN , J. Differential Equations 246 (2009) 1288–1311. [2] T. Bartsch, Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN , Comm. Partial Differential Equations 20 (1995) 1725–1741. [3] V. Coti Zelati, P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on RN , Comm. Pure Appl. Math. XIV (1992) 1217–1269. [4] Y. Ding, C. Lee, Multiple solutions of Schröinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations 222 (2006) 137–163. [5] Y. Ding, A. Szulkin, Bound states for semilinear Schröinger equations with sign-changing potential, Calc. Var. Partial Differential Equations 29 (3) (2007) 397–419. [6] L. Jeanjean, K. Tanaka, A positive solution for asymptotically linear elliptic problem on RN autonomous at infinity, ESAIM Control Optim. Calc. Var. 7 (2002) 597–614. [7] Y.Q. Li, Z.-Q. Wang, J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 829–837. [8] Z.L. Liu, Z.-Q. Wang, On the Ambrosetti–Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (2004) 561–572. [9] P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992) 270–291. [10] A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (12) (2009) 3802–3822. [11] X.H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl. 401 (2013) 407–415. [12] M. Yang, Ground state solutions for a periodic Schröinger equation with superlinear nonlinearities, Nonlinear Anal. 72 (5) (2010) 2620–2627. [13] W.M. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358. [14] C.O. Alves, G.M. Figueiredo, Existence and multiplicity of positive solutions to a p-Laplacian equation in RN , Differential Integral Equations 19 (2006) 143–162. [15] A. El Khalil, S. El Manouni, M. Ouanan, On some nonlinear elliptic problems for p-Laplacian in RN , NoDEA Nonlinear Differential Equations Appl. 15 (2008) 295–307. [16] S.B. Liu, Existence of solutions to a superlinear p-Laplacian equation, Electron. J. Differential Equations 66 (2001) 6. [17] S.B. Liu, On ground states of superlinear p-Laplacian equations in RN , J. Math. Anal. Appl. 361 (2010) 48–58. [18] E.R. Fadell, P.H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal. 26 (1) (1977) 48–67. [19] M. Degiovanni, S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 907–919. [20] T. Bartsch, Z.L. Liu, On a superlinear elliptic p-Laplacian equation, J. Differential Equations 198 (2004) 149–175. [21] F. Fang, S.B. Liu, Nontrivial solutions of superlinear p-Laplacian equations, J. Math. Anal. Appl. 351 (2009) 138–146. [22] K. Perera, Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2003) 301–309. [23] C. Liu, Y. Zheng, Existence of nontrivial solutions for p-Laplacian equations in RN , J. Math. Anal. Appl. 380 (2011) 669–679. [24] L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem set on RN , Proc. Roy. Soc. Edinburgh 129 (1999) 787–809. [25] S.B. Liu, S.J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.) 46 (4) (2003) 625–630 (in Chinese). [26] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [27] T. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7 (1983) 241–273.
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence of infinitely many solutions for p-Laplacian equations in RN ✩ Xiaoyan Lin a,∗ , X.H. Tang b a
Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, PR China
b
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China
highlights • We give a direct sum decomposition of the Sobolev space corresponding to the quasilinear elliptic problem. • We obtain the existence of infinitely many solutions of the quasilinear elliptic problem. • We establish new theorems under weaker assumptions on the nonlinearity.
article
abstract
info
Article history: Received 27 January 2013 Accepted 22 June 2013 Communicated by Enzo Mitidieri
Based on the cohomological linking method for cones and a new direct sum decomposition of W 1,p (RN ), we study the existence and multiplicity of solutions of a p-Laplacian equation in RN with sign-changing potential and subcritical p-superlinear nonlinearity. Our assumptions are mild and different from those studied previously. © 2013 Elsevier Ltd. All rights reserved.
MSC: 35J20 35J60 Keywords: p-Laplacian equation p-superlinear Sign-changing potential
1. Introduction Consider the following p-Laplacian equation
− △p u + V (x)|u|p−2 u = f (x, u), u ∈ W 1,p (RN ),
x ∈ RN ,
(1.1)
where △p u := div(|∇ u|p−2 ∇ u) is the p-Laplacian operator with p > 1, V : RN → R and f : RN × R → R. For p = 2, (1.1) turns into the following semilinear Schrödinger equation
−△u + V (x)u = f (x, u), u ∈ H 1 (RN ),
x ∈ RN ,
(1.2)
✩ This work is partially supported by the NNSF (No: 11171351) of China and supported by Scientific Research Fund of Hunan Provincial Education Department (08A053) and supported by Hunan Provincial Natural Science Foundation of China (No. 11JJ2005). ∗ Corresponding author. Tel.: +86 13466651234. E-mail addresses: [email protected] (X. Lin), [email protected] (X.H. Tang).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.06.011
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
73
which has been studied extensively; see for example [1–13] and the references quoted in them. Most of them deal with the problem (1.2) with a constant sign potential V (x), and the others treat the case where V (x) is sign-changing. For the second case, the classical proof is based on the fact that the spectrum σ (−△ + V ) of self-adjoint operator −△ + V induces a suitable direct sum decomposition of H 1 (RN ). For general case p > 1, many results for (1.2) can be extended to (1.1) when the potential V (x) is of constant sign; see for example [14–17] and the references therein. However, if V (x) is sign-changing, the quasilinear problem (1.1) is far more difficult as − △p is no longer a self-adjoint operator and so a complete description of its spectrum is not available. For these reasons, few papers have treated this case so far. Using the cohomological index of Fadell and Rabinowitz [18], Degiovanni and Lancelotti [19] developed a new linking theorem over cones by constructing an unbounded sequence of minimax eigenvalues; then, making use of the linking theorem, they have proved that the quasilinear elliptic boundary value problem admits a nontrivial solution on a bounded domain. Some related results can be found in [20–22] for a similar problem on a bounded domain. In a recent paper [23], Liu and Zheng obtained a theorem on the existence of a nontrivial solution of (1.1) by applying a linking theorem over cones which was mainly developed in [19], where the following assumptions on V and f are introduced: (V1) V ∈ C (RN , R) and inf RN V (x) > −∞; (V2) for any M > 0, meas x ∈ RN : V (x) ≤ M < ∞, where meas(·) denotes the Lebesgue measure in RN ; (S0) f ∈ C (RN +1 , R), and there exist constants c1 , c2 > 0 and q ∈ (p, p∗ ) such that
|f (x, t )| ≤ c1 |t |p−1 + c2 |t |q−1 ,
∀ (x, t ) ∈ RN × R,
(1.3)
where p = ∞ if N ≤ p and p = pN /(N − p) if N > p; (S1) f (x, t ) = o(|t |p−1 ), as |t | → 0, uniformly in x ∈ RN ; t F (x,t ) (S2) F (x, t ) := 0 f (x, s)ds ≥ 0, ∀ (x, t ) ∈ RN × R, lim|t |→∞ |t |p = ∞, uniformly in x ∈ RN ; (S3) there exists θ ≥ 1 such that ∗
∗
θ F (x, t ) ≥ F (x, st ), where F (x, t ) =
1 tf p
∀ (x, t ) ∈ RN × R, s ∈ [0, 1],
(x, t ) − F (x, t ).
Now, we are able to state a main result in [23]. Theorem 1.1 ([23, Theorem 1.1]). Assume that V and f satisfy (V1), (V2), (S0), (S1), (S2) and (S3). Then problem (1.1) possesses a nontrivial solution. Condition (V2) was introduced in [2], and then was used by many authors, for example, [13]. Condition (S3) is due to Jeanjean [24]. In [6], this condition is also used together with a Cerami type argument in singularly perturbed elliptic problems in RN with autonomous nonlinearity. As shown in [25], condition (S3) is somewhat weaker than the condition that f (x, t )/|t |p−1 is nondecreasing in t ∈ (−∞, 0) ∪ (0, +∞) for all x ∈ RN . Moreover, there are many functions (e.g. f (x, t ) = a|t |p−2 t ln(1 + |t |), a > 0) which satisfy (S3), but do not satisfy the following classical condition which was introduced by Ambrosetti and Rabinowitz in [26]: (AR) there exists µ > p such that 0 < µF (x, t ) ≤ tf (x, t ),
t ̸= 0.
However, (AR)-condition does not imply condition (S3); see the example in [11]. Therefore, (S2) and (S3) are complementary super-quadratic conditions to (AR). In this paper, we are interested in the existence of a nontrivial solution or infinitely many nontrivial solutions of (1.1) with sign-changing potential and subcritical p-superlinear nonlinearity. By using a linking theorem over cones in [19], we will establish theorems on the existence of a nontrivial solution of (1.1) with mild assumptions deeply different from those studied in previous related works. For any R > 0, the Sobolev embedding theorem implies W 1,p (BR ) ↩→ L2 (BR ), where BR = {x ∈ RN : |x| < R}. Based on the above fact, we can construct a direct sum decomposition of W 1,p (RN ). Then a Z2 -version of the Mountain Pass Theorem is applied to obtain the existence theorem of infinitely many nontrivial solutions of (1.1). As far as we are aware there were no such multiplicity results in this situation. To state our results, we first introduce the following assumptions:
(V2′ ) there exists a constant d0 > 0 such that lim meas x ∈ RN : |x − y| ≤ d0 , V (x) ≤ M = 0, |y|→+∞
∀M > 0,
where meas(·) denotes the Lebesgue measure in RN ; (S2 ) lim|t |→∞ |F |(tx|,pt )| = ∞, a.e. x ∈ RN , and there exists r0 ≥ 0 such that ′
F ( x, t ) ≥ 0 ,
∀ (x, t ) ∈ RN × R,
(S4) f (x, −t ) = −f (x, t ), ∀ (x, t ) ∈ R × R; N
|t | ≥ r0 ;
74
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
(S5) F (x, t ) :=
1 tf p
(x, t ) − F (x, t ) ≥ 0, and there exist c0 > 0 and κ > max{1, N /p} such that
|F (x, t )|κ ≤ c0 |t |pκ F (x, t ),
∀ (x, t ) ∈ RN × R,
|t | ≥ r0 ;
(S6) there exist µ > p and ϱ > 0 such that
µF (x, t ) ≤ tf (x, t ) + ϱt p ,
∀ (x, t ) ∈ RN × R;
(S7) there exist µ > p and r1 > 0 such that
µF (x, t ) ≤ tf (x, t ),
∀ (x, t ) ∈ RN × R,
|t | ≥ r1 .
Now, we are ready to state the main results of this paper. Theorem 1.2. Assume that V and f satisfy (V1), (V2′ ), (S0), (S1), (S2) and (S5). Then problem (1.1) possesses a nontrivial solution. Theorem 1.3. Assume that V and f satisfy (V1), (V2′ ), (S0), (S1), (S2) and (S6). Then problem (1.1) possesses a nontrivial solution. It is easy to check that (S0) and (S7) imply (S6). Thus, we have the following corollary. Corollary 1.4. Assume that V and f satisfy (V1), (V2′ ), (S0), (S1), (S2) and (S7). Then problem (1.1) possesses a nontrivial solution. Theorem 1.5. Assume that V and f satisfy (V1), (V2′ ), (S0), (S2′ ), (S4) and (S5). Then problem (1.1) possesses infinitely many nontrivial solutions. Theorem 1.6. Assume that V and f satisfy (V1), (V2′ ), (S0), (S2′ ), (S4) and (S6). Then problem (1.1) possesses infinitely many nontrivial solutions. Corollary 1.7. Assume that V and f satisfy (V1), (V2′ ), (S0), (S2′ ), (S4) and (S7). Then problem (1.1) possesses infinitely many nontrivial solutions. Remark 1.8. Obviously, (V2′ ), (S2′ ) and (S5) are weaker than (V2), (S2) and (AR), respectively. In particular, we remove the usual condition (S1), and F (x, t ) is allowed to be sign-changing in Theorems 1.5, 1.6 and Corollary 1.7. It is easy to check that function f (x, t ) = a|t |p−2 t ln (1 + |t |) with a > 0 satisfies (S0), (S1), (S2), (S3) and (S5). However, the following function f (x, t ) = 3|t |
2
t
|s|1+sin s sds + |t |4+sin t t ,
(1.4)
0
satisfies (S5) not (S3) for p = 2; see [11, Section 3]. In addition, the following functions f (x, t ) = a(x)|t |p−1 t (p + 3)t 2 − 2(p + 2)t + p + 1 ,
(1.5)
f (x, t ) = a(x)|t |p−2 t 4|t |3 + 2t sin t − 4 cos t
(1.6)
and f (x, t ) = a(x)
m
bi |t |βi t ,
(1.7)
i=1
satisfy (S0), (S2′ ), (S4) and (S7), where b1 > 0, bi ∈ R, i = 2, 3, . . . , m; β1 > β2 > · · · > βm ≥ p − 2, a ∈ C (RN , R), and 0 < infRN a ≤ supRN a < ∞. One can see that they satisfy neither (AR) nor (S3). 2. Existence of a nontrivial solution By (V1), V (x) is bounded from below and so there is an a0 > 0 such that V (x) + a0 ≥ 1,
∀ x ∈ RN .
Define
E=
u ∈ W 1,p (RN ) :
RN
|∇ u|p + [V (x) + a0 ]|u|p dx < +∞ .
(2.1)
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
75
Then E is a reflexive and separable Banach space with the norm
∥ u∥ = RN
|∇ u|p + [V (x) + a0 ]|u|p dx
1/p
,
u ∈ E.
Evidently, E is continuously embedded into W 1,p (RN ) and hence continuously embedded into Ls (RN ) for p ≤ s ≤ p∗ , i.e., there exists γs > 0 such that
∥u∥s ≤ γs ∥u∥,
∀ u ∈ E,
(2.2)
where ∥u∥s denotes the usual norm in L (R ) for all p ≤ s ≤ p . In fact we further have the following lemma due to [2]. s
∗
N
Lemma 2.1 ([2, Lemma 3.1]). Suppose that (V1) and (V2′ ) hold. Then the embedding from E into Ls (RN ) is compact for p ≤ s < p∗ . Now we define a functional Φ on E by
Φ ( u) =
1 p
RN
|∇ u|p + V (x)|u|p dx −
RN
F (x, u)dx,
∀ u ∈ E.
(2.3)
Under assumptions (V1) and (S0), the functional Φ defined by (2.3) is of class C 1 (E , R). Moreover,
Φ (u) =
1 p
p
∥ u∥ −
a0 p
p p
∥ u∥ −
⟨Φ ′ (u), v⟩ = ⟨H ′ (u), v⟩ −
RN
F (x, u)dx,
∀ u ∈ E,
a0 |u|p−2 u + f (x, u) v dx,
RN
(2.4)
∀ u, v ∈ E ,
(2.5)
where H ( u) =
1
p
RN
|∇ u|p + (V (x) + a0 )|u|p dx,
∀ u ∈ E.
(2.6)
We say that ϕ ∈ C 1 (X , R) satisfies (C)c -condition if any sequence {un } such that
ϕ(un ) → c ,
∥ϕ ′ (un )∥(1 + ∥un ∥) → 0
(2.7)
has a convergent subsequence. Lemma 2.2 ([19, Corollary 2.9], [23, Lemma 2.2]). Let X be a real Banach space and C− and C+ two symmetric closed cones in X , C− ∩ C+ = {0} and Index(C− \ {0}) = Index(X \ C+ ) < ∞,
(2.8)
where and in the sequel, Index is the Z2 -cohomological index of [18]. Let r− > r+ > 0 and e ∈ X \ C− with ∥e∥ = 1. If ϕ ∈ C 1 (X , R) satisfies (C)c -condition for all c > 0, and inf ϕ(S+ ) > sup ϕ(∂ Q ),
sup ϕ(Q ) < ∞,
where S+ = {u ∈ C+ : ∥u∥ = r+ }, Q = {u + te : u ∈ C− , t ≥ 0, ∥u + te∥ ≤ r− }, and
∂ Q = {u ∈ C− : ∥u∥ ≤ r− } ∪ {u + te : u ∈ C− , t ≥ 0, ∥u + te∥ = r− }, then ϕ has a critical point with value c ≥ infS+ ϕ > 0. Set M = {u ∈ E : ∥u∥p = 1} and
λn = inf sup ∥u∥ : A ⊆ M, A is symmetric and Index(A) ≥ n , p
n ∈ Z.
(2.9)
u∈A
Lemma 2.3 ([23, Theorem 3.8]). Suppose that (V1) and (V2′ ) are satisfied. Then 0 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · and λn → +∞ as n → ∞.
76
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
By Lemma 2.3, we can choose an integer m ≥ 1 such that max{λm , a0 } < λm+1 . Set C− = {u ∈ E : ∥u∥p ≤ λm ∥u∥pp },
C+ = {u ∈ E : ∥u∥p ≥ λm+1 ∥u∥pp }.
(2.10)
Lemma 2.4 ([23, Theorem 3.10]). Suppose that (V1) and (V2′ ) are satisfied. Then C− and C+ are two symmetric closed cones in E , C− ∩ C+ = {0} and Index(C− \ {0}) = Index(E \ C+ ) = m.
(2.11)
Lemma 2.5. Suppose that (V1), (V2′ ), (S0) and (S1) are satisfied. Then there exist constants r+ , α > 0 such that Φ (u) ≥ α for u ∈ C+ and ∥u∥ = r+ . Proof. Choose ε ∈ (0, (λm+1 − a0 )/2), in view of conditions (S0) and (S1), there exists a constant Cε > 0 such that
|F (x, u)| ≤
ε p
|u|p + Cε |u|q ,
∀ (x, u) ∈ RN × R.
(2.12)
From (2.4), (2.10) and (2.12), one has
Φ (u) =
≥
1 p 1 p
∥ u∥ p − ∥ u∥ p −
a0
∥u∥pp −
p a0 + ε p
RN
F (x, u)dx
∥u∥pp − Cε ∥u∥qq
λm+1 − (a0 + ε) ∥u∥p − Cε γqq ∥u∥q pλm+1 λm+1 − a0 ∥u∥p − Cε γqq ∥u∥q . ≥ 2pλm+1
≥
Since p < q, the assertion follows.
Lemma 2.6. Suppose that (V1), (V2′ ), (S0) and (S2) are satisfied. Let e ∈ E \ C− with ∥e∥ = 1. Then there exists a constant r− > r+ such that sup Φ (Q ) < ∞ and sup Φ (∂ Q ) ≤ 0, where Q = {u + te : w ∈ C− , t ≥ 0, ∥u + te∥ ≤ r− } .
(2.13)
Proof. It is sufficient to show that Φ (u) → −∞ as u ∈ C− + R+ e and ∥u∥ → ∞. Arguing indirectly, assume that for some sequence {un + tn e} ⊂ C− + Re with ∥un + tn e∥ → ∞, there is M > 0 such that Φ (un + tn e) ≥ −M for all n ∈ N. Set + tn e = wn + sn e, then ∥vn ∥ = 1. According to [19, Proposition 2.12], there exists a constant β ≥ 1 such that vn = ∥uun + t e∥ n
n
∥u∥ + ∥e∥ ≤ β∥u + e∥,
∀ u ∈ C− .
(2.14)
It follows from (2.14) and ∥wn + sn e∥ = 1 that
∥wn ∥ + sn = ∥wn ∥ + |sn |∥e∥ ≤ β∥wn + sn e∥ = β.
(2.15)
Hence, passing to a subsequence, we may assume that sn → s0 ≥ 0, wn ⇀ w in E, wn → w a.e. on R , and so, by Lemma 2.1, N
wn → w in Lp (RN ). So, we have
1/p 1 = ∥wn + sn e∥ ≤ ∥wn ∥ + sn ≤ λ1m/p ∥wn ∥p + sn → λm ∥w∥p + s0 .
This shows that w + s0 e ̸= 0. Let A := {x ∈ RN : w(x) + s0 e(x) ̸= 0}, then meas(A) > 0. For a.e. x ∈ A, we have limn→∞ |un (x)| = ∞. Hence, it follows from (2.4), (S2) and Fatou’s lemma that
−M Φ (un + tn e) ≤ lim p n →∞ ∥ un + t n e ∥ ∥un + tn e∥p 1 a0 F (x, un + tn e) p lim − ∥vn ∥pp − |v | dx n n→∞ p p |un + tn e|p RN 1 F (x, un + tn e) − lim inf |wn + sn e|p dx n →∞ p |un + tn e|p RN 1 F (x, un + tn e) − lim inf |wn + sn e|p dx p n →∞ N p | u + t e | n n R −∞,
0 = lim
n→∞
= ≤ ≤ =
which is a contradiction.
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
77
Lemma 2.7. Suppose that (V1), (V2′ ), (S0), (S2′ ) and (S5) are satisfied. Then any sequence {un } ⊂ E satisfying
Φ (un ) → c > 0,
⟨Φ ′ (un ), un ⟩ → 0
(2.16)
is bounded in E. Proof. To prove the boundedness of {un }, arguing by contradiction, suppose that ∥un ∥ → ∞. Let vn = un /∥un ∥. Then ∥vn ∥ = 1 and ∥vn ∥s ≤ γs ∥vn ∥ = γs for p ≤ s ≤ p∗ . Observe that for n large 1 F (x, un )dx. (2.17) c + 1 ≥ Φ (un ) − ⟨Φ ′ (un ), un ⟩ = p
RN
For 0 ≤ a < b, let
Ωn (a, b) = x ∈ RN : a ≤ |un (x)| < b .
(2.18)
Passing to a subsequence, we may assume that vn ⇀ v in E, then by Lemma 2.1, vn → v in Ls (RN ), p ≤ s < p∗ , and vn → v a.e. on RN . If v = 0, then vn → 0 in Ls (RN ), p ≤ s < p∗ , vn → 0 a.e. on RN . Hence, it follows from (2.4) and (2.16) that 1 p
|F (x, un )| dx. ∥ un ∥ p
≤ lim sup n→∞
RN
(2.19)
On the other hand, by virtue of (S0), one has
Ωn (0,r0 )
|F (x, un )| |vn |p dx ≤ | un | p
p
≤
c1
c1 p
+ +
c2 q c2 q
q −2 r0
q −2
Ωn (0,r0 )
r0
|vn |p dx
|vn |p dx → 0.
RN
(2.20)
Set κ ′ = κ/(κ − 1). Since κ > max{1, N /p}, one sees that pκ ′ ∈ (p, p∗ ). Hence, from (S5) and (2.17), one has
Ωn (r0 ,∞)
|F (x, un )| |vn |p dx ≤ | un | p
Ωn (r0 ,∞)
1/κ
≤ c0
|F (x, un )| |un |p
1/κ
κ
Ωn (r0 ,∞)
1/κ
Ωn (r0 ,∞)
≤ [c0 (c + 1)]1/κ
F (x, un )dx
′
|vn |pκ dx ′
1/κ ′
1/κ ′
RN
|vn |pκ dx RN
|vn |pκ dx ′
dx
1/κ ′
→ 0.
(2.21)
Combining (2.20) with (2.21), we have
RN
|F (x, un )| dx = ∥ un ∥ p
Ωn (0,r0 )
|F (x, un )| |vn |p dx + |un |p
Ωn (r0 ,∞)
|F (x, un )| |vn |p dx → 0, |un |p
which contradicts (2.19). Set A := {x ∈ RN : v(x) ̸= 0}, then meas(A) > 0. For a.e. x ∈ A, we have limn→∞ |un (x)| = ∞. Hence A ⊂ Ωn (r0 , ∞) for large n ∈ N, it follows from (S0), (S2′ ), (2.4) and Fatou’s lemma that
Φ (un ) ∥un ∥ ∥un ∥p 1 a0 F ( x , un ) p = lim − ∥vn ∥pp − |v | dx n n→∞ p p | un | p RN 1 F (x, un ) F (x, un ) p p ≤ lim − |v | dx − |v | dx n n p p n→∞ p Ωn (0,r0 ) |un | Ωn (r0 ,∞) |un | 1 c1 c2 F (x, un ) p ≤ lim sup + + r0q−2 |vn |p dx − |v | dx n p p p q n→∞ RN Ωn (r0 ,∞) |un | 1 c1 c2 F ( x , un ) ≤ + + r0q−2 γpp − lim inf |vn |p dx p n→∞ p p q Ωn (r0 ,∞) |un | 1 c1 c2 |F (x, un )| + r0q−2 γpp − lim inf [χΩn (r0 ,∞) (x)]|vn |p dx = + n →∞ p p q |un |p RN
0 = lim
n→∞
c + o(1) p
= lim
n→∞
78
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
≤
1 p
+
c1 p
+
c2 q
q −2
r0
γpp −
lim inf RN
F (x, un )
|un |p
n→∞
[χΩn (r0 ,∞) (x)]|vn |p dx
= −∞,
(2.22)
which is a contradiction. Thus {un } is bounded in E.
Lemma 2.8. Suppose that (V1), (V2′ ), (S0) and (S2′ ) are satisfied. Then any sequence {un } ⊂ E satisfying (2.16) has a convergent subsequence in E. Proof. Lemma 2.7 implies that {un } is bounded in E. Going if necessary to a subsequence, we can assume that un ⇀ u in E. Since E is a reflexive Banach space, it is isometrically isomorphic to a locally uniformly convex space. So the weak convergence and norm convergence imply strong convergence. Therefore we only need to show that ∥un ∥ → ∥u∥. Using Lemma 2.1, un → u in Ls (RN ) for p ≤ s < p∗ , which, together with [11, Lemma 2.3], one has
RN
|un |p−2 un − |u|p−2 u |un − u|dx → 0
(2.23)
|f (x, un ) − f (x, u)| |un − u|dx → 0.
(2.24)
and
RN
Observe that
⟨H ′ (un ) − H ′ (u), un − u⟩ = ⟨Φ ′ (un ) − Φ ′ (u), un − u⟩ + a0 + RN
RN
p−2 |un | un − |u|p−2 u (un − u)dx
[f (x, un ) − f (x, u)](un − u)dx.
(2.25)
It is clear that
⟨Φ ′ (un ) − Φ ′ (u), un − u⟩ → 0.
(2.26)
From (2.23)–(2.26), we have
⟨H ′ (un ) − H ′ (u), un − u⟩ → 0.
(2.27)
In virtue of [23, Lemma 3.1], one has
⟨H ′ (un ) − H ′ (u), un − u⟩ ≥ (∥un ∥p − ∥u∥p )(∥un ∥ − ∥u∥) ≥ 0. Hence ∥un ∥ → ∥u∥ as n → ∞ and the assertion follows.
(2.28)
Lemma 2.9. Suppose that (V1), (V2′ ), (S0), (S2′ ) and (S6) are satisfied. Then any sequence {un } ⊂ E satisfying (2.16) has a convergent subsequence in E. Proof. Employing Lemma 2.8, we only prove that {un } is bounded in E. To this end, arguing by contradiction, suppose that ∥un ∥ → ∞. Let vn = un /∥un ∥. Then ∥vn ∥ = 1 and ∥vn ∥s ≤ γs ∥vn ∥ = γs for p ≤ s ≤ p∗ . Passing to a subsequence, we may assume that vn ⇀ v in E, then by Lemma 2.1, vn → v in Ls (RN ), p ≤ s < p∗ , and vn → v a.e. on RN . By (2.4), (2.5), (2.16) and (S6), one has c + 1 ≥ Φ ( un ) −
= ≥
1
µ
⟨Φ ′ (un ), un ⟩
µ−p (µ − p)a0 ∥ un ∥ p − ∥un ∥pp + pµ pµ
RN
µ−p (µ − p)a0 + pϱ ∥ un ∥ p − ∥un ∥pp , pµ pµ
1
µ
f (x, un )un − F (x, un ) dx
for large n ∈ N,
(2.29)
which implies 1≤
(µ − p)a0 + pϱ (µ − p)a0 + pϱ lim sup ∥vn ∥pp = lim sup ∥v∥pp . µ−p µ−p n→∞ n→∞
(2.30)
Hence, it follows from (2.30) that v ̸= 0. By a similar fashion as (2.22), we can conclude a contradiction. Thus, {un } is bounded in E. Proof of Theorem 1.2. Let X = E , C− and C+ be defined by (2.10). Then Lemmas 2.4–2.6 and 2.8 imply that Φ satisfies all conditions of Lemma 2.2. Thus, problem (1.1) possesses a nontrivial solution.
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
79
Proof of Theorem 1.3. Let X = E , C− and C+ be defined by (2.10). Then Lemmas 2.4–2.6 and 2.9 imply that Φ satisfies all conditions of Lemma 2.2. Thus, problem (1.1) possesses a nontrivial solution. 3. Existence of infinity many solutions In this section, we are concerned with the existence of infinity many solutions for (1.1). By the Sobolev embedding theorem, we have the following lemma.
√
Lemma 3.1. Let Q0 := (x1 , x2 , . . . , xN ) ∈ RN : 0 ≤ xi < d0 / N , i = 1, 2, . . . , N . Then there exists a constant C0 > 0 such that
p/p′
p′
≤ C0
|u| dx
|∇ u|p + |u|p dx,
(3.1)
Q0
Q0
where p′ = 2p if p ≥ N and p′ = (N − 1)p/(N − p) if p < N. Lemma 3.2. Suppose that (V1) and (V2′ ) are satisfied. Then for any r > 0 and M > 0
p
|u| dx ≤ |x|>r
1
+ C0 [εr (M )]
M
(p′ −p)/p′
∥u∥p ,
∀ u ∈ E,
(3.2)
where
εr (M ) := sup meas x ∈ RN : |x − y| ≤ d0 , V (x) ≤ M . |y|≥r
Proof. Let Q0 be defined as in Lemma 3.1. Then Q0 ⊂ Bd0 . Choose a sequence {z (i) }i∈N ⊂ RN such that
RN =
∞
z (i) + Q0 ,
z (i) + Q0 ∩ z (j) + Q0 = ∅,
∀ i ̸= j.
(3.3)
i =1
Let Θi = z (i) + Q0 . For any r > 0 and M > 0, let A(r , M ) := {x ∈ RN : |x| > r , V (x) > M } and B(r , M ) := {x ∈ RN : |x| > r , V (x) ≤ M }. Then
p
|u| dx ≤ A(r ,M )
1
M
1
V (x)|u| dx ≤ p
A(r ,M )
M
RN
V (x)|u|p dx ≤
1 M
∥ u∥ p .
(3.4)
On the other hand, from (3.1), (3.3) and the Hölder inequality, one has
|u|p dx = B(r ,M )
∞ i =1
|u|p dx B(r ,M )∩Θi
∞ (p′ −p)/p′ ≤ [meas(B(r , M ) ∩ Θi )] i =1
≤ [εr (M )]
(p′ −p)/p′
∞
′
′
′
′
p/p′
∞ |∇ u|p + |u|p dx
Θi
|∇ u|p + |u|p dx RN
≤ C0 [εr (M )](p −p)/p ∥u∥p . ′
|u| dx B(r ,M )∩Θi
|u| dx
i=1
≤ C0 [εr (M )](p −p)/p
p/p′
Θi
i =1
≤ C0 [εr (M )](p −p)/p
p′
p′
′
Both (3.4) and (3.5) imply that (3.2) holds.
(3.5)
80
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
Lemma 3.3 ([27]). Let X be an infinite dimensional Banach space, X = Y ⊕ Z , where Y is finite dimensional. If I ∈ C 1 (X , R) satisfies (C)c -condition for all c > 0, and (I1) I (0) = 0, I (−u) = I (u) for all u ∈ X ; (I2) there exist constants ρ, α > 0 such that I |∂ Bρ ∩Z ≥ α ;
(I3) for any finite dimensional subspace X˜ ⊂ X , there is R = R(X˜ ) > 0 such that I (u) ≤ 0 on X˜ \ BR ; then I possesses an unbounded sequence of critical values. Lemma 3.4. Suppose that (V1), (V2′ ), (S0) and (S2′ ) are satisfied. Then for any finite dimensional subspace E˜ ⊂ E, there holds
Φ (u) → −∞,
u ∈ E˜ .
∥u∥ → ∞,
(3.6)
Proof. Arguing indirectly, assume that for some sequence {un } ⊂ E˜ with ∥un ∥ → ∞, there is M > 0 such that Φ (un ) ≥ −M for all n ∈ N. Set vn = un /∥un ∥, then ∥vn ∥ = 1. Passing to a subsequence, we may assume that vn ⇀ v in E. Since E˜ is finite dimensional, then vn → v ∈ E˜ in E, vn → v a.e. on RN , and so ∥v∥ = 1. Hence, we can deduce a contradiction in a similar way as (2.22). Corollary 3.5. Suppose that (V1), (V2′ ), (S0) and (S2′ ) are satisfied. Then for any finite dimensional subspace E˜ ⊂ E, there is R = R(E˜ ) > 0 such that
Φ (u) ≤ 0,
∀ u ∈ E˜ ,
∥u∥ ≥ R.
(3.7)
Since εr (M ) → 0 as r → ∞, in view of Lemma 3.2, we can choose R0 > 0 such that
|u|p dx ≤ |x|>R0
1 2(2a0 + pc1 )
∥ u∥ p ,
∀ u ∈ E.
(3.8)
Let {ej } is a total orthonormal basis of L2 (BR0 ) and define Xj = Rej , j ∈ N, Yk = ⊕kj=1 Xj ,
Zk = ⊕∞ j=k+1 Xj ,
k ∈ N.
(3.9)
Lemma 3.6. Suppose that (V1) and (V2′ ) are satisfied. Then for p ≤ s < p∗
βk :=
sup
u∈Zk , ∥u∥ 1,p =1 W (BR )
∥u∥Ls (BR0 ) → 0,
k → ∞.
(3.10)
0
Proof. Note that W 1,p (BR0 ) ↩→ Ls (BR0 ) for 1 ≤ s ≤ p∗ . Thus 0 < βk+1 ≤ βk < ∞, and so that βk → β ≥ 0, k → ∞. For k ≥ 0, there exists uk ∈ Zk such that ∥uk ∥W 1,p (BR ) = 1 and ∥uk ∥Ls (BR ) > βk /2. By definition of Zk , uk ⇀ 0 in L2 (BR0 ), and so 0
0
uk ⇀ 0 in W 1,p (BR0 ). Hence Lemma 2.1 implies that uk → 0 in Ls (BR0 ). Thus β = 0.
By Lemma 3.6, we can choose an integer m ≥ 1 such that
|u| dx ≤ |x|≤R0
1
p
2(2a0 + pc1 ) |x|≤R0
∀ u ∈ Zm ∩ W 1,p (BR0 ).
|∇ u|p + |u|p dx,
(3.11)
Let η(x) = 0 if |x| ≤ R0 and η(x) = 1 if |x| > R0 . Define Y = {(1 − η)u : u ∈ E , (1 − η)u ∈ Ym }
(3.12)
Z = {(1 − η)u : u ∈ E , (1 − η)u ∈ Zm } + {ηv : v ∈ E }.
(3.13)
and
Then Y and Z are subspaces of E, and E = Y ⊕ Z . Lemma 3.7. Suppose that (V1), (V2′ ) and (S0) are satisfied. Then there exist constants ρ, α > 0 such that Φ |∂ Bρ ∩Z ≥ α . Proof. By (3.8), (3.11) and (3.13), we have
p p
p
∥ u∥ = |x|≤R0
≤
|u| dx +
1
|u|p dx |x|>R0
|∇ u|p + |u|p dx +
2(2a0 + pc1 ) |x|≤R0 1 ≤ ∥u∥p , ∀u ∈ Z . 2a0 + pc1
1 2(2a0 + pc1 )
∥u∥p
X. Lin, X.H. Tang / Nonlinear Analysis 92 (2013) 72–81
81
Hence it follows from (S0), (2.2), (2.4) and the above inequality that
Φ (u) =
≥ ≥ ≥
1 p 1 p 1 p
∥ u∥ p − ∥ u∥ p − ∥ u∥ p −
1 2p
a0
p 2a0 + pc1 2p 1 2p
∥u∥pp −
RN
F (x, u)dx
∥u∥pp −
c2 q
∥u∥qq
q
∥ u∥ p −
c2 γq q
∥ u∥ q
q
∥ u∥ p −
c2 γq q
∥ u∥ q .
Since p < q, the assertion follows.
Proof of Theorem 1.5. Let X = E , Y and Z be defined by (3.12) and (3.13), respectively. Lemmas 2.7, 2.8 and 3.7 and Corollary 3.5 imply that Φ satisfies all conditions of Lemma 3.3. Thus, problem (1.1) possesses infinitely many nontrivial solutions. By a similar fashion, instead of Lemmas 2.7 and 2.8, by Lemma 2.9, we can prove Theorem 1.6. References [1] C.O. Alves, G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN , J. Differential Equations 246 (2009) 1288–1311. [2] T. Bartsch, Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN , Comm. Partial Differential Equations 20 (1995) 1725–1741. [3] V. Coti Zelati, P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on RN , Comm. Pure Appl. Math. XIV (1992) 1217–1269. [4] Y. Ding, C. Lee, Multiple solutions of Schröinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations 222 (2006) 137–163. [5] Y. Ding, A. Szulkin, Bound states for semilinear Schröinger equations with sign-changing potential, Calc. Var. Partial Differential Equations 29 (3) (2007) 397–419. [6] L. Jeanjean, K. Tanaka, A positive solution for asymptotically linear elliptic problem on RN autonomous at infinity, ESAIM Control Optim. Calc. Var. 7 (2002) 597–614. [7] Y.Q. Li, Z.-Q. Wang, J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 829–837. [8] Z.L. Liu, Z.-Q. Wang, On the Ambrosetti–Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (2004) 561–572. [9] P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992) 270–291. [10] A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (12) (2009) 3802–3822. [11] X.H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl. 401 (2013) 407–415. [12] M. Yang, Ground state solutions for a periodic Schröinger equation with superlinear nonlinearities, Nonlinear Anal. 72 (5) (2010) 2620–2627. [13] W.M. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358. [14] C.O. Alves, G.M. Figueiredo, Existence and multiplicity of positive solutions to a p-Laplacian equation in RN , Differential Integral Equations 19 (2006) 143–162. [15] A. El Khalil, S. El Manouni, M. Ouanan, On some nonlinear elliptic problems for p-Laplacian in RN , NoDEA Nonlinear Differential Equations Appl. 15 (2008) 295–307. [16] S.B. Liu, Existence of solutions to a superlinear p-Laplacian equation, Electron. J. Differential Equations 66 (2001) 6. [17] S.B. Liu, On ground states of superlinear p-Laplacian equations in RN , J. Math. Anal. Appl. 361 (2010) 48–58. [18] E.R. Fadell, P.H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal. 26 (1) (1977) 48–67. [19] M. Degiovanni, S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 907–919. [20] T. Bartsch, Z.L. Liu, On a superlinear elliptic p-Laplacian equation, J. Differential Equations 198 (2004) 149–175. [21] F. Fang, S.B. Liu, Nontrivial solutions of superlinear p-Laplacian equations, J. Math. Anal. Appl. 351 (2009) 138–146. [22] K. Perera, Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2003) 301–309. [23] C. Liu, Y. Zheng, Existence of nontrivial solutions for p-Laplacian equations in RN , J. Math. Anal. Appl. 380 (2011) 669–679. [24] L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem set on RN , Proc. Roy. Soc. Edinburgh 129 (1999) 787–809. [25] S.B. Liu, S.J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.) 46 (4) (2003) 625–630 (in Chinese). [26] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [27] T. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7 (1983) 241–273.