Multiple positive solutions for Schrödinger problems with concave and convex nonlinearities

Multiple positive solutions for Schrödinger problems with concave and convex nonlinearities

Xiaofei Cao

and Junxiang Xu

1Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, 223003, China

2Faculty of Mathematics, Southeast University, Nanjing 210096, China

Received 19 November 2017, appeared 5 August 2018 Communicated by Petru Jebelean

Abstract. In this paper, we consider the multiplicity of positive solutions for a class of Schrödinger equations involving concave-convex nonlinearities in the whole space.

With the help of the Nehari manifold, Ekeland variational principle and the theory of Lagrange multipliers, we prove that the Schrödinger equation has at least two positive solutions, one of which is a positive ground state solution.

Keywords: Schrödinger problem, Nehari manifold, Ekeland variational principle.

2010 Mathematics Subject Classification: 35A01, 35A15.

1 Introduction and main results

This paper concerns the multiplicity of positive solutions for the following Schrödinger equa-

tion (

−4u+V(x)u= f(x)|u|^q−2u+g(x)|u|^p−2u inR^N, u∈ H¹ R^N

, (1.1)

where 1 < _q < ₂ < _p < ₂^∗ ₍₂^∗ = _{∞ if N} = _{1, 2 and 2}^∗ = _2N/(_N−2) _{if N} ≥ 3) and V(x),f(x),g(x)satisfy suitable conditions.

There are many works on nonlinearity of concave-convex type under various conditions on potential V(x). WhenV(x) ≡ 0, Equation (1.1) is considered in a bounded domain. This problem can date back to the famous work of Ambrosetti–Brezis–Cerami in [1], where the authors considered the following problem

(−4u=λ|u|^q−2u+|u|^p−2u in Ω,

u∈ H₀¹(_Ω), (1.2)

where Ω ⊂ _R^N is a bounded domain, 1 < q < 2 < p ≤ 2^∗. They proved that Equation (1.2) has at least two positive solutions for suffciently small λ > 0. In this case, the compact embedding H₀¹(_Ω),→ L^p(_Ω) (p ∈ [2, 2^∗))plays an important role; for more general results

BCorresponding author. Email: caoxiaofei258@126.com

in bounded domains see [4,5,8,13,18,24,27] and their references. In the whole space R^N some authors concerned Equation (1.1) withV(x)satisfying suitable conditions such that the embedding

X:=

u∈ H¹ R^N

: Z

R^NV(x)|u(x)|²dx< +_∞

,→L^p

R^N

, p∈ [2, 2^∗), (1.3) is compact. For example, Bartsch and Wang [2] first introduced the following weaker condition

(V) V(x) ∈ C R^N,R

,V₀ := inf_RNV(x) > 0 and for any M > 0, there exists a constant r₀ >0 such that meas({x∈ B_r0(y)_:V(x)≤ M}) → 0 as |y| → +_{∞, where} B_r0(y) _de- notes the ball centered atywith radiusr₀and meas the Lebesgue measure inR^N. For some results in this area, we also refer to [14,21].

If the potential functionV(x)is bounded, the embedding (1.3) is not compact; in the case of the constant potential, i.e., V(x) is a positive constant in Equation (1.1), we can refer to [25,26,28]. However, we do not know any results for Equation (1.1) with bothV(x)andg(x) bounded functions. A direct extension to the caseV(x)and g(x)bounded functions is faced with difficulties. On the one hand, because the nonlinearity is a combination of the concave and convex terms, estimating the critical value by suitable autonomous equation becomes complex. On the other hand, since bothV(x) and g(x)are bounded functions, the proof of the(PS)condition satisfied for the critical value in suitable range becomes delicate. In this paper, we are concerned about Equation (1.1) with both V(x) and g(x) bounded functions on the basis of variational arguments. IfV(x), f(x)and g(x)satisfy the suitable conditions, we prove multiple positive solutions for equation (1.1) under the quantitative assumption.

Up to now, there is a lot of papers considered different problems and obtained the relevant results under the quantitative assumption, see [6,7,12,29] for Kirchhoff problems, [15,26,27]

for Schrödinger problems and [16] for Schrödinger–Maxwell problems. For example, Wu [27]

considered the following Schrödinger problem:

(−4u= f(x)|u|^q−2u+ (1−g(x))|u|^2∗−2u inΩ,

u=0, in∂Ω,

where 1 < q < 2, 2^∗ = 2N/(N−2)(N ≥ 3), Ω ⊂ _R^N is a bounded domain with smooth boundary and the weight functions f,g ∈ C(_Ω) satisfy the suitable conditions. Then there exists λ₀ > 0 such that if kf⁺k_Lq∗ < λ₀, this problem has three positive solutions, where q^∗ =2^∗/(2^∗−q)and f⁺=max{f, 0} 6=0.

To state our main result, we introduce precise conditions onV(x), f(x)andg(x): (V) V(x)∈C R^N,R

, 0<V₀ := inf

x∈_R^NV(x)≤V(x)≤V_∞:= lim

|x|→+_∞V(x)<+_∞, (f) f is positive, continuous and belongs to L^q∗ R^N

, where q^∗ is conjugate to p/q (i.e.

q^∗ = p/(p−q)), (g) g(x)∈ C R^N

∩L^∞ R^N

, 0<g_∞ := lim

|x|→+_∞g(x)≤ g(x)≤ sup

x∈_R^N

g(x)<+_∞.

Our main result is as follows.

Letσ := (p−2)(2−q)^{(2−q)/(p−2)} _p^S₋^p_q^{(p−q)/(p−2)}and 0<σ^∗ =qσ/2<σ, whereS_p is the best Sobolev constant described in the following Lemma2.2.

Theorem 1.1. Under the assumptions (V), (f) and(g), if |f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈ (0,σ^∗), Equation (1.1) has at least two positive solutions, which correspond to negative energy and positive energy, respectively; in particular, the one with negative energy is a positive ground state solution.

The combined effects of a sub-linear and a super-linear terms change the structure of the solution set. According to the behaviour of nonlinearities and to the results we want to prove, the method of the decomposition of Nehari manifold turns out to be more appropriate. With the help of suitable autonomous equation, the Ekeland variational principle and the theory of Lagrange multipliers, we can prove that Equation (1.1) has at least two positive solutions, one of which is a positive ground state solution. In addition, the condition(V)can be replaced by other forms.

Remark 1.2. Assume that V

,(f) and (g), if |f|_q^∗|g|⁽_∞^{2−q)/(p−2)} is sufficiently small, then Theorem1.1still holds.

Remark 1.3. Assume thatV(x)≡C,(f)and(g), if|f|_q^∗|g|⁽_∞^{2−q)/(p−2)}is sufficiently small, then Theorem1.1still holds, whereCis a positive constant.

The rest of this paper is organized as follows: Section 2 is dedicated to our variational framework and some preliminary results. Section 3 concerns with the proof of Theorem1.1.

Throughout this paper,CandC_i denote distinct constants. L^p R^N

is the usual Lebesgue space endowed with the standard norm |u|_p = R

R^N|u|^pdx1/p

for 1 ≤ p < _∞ and |u|_∞ = sup_x∈RN|u(x)|forp=_∞. When it causes no confusion, we still denote by{u_n}a subsequence of the original sequence{un}.

2 Preliminary results

With the fact that the problem (1.1) has a variational structure, the proof is based on the variational approach and the use of the Nehari manifold technique. So, we will first recall some preliminaries and establish the variational setting for our problem in this section.

Define

E:=

u∈ H¹ R^N

\{0}

Z

R^NV(x)|u|²dx<+_∞

with the associate norm

kuk= _Z

R^N(|∇u|²+V(x)u²)dx ¹₂

.

Under the assumption (V), we know that the norm k · k is equivalent to the usual norm in H¹ R^N

. The energy functional corresponding to Equation (1.1) is I(u) = ¹

2 Z

R^N |∇u|²+V(x)|u|² dx−¹ q

Z

R^N f(x)|u|^qdx−¹ p

Z

R^Ng(x)|u|^pdx, u∈ E. (2.1) Lemma 2.1. If(V),(f)and(g)hold, then the functional I∈ C¹(E,R)and for any u,v∈ E

D

I⁰(u)_,vE

=

Z

R^N∇u∇v dx+

Z

R^NV(x)uv dx

−

Z

R^N f(x)|u|^q−2uv dx−

Z

R^Ng(x)|u|^p−2uv dx. (2.2) Furthermore, I⁰ is weakly sequentially continuous in E.

Proof. The proof is a direct computation. Here we omit details and refer to [23].

Lemma 2.2 ([23]). Under the assumption (V), the embedding E ,→ L^p R^N

is continuous for p∈[2, 2^∗]. Let

S_p = inf

u∈E\{0}

kuk² R

R^N|u|^pdx2/p >0, then

|u|_p ≤S⁻_p¹²kuk, ∀u∈ E.

It is well-known that seeking a weak solution of Equation (1.1) is equivalent to finding a critical point of the corresponding functional I. In the following, we are devoted to finding the critical point of the corresponding functionalI.

As usual, some energy functional such as I in (2.1) is not bounded from below on E but, as we will see, is bounded from below on an appropriate subset ofEand a minimizer on this set (if it exists) may give rise to a solution of corresponding differential equation (see [22]). A good exemplification for an appropriate subset ofEis the so-called Nehari manifold

N :=ⁿu∈ H¹(_R^N)\{0} | I⁰(u),u

=0o ,

where h, idenotes the usual duality between E andE^∗. It is clear to see that u ∈ N if and only if foru ∈H¹ R^N

\{0}, kuk²=

Z

R^N f(x)|u|^qdx+

Z

R^N g(x)|u|^pdx. (2.3) Obviously,N contains all nontrivial solutions of Equation (1.1). Below, we shall use the Nehari manifold methods to find critical points for the functional I.

The Nehari manifoldN is closely linked to the behavior of functions of the formK_u:t → I(tu)_for t > 0. Such maps are known as fibering maps, which were introduced by Drábek and Pohozaev in [9]. Foru∈ E, let

K_u(t) =I(tu) = ¹

2t²kuk²−¹ qt^q

Z

R^N f(x)|u|^qdx− ¹ pt^p

Z

R^Ng(x)|u|^pdx;

K_u⁰(t) =tkuk²−t^q−1 Z

R^N f(x)|u|^qdx−t^p−1 Z

R^Ng(x)|u|^pdx;

K_u⁰⁰(t) =kuk²−(q−₁)t^q−2 Z

R^N f(x)|u|^qdx−(p−₁)t^p−2 Z

R^Ng(x)|u|^pdx.

Lemma 2.3. Let u∈ E and t>0. Then tu∈ N if and only if K⁰_u(t) =0, that is, the critical points of K_u(t)correspond to the points on the Nehari manifold. In particular, u∈ N if and only if K⁰_u(₁) =_0.

Proof. The result is an immediate consequence of the fact:

K⁰_u(t) =I⁰(tu),u

= ¹ t

I⁰(tu),tu .

Thus, it is natural to split N into three parts corresponding to local minima, points of inflection and local maxima. Accordingly, we define

N⁺={u∈ N |K⁰⁰_u(₁)>₀}_, N⁰={u∈ N |K⁰⁰_u(₁) =₀} _and N⁻={u∈ N | K⁰⁰_u(₁)<₀}_.

It is easy to see that

K⁰⁰_u(₁) =kuk²−(q−1)

Z

R^N f(x)|u|^qdx−(p−1)

Z

R^Ng(x)|u|^pdx. (2.4) Define

Ψ(u) =K_u⁰(1) =I⁰(u),u

= kuk²−

Z

R^N f(x)|u|^qdx−

Z

R^Ng(x)|u|^pdx. (2.5) Then foru∈ N,

d

dtΨ(tu) t=1

=_Ψ⁰(u)_,u

=_Ψ⁰(u)_,u

−I⁰(u)_,u

=K_u⁰⁰(₁)

=kuk²−(q−1)

Z

R^N f(x)|u|^qdx−(p−1)

Z

R^Ng(x)|u|^pdx.

For eachu∈ N,Ψ(u) =K⁰_u(1) =0. Thus, for eachu∈ N, we have K_u⁰⁰(1) =K_u⁰⁰(1)−(q−1)_Ψ(u) = (2−q)kuk²−(p−q)

Z

R^N g(x)|u|^pdx (2.6) and

K_u⁰⁰(1) =K⁰⁰_u(1)−(p−1)_Ψ(u) = (2−p)kuk²+ (p−q)

Z

R^N f(x)|u|^qdx. (2.7) In order to ensure the Nehari manifold N to be a C¹-manifold, we need the following proposition.

Proposition 2.4. Letσ := (p−2)(2−q)^{(2−q)/(p−2)} _p^S₋^p_q^{(p−q)/(p−2)}, where S_p is the best Sobolev constant described in Lemma2.2. If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈(0,σ), then the setN⁰= _∅.

Proof. Suppose, on the contrary, there exists au ∈ N such thatK_u⁰⁰(1) =0. By Lemma2.2, Z

R^Ng(x)|u|^pdx≤ |g|_∞S⁻

p

p2kuk^p. (2.8)

Noting that 2< p <2^∗, from (2.6) we have

(2−q)kuk² ≤(p−q)|g|_∞S⁻

p

p 2kuk^p, so

kuk ≥





(2−q)S

p

p2

(p−q)|g|_∞





1 p−2

. (2.9)

Moreover, by the Hölder inequality and Lemma2.2, we have Z

R^N f(x)|u|^qdx≤ _Z

R^N|f(x)|^q∗dx _q¹_{∗ Z}

R^N|u|^pdx ^q_p

=|f|_q∗|u|^q_p ≤ |f|_q∗S⁻

q

p2kuk^q. (2.10) From (2.7) we have

(p−2)kuk²≤ (p−q)|f|_q∗S⁻

q

p2kuk^q, which implies that

kuk ≤





(p−q)|f|_q^∗ (p−2)S

q

p2





1 2−q

. (2.11)

This with (2.9) and (2.11) implies that

|f|_q^∗|g|

2−q p−2

∞ >





(2−q)S

p

p2

p−q





2−q p−2

p−₂ p−qS

q

p2 = (_p−2)(₂−q)^2p−−q2 S_p

p−q ^p_p⁻₋₂^q

=_σ,

which contradicts with the condition.

Proposition 2.5. Suppose that|f|_q^∗|g|⁽_∞^{2−q)/(p−2)}∈ (0,σ)and u ∈E. Then, there are unique t⁺and t⁻with0<t⁺ <t_max <t⁻such that t⁺u∈ N⁺, t⁻u∈ N⁻and

I(t⁺u) = inf

0≤t≤tmax

I(tu), I(t⁻u) = sup

t≥tmax

I(tu). Proof. Let

h(t) =t^2−qkuk²−t^p−q Z

R^Ng(x)|u|^pdx, then we have

K⁰_u(t) =t^q−1

h(t)−

Z

R^N f(x)|u|^qdx

. (2.12)

Clearly,h(₀) =_{0 andh}(_t)→ −_∞ast →_{∞. From 1}< _q<₂< _p<₂^∗_and h⁰(t) =t^p−q−1

(2−q)t^2−pkuk²−(p−q)

Z

R^Ng(x)|u|^pdx

=0,

we can infer that there is a unique t_max > 0 such that h(t) achieves its maximum at t_max, increasing fort ∈[0,tmax)and decreasing fort∈ (tmax,∞)with limt→_∞h(t) =−_{∞, where}

tmax= (2−q)kuk² (p−q)R

R^Ng(x)|u|^pdx

!_p−¹₂ . It follows

h(tmax) =kuk^q kuk^p R

R^Ng(x)|u|^pdx

!²_p⁻₋^q₂

2−q p−q

²_p⁻₋^q₂ p−2 p−q

≥ kuk^q





kuk^p

|g|_∞S⁻

p

p2kuk^p





2−q p−2

2−q p−q

²_p⁻₋^q₂ p−2 p−q

=kuk^q





(2−q)S

p

p2

|g|_∞(p−q)





2−q p−2

p−2 p−q >0.

(2.13)

From|f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈(0,σ), (2.10) and (2.13) we also have Z

R^N f(x)|u|^qdx< kuk^q





(2−q)S

p

p2

|g|_∞(p−q)





2−q p−2

p−2

p−q < h(t_max). (2.14) Moreover, fortu∈ N,K_u⁰(t) =0. By (2.12) we obtain that

K_u⁰⁰(t) =t^q−1h⁰(t)_.

By (2.12) and (2.14) we know there are unique t⁺ andt⁻ with 0 < t⁺ < t_max < t⁻ such that K⁰_t+u(1) = 0, K⁰_t−u(1) = 0, that is t⁺u,t⁻u ∈ N. From K⁰⁰_u(t) = t^q−1h⁰(t) and h⁰(t⁺) > 0 >

h⁰(t⁻), one arrives at the conclusion.

The forthcoming lemma is to obtain the minimizing sequence of the energy functional I on the Nehari manifoldN.

Lemma 2.6. The energy functional I is coercive and bounded from below onN. Proof. Foru∈ N, then, by the Hölder inequality and Lemma2.2,

I(u) = I(u)− ¹ p

I⁰(u),u

= 1

2 − ¹ p

kuk²− 1

q− ¹ p

Z

R^N f(x)|u|^qdx

≥ 1

2 − ¹ p

kuk²− 1

q− ¹ p

|f|_q^∗S⁻

q

p2kuk^q. This completes the proof.

Lemma 2.7. Under the assumptions(V)_,(f)and(g), the following results hold.

(i) If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)}∈ (0,σ), then c₁=inf_u∈N⁺I(u)<0;

(ii) If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈(0,σ^∗), then c₂ =inf_u∈N⁻ I(u)>0,whereσ^∗ =qσ/2andσdescribed in Proposition2.4.

Proof. (i)For each u∈ N⁺,K⁰⁰_u(1)>0. From (2.7), we have (p−q)

Z

R^N f(x)|u|^qdx>(p−2)kuk². If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)}∈ (0,σ), then

I(u) =I(u)− ¹ p

I⁰(u),u

= ^p−2

2p kuk²− ^p−q pq

Z

R^N f(x)|u|^qdx

< ^p−2

2p kuk²− ^p−2

pq kuk² = (p−2)(q−2)

2pq kuk²<0.

(2.15)

Thus, inf_u∈N⁺ I(u)<0.

(ii) For each u ∈ N⁻, K_u⁰⁰(1) < 0. From (2.9) and (2.10), we have if |f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈ (0,σ^∗), then

I(u) = I(u)− ¹ p

I⁰(u)_,u

= 1

2 − ¹ p

kuk²− 1

q− ¹ p

Z

R^N f(x)|u|^qdx

≥ 1

2− ¹ p

kuk²− 1

q− ¹ p

|f|_q∗S⁻

q

p2kuk^q

=kuk^q 1

2− ¹ p

kuk^2−q− 1

q− ¹ p

|f|_q∗S⁻

q

p2

≥





(2−q)S

p

p2

(p−q)|g|_∞





q p−2 



 1

2− ¹ p





(2−q)S

p

p2

(p−q)|g|_∞





2−q p−2

− 1

q− ¹ p

|f|_q^∗S⁻

q

p2





 >0.

Lemma 2.8. If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈(0,σ), then the setN⁻is closed in E.

Proof. Let{u_n} ⊂ N⁻ such thatu_n →uin E. In the following we prove u∈ N⁻. Indeed, for anyu∈ N⁻, from (2.6) we have

(2−q)kuk²<(p−q)

Z

R^Ng(x)|u|^pdx.

Similar to the proof of (2.9), we have

kuk ≥





(2−q)S

p

p2

(p−q)|g|_∞





1 p−2

. (2.16)

HenceN⁻ is bounded away from 0.

By hI⁰(un),uni = 0 and Lemma2.1, we have hI⁰(u),ui = 0. (2.6) implies that K_u⁰⁰_n(1) → K⁰⁰_u(1). FromK⁰⁰_un(1)<0, we haveK⁰⁰_u(1)≤0. By Proposition2.4we know, if|f|_q^∗|g|⁽_∞^{2−q)/(p−2)}∈ (0,σ), thenK⁰⁰_u(1)<0. Thus we deduceu ∈ N⁻.

The following lemma is used to extract a(PS)_{c1 (or}(PS)_c2) sequence from the minimizing sequence of the energy functionalI on the Nehari manifoldN⁺(orN⁻).

Lemma 2.9. If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈(0,σ), then for every u ∈ N⁺, there existe>0and a differen- tiable functionϕ⁺ :B_e(₀)→_R+:= (_0,+_∞)such that

ϕ⁺(₀) =_1, ϕ⁺(_w)(_u−w)∈ N⁺, ∀w∈ Be(₀) and

h(ϕ⁺)⁰(0),wi= L(u,w)/K⁰⁰_u(1), (2.17) where

L(u,w) =2hu,wi −q Z

R^N f(x)|u|^q−2uw dx−p Z

R^Ng(x)|u|^p−2uw dx.

Moreover, for any C₁,C2 >0, there exists C>0such that if C₁≤ kuk ≤C2,then

D

ϕ⁺0

(0),wE

≤Ckwk. Proof. We defineF:R×E→_Rby

F(t,w) =K⁰_u−w(t), it is easy to see thatFis differentiable. Since F(1, 0) =0 and

F_t(1, 0) =K_u⁰⁰(1)>0,

we apply the implicit function theorem at point (1, 0) to obtain the existence of e > 0 and differentiable function ϕ⁺:Be(0)→_R+:= (0,+_∞)such that

ϕ⁺(0) =1, F ϕ⁺(w),w

=0, ∀w∈Be(0). Thus,

ϕ⁺(w)(u−w)∈ N_, ∀w∈ B_e(₀)_.

Next, we prove for any w ∈ B_e(0), ϕ⁺(u−w) ∈ N⁺. Indeed, by u ∈ N⁺ and the set N⁻∪ N⁰ is closed, we know dist u,N⁻∪ N⁰ > 0. Since ϕ⁺(w)(u−w)is continuous with respect to w, we know wheneis small enough, forw∈ Be(0), then

ϕ⁺(w)(u−w)−u < ¹

2dist u,N⁻∪ N⁰, so

ϕ⁺(w)(u−w)− N⁻∪ N⁰≥dist u,N⁻∪ N⁰−dist ϕ⁺(w)(u−w),u

> ¹

2dist u,N⁻∪ N⁰>0.

Thus, forw∈ Be(0), then ϕ⁺(w)(u−w)∈ N⁺.

Besides, by the differentiability of implicit function theorem, we have (ϕ⁺)⁰(0),w

=−hFw(1, 0),wi F_t(1, 0) ^.

Note that L(u,w) =− hF_w(1, 0),wiandK⁰⁰_u(1) = F_t(1, 0). Therefore (2.17) holds.

In the following we prove that there existsδ >0 such thatK_u⁰⁰(₁)≥δ >_{0 with}C₁≤ kuk ≤ C₂, u ∈ N⁺, where C₁,C₂ > 0. On the contrary, if there exists a sequence {un} ∈ N⁺ with C₁ ≤ ku_nk ≤C₂, such that for anyδ_nsufficiently small,K⁰⁰_un(1)≤ δ_n, δ_n →0 asn →_{∞. From} (2.6) we have

(2−q)ku_nk²= (p−q)

Z

R^N g(x)|u_n|^p dx+O(δ_n),

where O(δ_n)→ 0 asδ_n →0. Noting that 1< q <2 < p <2^∗,C₁ ≤ kunk ≤ C2 and (2.8), we have

(2−q)ku_nk²≤(p−q)|g|_∞S⁻

p

p 2 ku_nk^p+O(δ_n), and so

kunk ≥ (2−q)S^p/2_p (p−q)|g|_∞

!1/(p−2)

+O(δ_n). (2.18)

From (2.7) we also have

(p−2)kunk² = (p−q)

Z

R^N f(x)|un|^q dx+O(δn). In view of (2.10), we have

(p−2)kunk² ≤(p−q)|f|_q∗S⁻

q

p2 kunk^q+O(δn), which implies that

ku_nk ≤ (p−q)|f|_q∗ (p−2)S^q/2_p

!1/(2−q)

+O(δ_n). (2.19)

Letn→∞, from (2.18) and (2.19) we deduce a contradiction.

Thus ifC₁≤ kuk ≤C₂, then there existsC >0 such that

(ϕ⁺0

(0),wE

| ≤Ckwk. This completes the proof.

Similarly, we establish the following lemma.

Lemma 2.10. If |f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈ (0,σ), then for every u ∈ N⁻, there exist e > 0 and a differentiable functionϕ⁻: B_e(₀)→_R+ := (_0,+_∞)

ϕ⁻(0) =1, ϕ⁻(w)(u−w)∈ N⁻, ∀w∈Be(0)

and D

ϕ⁻0

(0),wE

= L(u,w)/K_u⁰⁰(1), where

L(u,w) =2hu,wi −q Z

R^N f(x)|u|^q−2uw dx−p Z

R^Ng(x)|u|^p−2uw dx.

Moreover, for any C₁,C₂ >0, there exists C>₀such that if C₁≤ kuk ≤C₂,

|h(ϕ⁻)⁰(0),wi| ≤Ckwk.

From above, we can extract a (PS)_{c1 (or} (PS)_c2) sequence from the minimizing sequence of the energy functionalI on the Nehari manifoldN⁺(orN⁻).

Lemma 2.11. If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)}∈ (0,σ), then the minimizing sequence{un} ⊂ N⁺is the(PS)_c1 sequence in E.

Proof. By Lemma2.10and the Ekeland Variational Principle [10,23] onN⁺∪ N⁰, there exists a minimizing sequence{u_n} ⊂ N⁺∪ N⁰ such that

inf

u∈N⁺∪N⁰I(u)≤ I(u_n)< inf

u∈N⁺∪N⁰I(u) + ¹

n, (2.20)

I(un)− ¹

nkv−unk ≤ I(v), ∀v∈ N⁺∪ N⁰. (2.21) From Proposition 2.5, we know for each u ∈ E\{0}, there is a unique t⁺ such that t⁺u ∈ N⁺, then inf_u∈N⁺I ≤ I(t⁺u). By Lemma 2.7 and I(0) = 0, we get that inf_u∈N⁺_∪N0 I(u) = inf_u∈N⁺ I(u) = c₁. Thus we may assume u_n ∈ N⁺, I(u_n) → c₁ < 0. By Lemma 2.9, since

|f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈ (0,σ), we can find en > 0 and differentiable function ϕ⁺_n = ϕ⁺_n(w) > 0 such that

ϕ⁺_n(w) (u_n−w)∈ N⁺, ∀w∈B_en(0).

By the continuity of ϕ⁺_n(w) and ϕ⁺_n(0) = 1, without loss of generality, we can assume e_n is sufficiently small such that 1/2 ≤ ϕ⁺_n(w) ≤ 3/2 for kwk < e_n. From ϕ⁺_n(w) (un−w) ∈ N⁺ and (2.21) , we have

I ϕ⁺_n(w) (un−w)≥ I(un)− ¹ n

ϕ⁺_n(w) (un−w)−un

, which implies that

I⁰(u_n),ϕ⁺_n(w) (u_n−w)−u_n +o

ϕ⁺_n(w) (u_n−w)−u_n

≥ −¹ n

ϕ⁺_n(w) (u_n−w)−u_n . Consequently,

ϕ⁺_n(w)I⁰(u_n),w

+ ₁−ϕ⁺_n(w) I⁰(u_n),u_n

≤ ¹ n

ϕ⁺_n(w)−1

u_n−ϕ⁺_n(w)w +o

ϕ⁺_n(w) (u_n−w)−u_n

.

By the choice ofe_nand 1/2≤ ϕ⁺_n(w)≤3/2, we infer that there existsC₃>0 such that

I⁰(u_n),w ≤ ¹

n D

ϕ⁺_n0

(₀)_,wE u_n

+ ^C3

n kwk+o D

ϕ⁺_n0

(₀)_,wE

(ku_nk+kwk). Below we prove for {u_n} ⊂ N⁺, inf_nku_nk ≥ C₁ > 0, where C₁ is a constant. Indeed, if not, then I(u_n) would converge to zero, which contradicts I(u_n) → c₁ < 0. Moreover, by Lemma2.6we know thatI is coercive onN⁺,{un}is bounded inE. Thus, there existsC₂>0 such that 0<C₁≤ ku_nk ≤C₂. From Lemma 2.9,

(ϕ⁺_n)⁰(0),w

≤Ckwk. So

I⁰(u_n),w ≤ ^C

nkwk+ ^C

nkwk+o(kwk) and

I⁰(u_n)= sup

w∈E\{0}

|hI⁰(u_n),wi|

kwk ≤ ^C

n +o(1),

I⁰(un)→0, asn→_∞.

(2.22)

Thus, {u_n} ⊂ N⁺ is(PS)_c1 for I in E.

Lemma 2.12. If|f|_q^∗|g|⁽_∞^{2−q)/(p−2)} ∈(0,σ), then the minimizing sequence{u_n} ⊂ N⁻is the(PS)_c2 sequence in E.

Proof. From Lemma 2.8, N⁻ is closed in E. By Lemma 2.6, we know I is coercive on N⁻. So we use the Ekeland Variational Principle [23] on N⁻ to obtain a minimizing sequence {u_n} ⊂ N⁻ such that

u∈Ninf⁻I(u)≤ I(u_n)< inf

u∈N⁻I(u) + ¹ n, I(un)− ¹

nkv−unk ≤I(v), ∀v∈ N⁻.

In view of (2.15) and Lemma2.6, we know that there existC₁,C₂ >0 such that 0<C₁≤ kunk ≤C₂.

Hence by Lemma 2.10, in the same way as Lemma2.11, there exists a minimizing sequence {u_n} ⊂ N⁻ is the(PS)_c2 sequence inE.

The following lemmas aims at obtaining the critical points of I on the whole space from the critical points of I|_N⁺ and I|_N−, respectively.

Lemma 2.13. Suppose that u is a local minimizer for I onN⁺. Then I⁰(u) =_0.

Proof. If u 6= 0, u is a local minimizer for I on N⁺, then u is a nontrivial solution of the optimization problem

minimize I subject toΨ(u) =0, whereΨ(u)is described in (2.5). Then,u∈ N⁺ ⊂ N such that

I(u) =c₁ = inf

u∈N⁺I(u) = inf

u∈N I(u).

Note that Ψ⁰(u) 6= 0 and N⁺ is a local differential manifold. So by the theory of Lagrange multipliers, there existsµ∈_Rsuch thatI⁰(u) =µΨ⁰(u). Thus

I⁰(u),u

=µ

Ψ⁰(u),u .

Since u ∈ N⁺, we have hI⁰(u),ui = 0 and h_Ψ⁰(u),ui = K⁰⁰_u(1) 6= 0. Hence, µ = 0 and I⁰(u) =_0.

Lemma 2.14. Suppose that u is a nontrivial critical point of I|_N−, then it is a nontrivial critical point of I in E, i.e., I⁰(u) =0.

Proof. If u is a nontrivial critical point of I|_N−, i.e., u ∈ N⁻\{0}and (I|_N−)⁰(u) = 0. Note that N⁻ is a local differential manifold andΨ⁰(u) 6= 0, where Ψ(u)is described in (2.5). So by the theory of Lagrange multipliers, there existsµ∈_Rsuch that I⁰(u) =_µΨ⁰(u). Thus

I⁰(u),u

=µ

Ψ⁰(u),u .

Sinceu∈ N⁻, we havehI⁰(u),ui=0 andh_Ψ⁰(u),ui=K_u⁰⁰(1)6=0. Hence,µ=0 and I⁰(u) =0.

Thus the proof is complete.

3 Proof of Theorem 1.1

In order to obtain the nontrivial solutions, we bring in the following lemma.

Lemma 3.1(Lions [19,20,23]). Let r>0,q∈[2, 2^∗). If{un}is bounded in H¹(_R^N)and

nlim→_∞ sup

y∈_R^N Z

Br(y)

|u|^qdx=0, then we have un →0in L^p R^N

for p ∈ (2, 2^∗). Here2^∗ = 2N/(N−2)if N ≥ 3and2^∗ = _∞if N =1, 2.

Lemma 3.2. Let{un} ⊂E be a bounded(PS)_c sequence for I. Then either (i) un→0in E,or

(ii) there exist a sequence{y_n} ∈_R^N and constants r, δ>₀such that lim inf

n→_∞ Z

Br(yn)|u_n|² dx≥ δ>0.

Proof. Suppose the condition(ii)is not satisfied, i.e. for anyr >0, we have

nlim→_∞ sup

y∈_R^N Z

Br(y)

|u_n|² dx=0.

Then by Lemma3.1,u_n →0 in L^p(_R^N)for p∈(2, 2^∗). Therefore, 0≤

Z

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

Z

R^Ng(x)|u_n|^p dx

≤ |f|_q^∗|u_n|^q_p+|g|_∞|u_n|^p_p →0.

Since{un} ⊂Eis a bounded(PS)_c sequence for I, we have o(1) = I⁰(un)un=

Z

R^N

|∇un|²+V(x)|un|² dx− _Z

R^N f(x)|un|^qdx+

Z

R^Ng(x)|un|^p dx

, asn → ∞. It follows thatu_n →0 in Eas n→ ∞, i.e., the condition(i)is satisfied. Thus, the proof is complete.

To recover the compactness, we need to evaluate the critical value of Equation (1.1) through the critical value of a autonomous equation. Now, we consider the following autonomous equation

(−4u+V_∞u= g_∞|u|^p−2u inR^N,

u∈ H¹(_R^N), (3.1)

where 2 < p < 2^∗(2^∗ = _∞if N = 1, 2 and 2^∗ = 2N/(N−2) if N ≥ 3). The corresponding functional and the corresponding manifold are

I_∞(u) = ¹ 2

Z

R^N |∇u|²+V_∞|u|² dx− ¹ p

Z

R^Ng_∞|u|^pdx and

N_∞ =ⁿu∈ H¹ R^N

\ {0} I_∞⁰ (u),u

=0o .

Letw₀be the unique radially symmetric solution of Equation (3.1) such that I_∞(w₀) =c_∞, wherec_∞=inf_u∈N_∞I_∞(u)(see [3,17]).

In the following, we prove that when the critical value of Equation (1.1) is contained in the suitable range,(PS)_c condition holds.

Proposition 3.3. Let the assumptions of(V),(f)and(g)be satisfied, if|f|_q^∗|g|⁽_∞^{2−q)/(p−2)}∈ (0,σ^∗), then each (PS)_c sequence{un} ⊂ N (N = N⁺orN⁻)for I in E with c < c₁+c_∞ has a strongly convergent subsequence, where c₁is described in Lemma2.7.

Proof. Let{u_n} ⊂ N such that

I(un)→c and I⁰(un)→0 asn →_∞.

From Lemma 2.6 we know that the(PS)_c sequence {un} ⊂ N for I in E is bounded. Then, going if necessary to a subsequence, we have

un *u in E,

u_n →u in L^r_loc(_R^N), r ∈[2, 2^∗), u_n →u a.e. inR^N.

(3.2)

Set v_n := u_n−u, then there exists C > 0 such that kv_nk < C. It is sufficient to prove that vn→0 in Easn→_∞.

Note that

|u_n|^s− |u|^s ≤ |u_n−u|^s fors >_{1, (3.3)} we can infer that

Z

R^N f(x)|u_n|^q dx→

Z

R^N f(x)|u|^qdx and Z

R^N f(x)|v_n|^q dx→0 asn→_∞. (3.4) Indeed, from the condition (f), we have that for any e > 0, there exists R sufficiently large such that

_Z

|x|>R

|f(x)|^q∗dx 1/q^∗

<e.