Multiple solutions to elliptic equations on R N with combined nonlinearities

Multiple solutions to elliptic equations on R ^N with combined nonlinearities

Anran Li

and Chongqing Wei

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China Received 10 October 2014, appeared 7 October 2015

Communicated by Petru Jebelean

Abstract. In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equation

(−∆u+V(x)u=−λQ(x)|u|^q−2u+Q(x)f(u), x∈R^N,

u(x)→0, as|x| →+_∞, (P)_λ

where 1<q<2, λ∈R⁺, N≥3,V andQare radial positive functions, which can be vanishing or coercive at infinity, f is asymptotically linear or superlinear at infinity.

Keywords: weighted Sobolev embedding, sublinear, asymptotically linear, superlinear, critical point theory, variation methods.

2010 Mathematics Subject Classification: 35J10, 35J65, 58E05.

1 Introduction and main results

In this paper, we deal with the multiplicity of nontrivial radial solutions for the following elliptic equation

(−_∆u+V(x)u= −λQ(x)|u|^q−2u+Q(x)f(u), x∈_R^N,

u(x)→0, as|x| →+_∞, (P)λ

where 1 < q < 2, λ ∈ _R⁺, N ≥ 3, V and Q are radial positive functions, which can be vanishing or coercive at infinity.

WhenΩis a smooth bounded domain inR^N, the problem (−_∆u=±λ|u|^q−2u+ f(x,u)_, x ∈_Ω,

u(x) =0, x ∈_∂Ω, ^(P

0±)_λ where 1 < q < 2, λ > 0, N ≥ 3, has been widely studied in the literature. It plays a central role in modern mathematical sciences, in the theory of heat conduction in electrically

BCorresponding author. Email: anran0200.163.com

conductive materials, in the study of non-Newtonian fluids. However, it is not possible to give here a complete bibliography. Here we just list some representative results. In the case f is superlinear inunear infinity, problem (P⁰₊)_λ is the famous concave-convex problem, after the celebrated works [2,6], this kind of problem has been drawn much attention. In the case f is linear in u, in [12], at least two nonnegative solutions have been got for a more general question

(−_∆u= h(x)u^q+ f(x,u), x∈_Ω, 0≤u∈ H¹₀(_Ω)_{, 0}< _q<_1,

where the function h ∈ L^∞(_Ω) satisfies some additional conditions. For problem (P⁰₋)_λ, in the special case: f(u) = au+|u|^p, where 2 < p < _N^2N₋₂, one nonnegative solution for any a ∈_R andλ> 0 was found in [15] via the mountain pass theorem. Several papers have also been devoted to the study of nonlinearities with indefinite sign, for example [9,20] and the references therein.

WhenΩ=_R^N, there are a large number of papers devoted to the following equation,

−_∆u+V(x)u= f(x,u), withu∈W^1,2(_R^N). (Q) So far, in almost all the results concerning equation (Q), the nonlinear function f is assumed to be globally superlinear, that is to say, lim_|u|→0 ^f(x,u_u ⁾ = 0 and there exists θ > 2 such that 0< θF(x,u)≤ u f(x,u)for all(x,u)∈_R^N×(_R\{0}), where F(x,u) = Ru

0 f(x,t)dt. The case in whichV(x)→+_∞, |x| →_∞and fis globally superlinear was firstly studied by Rabinowitz in [16]. The assumptions in [16] ensure that the associated functional of the equation satisfies the Palais–Smale condition, this fact was observed in [4,5] where the results in [16] were generalized. For a radially symmetric Schrödinger equation with an asymptotically linear term, one radial solution has been obtained in [17,25] by Stuart and Zhou and their results were generalized to more general situations in [8,10,11,13].

Since the class Sobolev embedding isW^1,2(_R^N),→L^p(_R^N), p∈ (2,_N^2N₋₂), we cannot study the sublinear problems in W^1,2(_R^N) via variational methods directly. In order to overcome this obstacle, a regular way is to add some restrictions on potentialsV and Q. For example in [14], the authors obtained the existence of infinitely many nodal solutions for problem (Q), where V ∈ C(_R^N,R), V(x) ≥ 1, R

R^N 1

V(x)dx < +∞, the nonlinearity f is symmetric in the sense of being odd inu, and may involve a combination of concave and convex terms. There are also some other results about the concave and convex problem onR^N, such as [7,21,23,24]

and the references therein. However, to the best of our knowledge there is few result about problem (P)_λ with both sublinear terms and asymptotically linear terms.

Recently, in [18], the authors established a weighted Sobolev type embedding of radially symmetric functions which provides a basic tool to study quasilinear elliptic equations with sublinear nonlinearities. Motivated by the works of [18], we consider (P)_λ with more general potentials and nonlinearities. In this paper, we assume that

(V) V ∈C R^N,(_0,+_∞)is radially symmetric and there exists a₁∈ _{Rsuch that} lim inf

|x|→+_∞

V(x)

|x|^a1 >0;

(Q) Q∈ C R^N,(0,+_∞) is radially symmetric and there existsa₂∈_Rsuch that lim sup

|x|→+_∞

Q(x)

|x|^a2 < _∞.

It is clear that the indexes a₁ anda₂describe the behaviors ofV andQnear infinity. Ona₁,a₂, we assume that

(A₁) a₂≥ ²(N−1) +a₁

2 −N, N−2

2 −N≤a₂≤ −2;

(A2) a2< ²(N−₁) +a₁

4 −N, N−₂

2 −N≤a2≤ −2 ; (A₃) a₁≤ −2, N−2

2 −N< a₂ < ²(N−1) +a₁

2 −N;

(A₄) a2≤ ^N−₂

2 −N, 2(N−₁) +a₁

4 −N≤a2< ²(N−₁) +a₁

2 −N;

(A₅) a₁≥ −2, 2(N−1) +a₁

4 −N≤ a₂ < ²(N−1) +a₁

2 −N.

According to the indexes a₁,a₂, we define the bottom index 2∗,

2∗ =











2(a₂+N)

N−₂ ^{, if}(a₁,a₂)∈ A_i, i=1, 2, 3;

4(a2+N)

2(N−1) +a₁, if(a₁,a2)∈ A_i, i=4, 5.

LetC₀^∞(_R^N)denote the collection of smooth functions with compact support and C_0,r^∞(_R^N)_:=u∈C₀^∞(_R^N)|uis radial .

Denote byD^1,2_r (_R^N)the completion ofC_0,r^∞(_R^N)under the norm kuk_D1,2 =

_Z

R^N|∇u(x)|²dx ¹₂

. Define

W_r^1,2(_R^N;V):=

u∈ D^1,2_r (_R^N)

Z

R^NV(x)|u(x)|²dx<_∞

, which is a Hilbert space [1,19] equipped with the norm

kuk= _Z

R^N|∇u(x)|²+V(x)|u(x)|²dx ¹₂

. Let

L^p(_R^N;Q):=

u:R^N 7→ _R uis Lebesgue measurabe, Z

R^NQ(x)|u(x)|^pdx<_∞

, which is a Banach space equipped with the norm

kuk_Lp(_R^N;Q)= _Z

R^NQ(x)|u(x)|^pdx ¹_p

.

Following [18, Theorem 1.2], under the assumptions (V)_, (Q) _and (A_i)_, i = 1, . . . , 5, it holds that the embeddingW_r^1,2(_R^N;V),→ L^p(_R^N;Q)is compact forp∈ (2∗,_N^2N₋₂). We remark that the index 2∗ < 2 by (A_i), i = 1, . . . , 5, so it is possible to study (P)_λ with sublinear nonlinearities. We make the following assumptions on f:

(f₁) f ∈C(_R,R);

(f₂) there exists a positive constantC, such that|f(u)| ≤C(1+|u|^p−1), 2< p<2^∗:= _N^2N₋₂; (f₃) there exist one small positive constant r₀ and another positive constant C⁰, such that

|f(u)| ≤C⁰|u|, for|u| ≤r₀; (f₄) lim

|u|→_∞

2F(u)

|u|² =b.

Since under the assumptions (V), (Q)and(A_i), i= 1, . . . , 5, it holds that the embedding W_r^1,2(_R^N;V),→ L²(_R^N;Q)is compact, the eigenvalue problem

(−_∆u+V(x)u=µQ(x)u, x ∈_R^N,

u(x)→0, as|x| →+_∞, (P)_µ

has the eigenvalue sequence

0<µ₁<µ₂ ≤µ₃≤ · · · →+_∞.

Similar to the eigenvalue problem in bounded domain,µ₁ > 0 is simple, isolated and has an associated eigenfunctionφ₁which is positive inR^N.

Our main results are the following.

Theorem 1.1. Under the assumptions (V), (Q)and(A_i), i= 1, . . . , 5, if f satisfies(f₁), (f₃)and (f₄)withµ₁< b,(P)_λ has at least two nontrivial solutions.

Theorem 1.2. Let us assume that conditions(V), (Q)and(A_i), i=1, . . . , 5hold, and that f satisfies (f₁)and(f₄)withµ_k+₁ <b<+_∞for some k ∈_N, moreover,

(f3)⁰ F(u)≥ ^µm

2 u², u∈_R,lim sup

u→0

2F(u)

u² <µ_m+₁, for some m∈_N, m≤k;

(f5) lim

|u|→_∞H(u) = +_{∞, where H}(u) = ¹

2f(u)u−F(u)≥0, u∈_{R, F}(u) =

Z _u

0 f(s)ds.

Then, there existsλ^∗ >0, such that forλ∈ (0,λ^∗),(P)_λhas at least three nontrivial solutions.

In the caseb= +∞, we establish the following version of Theorem1.2.

Theorem 1.3. Under the assumptions(V), (Q)and(A_i), i=1, . . . , 5, if f satisfies(f₁),(f2)and (f₄)⁰ lim

|u|→+_∞

F(u)

|u|² = +_∞;

(f₆) there existsθ ≥ 1, such that θH(u)≥ H(su), (s,u)∈ [0, 1]×R. Moreover, F(u) ≥ ^µm 2 u², u∈_R,lim sup

u→0

F(u) u² < ¹

2µ_m+1, for some m∈_N.

Then, there existsλ^∗∗ >0, such that forλ∈(0,λ^∗∗),(P)_λhas at least three nontrivial solutions.

Remark 1.4. In Theorem1.1, f may be superlinear or asymptotically linear near zero, we can get two nontrivial solutions by the mountain pass theorem and the truncation technique. In Theorem1.2 and1.3, in order to guarantee the functional associated to problem (P)_λ enjoys linking structure, f has to satisfy some stricter growth condition near zero.

Remark 1.5. In Theorems1.2and1.3, we can get three nontrivial solutions: two mountain pass solutions and one linking solution. We can distinguish them by choosing special “paths”.

Remark 1.6. As we have known, there are few results about problems on R^N with both sublinear and asymptotically linear nonlinearities at the same time.

The paper is organized as follows. In Section 2, we give some preliminary results. The proof of our main results will be given in Section 3.

2 Preliminary

In this section we give some preliminaries that will be used to prove the main results of this paper. We begin with a special case of results on Sobolev embedding which is due to [18].

Lemma 2.1 ([18]). Let (V), (Q), (A_i), i = 1, . . . , 5be satisfied, the Sobolev space W_r^1,2(_R^N;V) is compactly embedded in L^p(_R^N;Q), for any p such that2∗ < p< _N^2N₋₂.

Foru∈W_r^1,2(_R^N;V), we denote I_λ(u) = ¹

2kuk²+^λ q

Z

R^NQ(x)|u(x)|^qdx−

Z

R^NQ(x)F(u(x))dx, (2.1) I_λ^±(u) = ¹

2kuk²+^λ q

Z

R^NQ(x)|u^±(x)|^qdx−

Z

R^NQ(x)F(u^±(x))dx, (2.2) where u⁺ = _max{u, 0}_, u⁻ = _min{u, 0}, then under the conditions (f₁)_–(f₃), by (2.1) and (2.2), I_λ andI_λ^±∈C¹(W_r^1,2(_R^N;V), R).

Recall that a sequence{u_n}is a (PS)_c sequence for a functional I, if I(u_n)→c, I⁰(u_n)→0, as n→_∞.

a sequence{u_n}is a (C)_c sequence for a functional I, if

I(u_n)→c, (1+ku_nk)I⁰(u_n)→0, asn→_∞.

Definition 2.2. AssumeXis a Banach space,I ∈C¹(X,R), we say thatI satisfies the (PS)_ccon- dition, if every (PS)csequence{un}has a convergent subsequence. I satisfies (PS) condition if I satisfies (PS)_cat anyc∈_R.

Definition 2.3. Assume X is a Banach space, I ∈ C¹(X,R), we say that I satisfies the (C)c

condition, if every (C)_c sequence{u_n}has a convergent subsequence. I satisfies (C) condition if I satisfies (C)_cat anyc∈_R.

Lemma 2.4 (Mountain pass theorem, Ambrosetti–Rabinowitz, 1973, [22]). Let X be a Banach space, I ∈C¹(X,R), e∈ X and r>0be such thatkek>r and

b:= inf

kuk=rI(u)> I(0)≥ I(e). If I satisfies the (PS)_c condition with

c:= inf

γ∈_Γmax

t∈[0,1]I(γ(t)),

Γ:={γ∈ C([0, 1],X)|γ(0) =0, γ(1) =e}. Then c is a critical value of I.

Lemma 2.5(Linking theorem, Rabinowitz, 1978, [22]). Let X = Y⊕Z be a Banach space with dimY <_{∞. Let R}>r>0and z∈ Z be such thatkzk=r. Define

M :={u= y+tz| kuk ≤R, t ≥0, y∈Y},

M₀ :={u= y+tz|y∈Y, kuk= R and t≥0orkuk ≤R and t=0}, N _:={u∈ Z| kuk=r}_.

Let I∈C¹(X,R)be such that

d:=_inf

N I >a:=_max

M₀ I.

If I satisfies the (PS)_ccondition with

c:= _inf

γ∈_Γmax

u∈MI(γ(u))_, Γ:={γ∈C(M,X)|γ|_M0 =id}. Then c is a critical value of I.

It is well known that the above two minimax theorems are still valid under (C)c condition.

In our paper, we denoteX:=W_r^1,2(_R^N;V),C denotes various positive constants.

Lemma 2.6. Under the assumptions (V), (Q)and(A_i), i = 1, . . . , 5, if f satisfies (f₁)–(f₃), then for givenλ>0, there existρ₁, β₁ >0, such that

u∈X,infkuk=ρ₁

I_λ⁺(u)≥β₁>0. (2.3)

Proof. By(f₁)–(f₃), there existsc>0, such that|F(u)| ≤c(|u|²+|u|^p). Then, I_λ⁺(u) = ¹

2kuk²+ ^λ q

Z

R^NQ(x)|u⁺(x)|^qdx−

Z

R^NQ(x)F(u⁺(x))dx

≥ ¹

2kuk²−c Z

R^NQ(x)|u⁺(x)|²dx−c Z

R^NQ(x)|u(x)|^pdx+^λ q

Z

R^NQ(x)|u⁺(x)|^qdx

≥ ¹

2kuk²−c Z

R^NQ(x)|u⁺(x)|²dx−c⁰kuk^p+ ^λ q Z

R^NQ(x)|u⁺(x)|^qdx.

Hence, forkuksmall enough I_λ⁺(u)≥ ¹

3kuk²−c Z

R^NQ(x)|u⁺(x)|²dx+ ^λ q Z

R^NQ(x)|u⁺(x)|^qdx. (2.4) Then, we can choose i ∈ _N, such that c ∈ ^µ₄ⁱ,^µi₄⁺¹

. Let X_j := span{φ_j}, j ∈ _N, and G_i := X₁⊕X2⊕ · · · ⊕X_i, where φ_j is the eigenfunction associated to the eigenvalue µ_j and

⊕ means the orthogonal sum of the subspace. We note that X := G_i⊕G_i^⊥, thus, u⁺ can be decomposed asu⁺=v+w, wherev∈ G_i, w∈ G_i^⊥. Observe that forv∈ G_i, there holds

kvk²≥µ₁ Z

R^NQ(x)v²dx, and forw∈G^⊥_i , there holds

kwk²≥µ_i+1 Z

R^NQ(x)w²dx.

Therefore, (2.4) implies that I_λ⁺(u)≥ ¹

12kuk²+¹ 4

1− ^4c µ_i+1

kwk²− ¹ 4

4c µ₁ −1

kvk²+ ^λ q Z

R^NQ(x)|u⁺(x)|^qdx

=: 1

12kuk²+ξkwk²−ηkvk²+ ^λ q Z

R^NQ(x)|u⁺(x)|^qdx.

(2.5)

It suffices to show that there existsρ₁ >0 small enough, such that for 0< kuk ≤ρ₁, I_λ1⁺(u)_:=ξkwk²−ηkvk²+ ^λ

q Z

R^NQ(x)|u⁺(x)|^qdx≥0. (2.6) Seeking a contradiction, we suppose that there exist u_n 6= 0 satisfying ku_nk → 0 asn → _∞, and I_λ1⁺(un)≤0. Decomposeu⁺_n =vn+wn, wherevn∈G_i, wn ∈G^⊥_i . We have

I_λ1⁺(u_n) =ξkw_nk²−ηkv_nk²+ ^λ q

Z

R^NQ(x)|u⁺_n(x)|^qdx<0. (2.7) Thenu⁺_n 6=0, in X. Letz_n= ^u+n

ku⁺_nk, up to a subsequence, we get that z_n*z, asn→_{∞, in}X,

z_n→z, asn→_{∞, in}L^s(_R^N;Q), 2∗ <s< ^2N N−₂^, zn(x)→z(x), asn→_∞, a.e. x∈_R^N.

Dividing both sides of (2.7) byku⁺_nk^q, ξkw_nk²−ηkv_nk²

ku⁺_nk² ku⁺_nk^2−q+^λ q

Z

R^NQ(x)|z_n(x)|^qdx≤0.

Let n → ∞, there holds R

R^NQ(x)|z(x)|^qdx ≤ 0, in view of ku⁺_nk → 0, as n → _{∞. Thus,} z(x) =0, a.e. x∈_R^N. Then, we have

kvnk²

ku⁺_nk² ≤ ^Ckv_nk²

L²(_R^N;Q)

ku⁺_nk² ≤Ckznk²_L2(_R^N;Q)→0, asn→_∞. (2.8) Using the equivalence of all norms on the finite dimensional space, choosingnsufficient large, we obtain that

I_λ1⁺(un) =ξkwnk²−ηkvnk²+^λ q

Z

R^NQ(x)|u⁺_n(x)|^qdx

≥ξku⁺_nk²−(ξ+η)kv_nk²

=

ξ−(ξ+η)kv_nk² ku⁺_nk²

ku⁺_nk²,

(2.9)

in view of (2.8), we get a contradiction from (2.9). Therefore, from (2.5) and (2.6) we can choose ρ₁ >0 small enough such that forkuk=ρ₁,

I_λ⁺(u)≥ ¹

12kuk²+I_λ1⁺(u)≥ ¹

12ρ²₁ =:β₁ >0.

Therefore, (2.3) is true.

Using a similar argument as in Lemma2.6, we have the following result.

Lemma 2.7. Under the assumptions (V), (Q)and(A_i), i = 1, . . . , 5, if f satisfies (f₁)–(f₃), then for givenλ>0, there existρ2, β2 >0, such that

u∈X,infkuk=ρ2

I_λ⁻(u)≥ β₂ >0.

Remark 2.8. In fact, for any λ > 0, 0 is a local minimizer of I_λ^±. In the bounded domain, it is easy to obtain this result by the fact that the local H₀¹(_Ω)-minimizer is also the local C₀¹(_Ω)-minimizer (see [3]).

3 Proof of main results

Proof of Theorem1.1. It is easy to see that I_λ⁺(₀) =0. We note that(_f1)_and(_f4)_withµ₁ <_b<

+_∞imply(f₂). Then, from Lemma2.6, givenλ>0, there existρ₁>0, β₁ >0, such that

u∈X,infkuk=ρ₁

I_λ⁺(u)≥β₁>0.

On the other hand,(f₁)_and(f₄)_{imply that}

|u|→+lim_∞ 2F(u)

|u|² =b> µ₁. (3.1)

Thus, choosingu=φ₁, I_λ⁺(tφ₁) = ¹

2t²kφ₁k²+ ^λt

q

q Z

R^NQ(x)|φ₁|^qdx−

Z

R^NQ(x)F(tφ₁)dx

=t² 1

2kφ₁k²+ ^λt

q−2

q Z

R^NQ(x)|φ₁|^qdx−

Z

R^NQ(x)^F(tφ₁) t²φ₁² ϕ²₁dx

.

By (3.1) and 1 < q< 2, we have I_λ⁺(tφ₁)→ −_{∞, as} t → +∞. Thus, for large enought₁, we havekt₁φ₁k>ρ₁, and I_λ⁺(t₁φ₁)<0. Define

c⁺= inf

γ∈_Γ⁺ max

0≤t≤1I_λ⁺(γ(t)) whereΓ⁺ :={γ∈C([0, 1], X)|γ(0) =0, γ(1) =t₁φ₁}.

Now, in order to apply Lemma 2.4 to prove Theorem 1.1, it is sufficient to verify that I_λ⁺ satisfies the (C)_c⁺ condition.

Lemma 3.1. Under the assumptions (V), (Q)and(A_i), i = 1, . . . , 5, if f satisfies (f₁), (f₃)and (f₄)withµ₁< b, then for any fixedλ>0, the functional I_λ⁺ satisfies the (C)_c⁺ condition.

Proof. For every (C)_c⁺ sequence {un}

I_λ⁺(un)→c⁺, asn→_∞, (3.2)

(₁+ku_nk)I_λ⁺⁰(u_n)→_{0, as}n→_{∞, (3.3)}

we claim that the sequence {u_n} is bounded in X. Seeking a contradiction, we suppose that kunk →_{∞. Let}zn= _k^u_uⁿ

nk, up to a subsequence, we get that z_n*z, asn→_{∞, in}X,

z_n→z, asn→_{∞, in}L^s(_R^N;Q), 2∗ <s< ^2N N−₂^, zn(x)→z(x), asn→_∞, a.e. x∈_R^N.

We claim thatz 6=0. otherwise,z=0, since by (3.3) o(1) =hI_λ⁺⁰(u_n), u_ni= ku_nk²+λ

Z

R^NQ(x)u⁺_n(x)^qdx−

Z

R^NQ(x)f(u⁺_n(x))u⁺_n(x)dx. (3.4) Dividing both sides of (3.4) byku⁺_nk²,

o(₁) =₁−

Z

R^NQ(x)^f(u⁺_n(x))u⁺_n(x)

kunk^{2 dx. (3.5)}

Assumptions (f₁),(f₃)and(f₄)withµ₁< b<+_∞imply that there existsC>0, such that

|f(u⁺_n(x))| ≤Cu⁺_n(x). (3.6) Combining (3.5) and (3.6), we have

1=

Z

R^NQ(x)^f(u⁺_n(x))u⁺_n(x)

ku_nk^{2 dx}+o(1)

≤C Z

R^NQ(x)|z⁺_n(x)|²dx+o(1). Lettingn→_∞, we get a contradiction. Thus,z 6=0 inX.

Set

P_n(x) =







f(u_n(x))

un(x) ^{, forx}∈_R^N, u_n(x)>0, 0, forx∈_R^N, u_n(x)≤0.

From I_λ⁺⁰(un) =o(1), we can get that Z

R^N∇u_n∇φdx+

Z

R^NV(x)u_nφdx+λ Z

R^NQ(x)(u⁺_n)^q−1φdx−

Z

R^NQ(x)f(u⁺_n)φdx =o(1), for all φ∈C_0,r^∞(_R^N). Dividingku_nkin both sides of the above equality, there holds

Z

R^N∇z_n∇φdx+

Z

R^NV(x)z_nφdx−

Z

R^NQ(x)P_n(x)z⁺_nφdx= o(1). (3.7) By (3.6),|Pn(x)| ≤Cfor x∈_R^N. Then we have

Z

{x∈_R^N|z⁺(x)=0}Q(x)Pn(x)z⁺_n(x)φ(x)dx

≤C Z

{x∈_R^N|z⁺(x)=0}Q(x)z⁺_n(x)|φ(x)|dx

= o(1) +C Z

{x∈_R^N|z⁺(x)=0}Q(x)z⁺(x)|φ(x)|dx=o(1).

(3.8)

On the other hand, sincez⁺_n(x) → z⁺(x) for a.e. x ∈ _R^N, we have lim_n→_∞u⁺_n(x) = +_∞ for a.e. x ∈ {x ∈ _R^N | z⁺(x) > 0}, which implies that limn→_∞Pn(x) =b, for a.e. x ∈ {x ∈ _R^N | z⁺(x)> 0}. Besides,|P_n(x)| ≤ C, fora.e.x ∈ _R^N. Using Lebesgue’s dominated convergence theorem, we obtain that

Z

{x∈_R^N| z⁺(x)>0}Q(x)(P_n(x)−b)z⁺_n(x)φ(x)dx

≤

Z

{x∈_R^N|z⁺(x)>0}Q(x)|P_n(x)−b|z⁺_n(x)|φ(x)|dx

≤C _Z

{x∈_R^N|z⁺(x)>₀}Q(x)|P_n(x)−b|²|φ(x)|dx ¹₂

=o(1).

(3.9)

By (3.8) and (3.9), Z

R^NQ(x)P_n(x)z⁺_n(x)φ(x)dx

=

Z

{x∈_R^N|z⁺(x)=0}Q(x)Pn(x)z⁺_n(x)φ(x)dx +

Z

{x∈_R^N|z⁺(x)>0}Q(x)P_n(x)z⁺_n(x)φ(x)dx

= o(1) +

Z

{x∈_R^N|z⁺(x)>0}Q(x)Pn(x)z⁺_n(x)φ(x)dx

= o(1) +b Z

R^NQ(x)z⁺(x)φ(x)dx.

(3.10)

Combining (3.7) and (3.10), lettingn→∞, there holds Z

R^N(∇z∇φ+V(x)zφ)dx =b Z

R^NQ(x)z⁺φdx. (3.11) We claim that meas{x ∈ _R^N |z⁺(x) 6= 0} >0. Otherwise z⁺ =0, taking φ= z in (3.11), we havez= 0, which is impossible. Taking φ= z⁻ in (3.11), we can getz ≥ 0. Moreover by the Hopf lemma, we also can getz>0 inR^N. Taking φ=φ₁in (3.11), we obtain

Z

R^N(∇z∇φ₁+V(x)zφ₁)dx= b Z

R^NQ(x)z⁺φ₁dx.

Sinceφ₁>0 is the eigenfunction associated toµ₁, andz≥0, we have Z

R^N(∇z∇φ₁+V(x)zφ₁)dx= µ₁ Z

R^NQ(x)zφ₁dx.

This is impossible, since b > µ₁. Then {u_n}is bounded in X. Since the embedding from X intoL^s(_R^N;Q), s∈ (2∗, _N^2N₋₂)is compact, there exists u ∈ X, such that un →u strongly inX.

Thus from (3.2) and (3.3), we can get that

I_λ⁺(u) =c⁺ ≥β₁, I_λ⁺⁰(u) =0.

Finally, we are now ready to conclude the proof of Theorem1.1. SinceI_λ⁺⁰(u)u⁻=0, then Z

R^N(∇u∇u⁻+V(x)uu⁻)dx=−λ Z

R^NQ(x)(u⁺)^q−1u⁻dx+

Z

R^NQ(x)f(u⁺)u⁻dx=0.

We haveu⁻=0,i.e.u≥0. Thus,uis a nonnegative solution for problem (P)_λ. Similarly, for I_λ⁻(u) = ¹

2kuk²+^λ q

Z

R^NQ(x)|u⁻|^qdx−

Z

R^NQ(x)F(u⁻)dx,

we can also get a nonpositive solution for problem (P)_λ. Thus, problem (P)_λ has at least two nontrivial solutions. The proof of Theorem1.1is complete.

Proof of Theorem1.2. In order to prove Theorem 1.2, we firstly verify that the functional I_λ enjoys the linking structure.

Lemma 3.2. Under the assumptions(V), (Q)and(A_i), i= 1, . . . , 5, if f satisfies the assumptions of Theorem1.2, then there exist positive numbers r, d, R andη=η(λ), such that

(i) for all u∈ X_m :=^Lm_j=1ker(−_∆+V−µ_jQ), we have I_λ(u)≤η(λ), lim

λ→0⁺η(λ) =0;

(ii) for all u∈ N _:= {u ∈X_m^⊥| kuk=r}, we have

I_λ(u)≥ d>0, for allλ>0;

(iii) for all u∈ X_m+1, andkuk ≥R, we have I_λ(u)≤0.

Proof. (i) Let u ∈ X_m, since F(u) > ¹₂µ_mu², u ∈ R, there exists ε₁ > 0, such that F(u) ≥

1

2(µ_m+ε)u², then

I_λ(u) = ¹

2kuk²+^λ q

Z

R^NQ(x)|u|^qdx−

Z

R^NQ(x)F(u)dx

≤ 1

2− ^µm+ε₁ 2µm

kuk²+ ^Cλ q kuk^q.

(3.12)

Let g(t) =¹₂− ^µm_2µ^+ε1

m

t²+^Cλ_q t^q, we have

maxt>0 g(t) =g(t₀), wheret₀= µm

ε₁ Cλ ₂₋¹_q

, and

g(t₀) =− ^ε1 2µ_m

µm

ε₁ C ₂₋²_q

λ

2

2−q + ^Cλ q

µm

ε₁ Cλ ₂₋^q_q

= − ^ε1 2µm

µ_m ε₁

₂₋²_q + ¹

q µ_m

ε₁

₂₋^q_q!

(Cλ)^2−2q. Then from (3.12), we can get that

I_λ(u)≤ − ^ε1 2µm

µ_m ε₁

₂₋²_q +¹

q µ_m

ε₁

₂₋^q_q!

(Cλ)^2−2q =:η(λ)→0, asλ→0.

(ii) Let u ∈ X_m^⊥, by (f₁)and(f₄) with µ_k+1 < b< +_∞, and lim sup_u→0 ^F_u^(u₂⁾ < ¹₂µ_m+1, we have that there existsε₂>0,C>0,p>2, such thatF(u)≤ ¹₂(µ_m+1−ε₂)u²+C|u|^p. Then

I_λ(u) = ¹

2kuk²+ ^λ q Z

R^NQ(x)|u|^qdx−

Z

R^NQ(x)F(u)dx

≥ ¹

2kuk²−¹

2(µ_m+1−ε₂)

Z

R^NQ(x)|u|²dx−C Z

R^NQ(x)|u|^pdx

≥ ¹ 2

1− ^µm+1−ε2

µ_m+1

kuk²−Ckuk^p,

(3.13)

thus, choosingr>0 small enough, (3.13) implies that inf

u∈X_m^⊥,kuk=rI_λ(u)≥d>0, independent of λ>0.

(iii) For any u ∈ X_m+1, set f(u) = bu+g(u), by (f₄), we have ^G_u^(u2⁾ → 0, as |u| → _∞, whereG(u) =Ru

0 g(s)ds. Then I_λ(u) = ¹

2kuk²+^λ q

Z

R^NQ(x)|u|^qdx− ^b 2 Z

R^NQ(x)|u|²dx−

Z

R^NQ(x)G(u)dx.

Sinceb>µ_m+1, for every z∈_span{φ_m+1}_, t∈ _R,w∈X_m, t²kzk²+kwk²−b

Z

R^NQ(tz+w)²dx<0. (3.14) Arguing by contradiction, we find a sequence {un}, satisfying kunk → _∞, un = tnz₀+wn, wherez₀ ∈span{φ_m+1}, t_n∈_R,w_n∈X_m, such that

I_λ(un) = ¹

2t²_nkz₀k²+ ¹

2kwnk²+ ^λ q

Z

R^NQ(x)|un|^qdx−

Z

R^NQ(x)F(un)dx≥0. (3.15) Dividing both sides of (3.15) byku_nk², there holds

I_λ(un) ku_nk² = ¹

2τ_n²kz0k²+ ¹

2kvnk²+ ^λ qku_nk²

Z

R^NQ(x)|un|^qdx−

Z

R^NQ(x)^F(un)

ku_nk^{2 dx}≥0, (3.16) where τ_n := _k^tn

unk, v_n := _k^wn

unk. Since τ_n²kz₀k²+kv_nk² = 1, after passing to a subsequence τ_n → τ, in R, v_n → v in X_m. Let u⁰ = τz₀+v, by (3.14), there exists a bounded domain Ω⊂_R^N, such that

τ²kz₀k²+kvk²−b Z

ΩQ(x)(τz₀+v)²dx <0. (3.17) AsF(u) = ¹₂bu²+G(u), it follows from (3.16) that

0≤ ¹

2τ_n²kz₀k²+¹

2kv_nk²−

Z

ΩQ(x)^F(un)

ku_nk^2dx+ ^λ qku_nk²

Z

R^NQ(x)|u_n|^qdx

= ¹

2τ_n²kz₀k²+¹

2kv_nk²−¹ 2b

Z

ΩQ(x)(τ_nz₀+v_n)²dx

−

Z

ΩQ(x)^G(un)

ku_nk^{2 dx}+ ^λ qku_nk²

Z

R^NQ(x)|u_n|^qdx.

Clearly, |G(u)| ≤ c₀u², for some c₀ > 0 and ^G_u^(u₂⁾ → 0, as |u| → _∞. Since τ_n → τ, in R, vn→v, inXm, then τnz0+vn→u⁰ =τz0+v, in L²(_R^N;Q). It is easy to see from Lebesgue’s dominated converge theorem that

Z

ΩQ(x)^G(u_n) ku_nk^{2 dx}=

Z

ΩQ(x)^G(u_n)

u²_n (τ_n²+v²_n)dx→0.

Hence, together with (3.17) 0≤ ¹

2τ²kz₀k²+¹

2kvk²−¹ 2b

Z

ΩQ(x)|tz₀+v|²dx<0, this is impossible.

Therefore, forY= X_m,Z= X^⊥_m, z∈span{φ_m+1}withkzk= r, M := {u =y+tz| kuk ≤R, t ≥0, y∈Y},

M₀ :={u=y+tz|y∈Y, kuk= Randt ≥0 orkuk ≤Randt=₀}, N := {u ∈Z| kuk=r}.

Lemma3.2implies that there existsλ^∗>0, such that for 0 <λ<λ^∗,

uinf∈N I_λ(u)> sup

u∈M₀

I_λ(u). Define

c_λ :=inf

γ∈_Γmax

u∈MI_λ(γ(u)), Γ:={γ∈ C(M,X) | γ|_M0 =id}. Next, we prove that the functional I_λ satisfies the (C)c_λ condition.

Lemma 3.3. Under the assumptions(V), (Q)and(A_i), i= 1, . . . , 5, if f satisfies the assumptions of Theorem1.2, then for any givenλ>0, the functional I_λsatisfies the (C)_cλ condition.

Proof. For every (C)c_λ sequence{un},

I_λ(u_n)→c_λ, asn→+_{∞. (3.18)}

(1+ku_nk)I_λ⁰(u_n)→0, asn→+_∞. (3.19) We just prove that{un}is bounded. Seeking a contradiction we suppose that kunk →_{∞. Let} w_n= _k^u_uⁿ

nk, up to a subsequence, we get that w_n*w inX,

wn→w inL^s(_R^N;Q), 2∗ <s< ^2N N−2, w_n(x)→w(x) a.e.x∈_R^N.

Now, we consider the two possible cases.

Case 1.w=0 in X. Fromo(1) =hI_λ⁰(u_n),u_ni, we have o(1) =kunk²+λ

Z

R^NQ(x)|un|^qdx−

Z

R^NQ(x)f(un)undx.