AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 256324 10.1155/2013/256324 256324 Research Article Existence of Nontrivial Solutions and High Energy Solutions for a Class of Quasilinear Schrödinger Equations via the Dual-Perturbation Method Chen Yu 1 Wu Xian 2 Mihǎilescu Mihai 1 Department of Mathematics Honghe University Mengzi Yunnan 661100 China uoh.edu.cn 2 Department of Mathematics Yunnan Normal University Kunming Yunnan 650092 China ynnu.edu.cn 2013 29 10 2013 2013 23 06 2013 12 09 2013 2013 Copyright © 2013 Yu Chen and Xian Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the quasilinear Schrödinger equation of the form Δu+VxuΔu2u=hx,u, xRN. Under appropriate assumptions on Vx and hx,u, existence results of nontrivial solutions and high energy solutions are obtained by the dual-perturbation method.

1. Introduction and Preliminaries

In this paper we consider the quasilinear Schrödinger equation of the form (1)-Δu+V(x)u-Δ(u2)u=h(x,u),xRN, where hC(RN×R,R) and VC(RN,R). Solutions of (1) are standing waves of the following quasilinear Schrödinger equation: (2)iψt+Δψ-V(x)ψ+kΔ(α(|ψ|2))α(|ψ|2)ψ+g(x,ψ)=0,xRN, where V(x) is a given potential, k is a real constant, and α and g are real functions. The quasilinear Schrödinger equations (2) are derived as models of several physical phenomena; for example, see . Several methods can be used to solve (1). For instance, the existence of a positive ground state solution has been proved in [6, 7] by using a constrained minimization argument; the problem is transformed to a semilinear one in  by a change of variables (dual approach); Nehari method is used to get the existence results of ground state solutions in [12, 13].

Recently, some new methods have been applied to these equations. In , the authors prove that the critical points are L functions by the Moser’s iteration; then the existence of multibump type solutions is constructed for this class of quasilinear Schrödinger equations. In , by analysing the behavior of the solutions for subcritical case, the authors pass to the limit as the exponent approaches to the critical exponent in order to establish the existence of both one-sign and nodal ground state solutions. Another new method which works for these equations is perturbations. In  4-Laplacian perturbations are led into these equations; then high energy solutions are obtained on bounded smooth domain.

In this paper, the perturbation, combined with dual approach, is applied to search the existence of nontrivial solution and sequence of high energy solutions of (1) on the whole space RN. For simplicity we call this method the dual-perturbation method.

We need the following several notations. Let Cc(RN) be the collection of smooth functions with compact support. Let (3)H1(R):={uL2(RN):|u|L2(RN)}, with the inner product (4)u,vH1=RN[u·v+uv]dx and the norm (5)uH1=u,uH11/2. Let the following assumption (V) hold:

VC(RN,R) satisfies infxRNV(x)a0>0 and lim|x|V(x)=+.

Set (6)E:={uH1(RN):NV(x)u2dx<+} with the inner product (7)u,vE=RN[u·v+V(x)uv]dx and the norm (8)uE=u,uE1/2. Then both H1(RN) and E are Hilbert spaces.

By the continuity of the embeddingELs(N) for s[2,2*] we know that, for each s[2,2*], there exists constant as>0 such that (9)usasuE,uE, where ·s denotes the Ls-norm. In the following, we use C or Ci to denote various positive constants. Moreover, we need the following assumptions:

there exist4<p<2(2*) if N3 and 4<p< if N=2 such that (10)|h(x,s)|C(1+|s|p-1),sR,

lims0h(x,s)/s=0 uniformly in xRN,

there exist μ>4 and r>0 such that (11)c0:=infxRN,|s|=rH(x,s)>0,μH(x,s)h(x,s)s

for all xRN and |s|r, where H(x,s)=0sh(x,t)dt.

By Lemma 3.4 in  we know that, under the assumption (V), the embedding ELs(RN) is compact for each 2s<2*.

Equation (1) is the Euler-Lagrange equation of the energy functional (12)J(u)=12RN[(1+2u2)|u|2+V(x)u2]dx-RNH(x,u)dx, where H(x,u)=0uh(x,t)dt. Due to the presence of the term RNu2|u|2dx, J(u) is not well defined in E. To overcome this difficulty, a dual approach is used in [9, 10]. Following the idea from these papers, let f be defined by (13)f(t)=11+2f2(t) on [0,+), f(0)=0 and f(-t)=-f(t) on (-,0]. Then f has the following properties:

f is uniquely defined C function and invertible;

0<f(t)1 for all tR;

|f(t)||t| for all tR;

limt0f(t)/t=1;

limt+f(t)/t=21/4, limt-f(t)/|t|=-21/4;

(1/2)f(t)tf(t)f(t) for all t0 and f(t)tf(t)(1/2)f(t) for all t0;

|f(t)|21/4|t| for all tR;

the function f2(t) is strictly convex;

there exists a positive C such that (14)|f(t)|{C|t|,|t|1,C|t|1/2,|t|1;

there exist positive constants C1 and C2 such that (15)|t|C1|f(t)|+C2|f(t)|2 for all tR;

|f(t)f(t)|1/2 for all tR;

for each ξ>0, there exists C(ξ)>0 such that f2(ξt)C(ξ)f2(t).

The properties (f1)(f11) have been proved in . It suffices to prove (f12).

Indeed, by (f1), (f4), and (f5), there exist δ>0 and M>0 such that, for |t|δ, (16)12t2f2(t)32t2, and for |t|M, (17)22|t|f2(t)322|t|. Since there exists a C0>0 such that f2(2t)C0f2(t) (see ), we can assume that 0<ξ<1. For |t|δ, we have |ξt|δ, and hence (18)f2(ξt)32ξ2t23ξ2f2(t); for |t|M/ξ>M, one has |ξt|M, and hence (19)f2(ξt)322ξ|t|3ξf2(t); and for δ|t|M/ξ, there exist m(ξ)>0 and M(ξ)>0 such that f2(ξt)M(ξ) and f2(t)m(ξ). Then we have (20)f2(ξt)M(ξ)M(ξ)m(ξ)f2(t). Hence f2(ξt)C(ξ)f2(t), where C(ξ)=max{3ξ2,M(ξ)/m(ξ)}.

After the change of variable, J(u) can be reduced to (21)I(v):=J(f(v))=12RN(|v|2+V(x)f2(v))dx-RNH(x,f(v))dx. From [8, 9, 11] we know that if vE is a critical point of I, that is, (22)I(v),φ=RNvφdx+RNV(x)f(v)f(v)φdx-RNh(x,f(v))f(v)φdx=0 for all φE, then u:=f(v) is a weak solution of (1). Particularly, if vH1(RN)C2(RN) is a critical point of I, then u:=f(v) is a classical solution of (1).

A sequence {un}E is called a Cerami sequence of J if {J(un)} is bounded and (1+un)J(un)0 in E*. We say that J satisfies the Cerami condition if every Cerami sequence possesses a convergent subsequence.

2. Some Lemmas

Consider the following perturbation functional Iθ defined by (23)Iθ(v)=I(v)+θ2RNV(x)v2dx, where θ(0,1]. We have the following lemmas.

Lemma 1.

If assumptions (V), (h1), and (h2) hold, then the functional Iθ is well defined on E and IθC1(E,R).

Proof.

By conditions (h1) and (h2), the properties (f2), (f3), (f7), and (f11) imply that there exists δ>0 such that (24)|h(x,f(v))f(v)||f(v)|f(v)|v| for|v|<δ,|h(x,f(v))f(v)|C|f(v)|p-1f(v)C|f(v)|p-2C|v|(p-2)/2for|v|δ. Hence (25)|h(x,f(v))f(v)|C(|v|+|v|(p/2)-1),(26)|H(x,f(v))|C(|v|2+|v|p/2) for all vR. By (26) and the continuity of the embedding ELs(RN) (s[2,2*]), (27)RNH(x,f(v))dx<+,vE. Hence Iθ is well defined in E.

Now, we prove that IθC1(E,R). It suffices to prove that (28)Ψ1(v)RNH(x,f(v))dxC1(E,R),Ψ2(v):=RNV(x)f2(v)dxC1(E,R).

For any v,ϕE and 0<|t|<1, by the mean value theorem, (25) and (f2)-(f3), we have (29)|H(x,f(v+tϕ))-H(x,f(v))||t|01|h(f(x,v+stϕ))f(v+stϕ)ϕ|dsC[|v||ϕ|+|ϕ|2+|v|(p-2)/2|ϕ|+|ϕ|p/2],|V(x)f2(v+tϕ)-V(x)f2(v)||t|201V(x)|f(v+stϕ)f(v+stϕ)ϕ|ds2V(x)01|v+stϕ||ϕ|ds2V(x)[|v||ϕ|+|ϕ|2]. The Hölder inequality implies that (30)C[|v||ϕ|+|ϕ|2+|v|(p-2)/2|ϕ|+|ϕ|p/2]L1(RN),2V(x)[|v||ϕ|+|ϕ|2]L1(RN). Hence, by the Lebesgue theorem, we have (31)Ψ1(v),ϕ=RNh(x,f(v))f(v)ϕdx,Ψ2(v),ϕ=2RNV(x)f(v)f(v)ϕdx for all ϕE. Now, we show that Ψi(·):EE*, i=1,2, are continuous. Indeed, if vnv in E, then vnv in Ls(RN) for all s[2,2*].

On the space Lp1(RN)Lp2(RN), we define the norm (32)vp1p2=vp1+vp2. Then (33)vnvinL2(RN)L(p/2)-1(RN). Moreover, on the space Lp1(RN)+Lp2(RN), we define the norm (34)vp1p2=inf{up1+wp2:v=u+w,uLp1(RN),wLp2(RN)up1}. By (25), we have (35)|h(x,f(v))f(v)|C(|v|+|v|(p/2)-1)C(|v|2/2+|v|q/r), where q=p/2 and r=p/(p-2). Then Theorem A.4 in  implies (36)h(x,f(vn))f(vn)-h(x,f(v))f(v)0in L2(RN)+Lr(RN) as n+. If h(x,f(vn))f(vn)-h(x,f(v))f(v)=yn+zn with ynL2(RN) and znLr(RN), one has (37)|RN[h(x,f(vn))f(vn)-h(x,f(v))f(v)]ϕdx|RN|yn||ϕ|+|zn||ϕ|dxC(yn2+znr)ϕE. Hence (38)|RN[h(x,f(vn))f(vn)-h(x,f(v))f(v)]ϕdx|Ch(x,f(vn))f(vn)-h(x,f(v))f(v)2rϕE, and hence (39)Ψ1(vn)-Ψ1(v)0 as n. Therefore, Ψ1C1(E,R).

Define (40)LVs(RN)={RNu:RNR:uis  measurableandRNV(x)usdx<} with the norm uLVs=(RNV(x)usdx)1/s. On the space LVp1(RN)LVp2(RN), we define the norm (41)vp1p2=vLVp1+vLVp2. On the space LVp1(RN)+LVp2(RN), we define the norm (42)vp1p2=inf{vp1p2=uLVp1+wLVp2:v=u+w,uLVp1(RN),wLVp2(RN)uLVp1}. From vnv in E, one has vn,vLV2(RN) and (43)vnvin  LV2(RN)LV2(RN) as n. Since |f(v)f(v)||v|, by the following Lemma 2, we have (44)f(vn)f(vn)f(v)f(v)in LV2(RN)+LV2(RN). If f(vn)f(vn)-f(v)f(v)=yn+zn with ynLV2(RN) and znLV2(RN), one has (45)|RNV(x)[f(vn)f(vn)-f(v)f(v)]ϕdx|RNV(x)|yn||ϕ|+V(x)|zn||ϕ|dx(ynLV2+znLV2)ϕE. Hence (46)|RNV(x)[f(vn)f(vn)-f(v)f(v)]ϕdx|f(vn)f(vn)-f(v)f(v)22ϕE, and hence (47)Ψ2(vn)-Ψ2(v)0 as n. Therefore, Ψ2C1(E,R). This completes the proof.

Lemma 2.

Assume that 1p,q,r,s<+, gC(RN×R) and (48)|g(x,v)|C(|v|p/r+|v|q/s). Then, for every vLVp(RN)LVq(RN), g(·,v)LVr(RN)+LVs(RN), and the operator (49)A:LVp(RN)LVq(RN)LVr(RN)+LVs(RN):vg(x,v) is continuous.

Proof.

Let η(s) be a smooth cut-off function such that η(s)=1 for |s|1 and η(s)=0 for |s|2. Define (50)g1(x,v)η(v)g(x,v),g2(x,v):=(1-η(v))g(x,v). We can assume that p/rq/s. Hence (51)|g1(x,v)|C|v|p/r,|g2(x,v)|C|v|q/s for all (x,v)RN×R. Assume vnv in LVp(RN)LVq(RN). Then vnv in LVp(RN) and g(·,vn)g(·,v) in LVr(RN). As in the proof of Lemma A.1 in , there exists a subsequence {wn} of {vn} and αLVp(RN) such that wn(x)v(x) and |v(x)|,|wn(x)|α(x) for a.e. xRN. Hence, from (51), one has (52)|g1(x,wn)-g1(x,v)|r2rC|α(x)|p a.e. on RN. It follows from the Lebesgue theorem that g1(·,wn)g1(·,v) in LVr(RN). Consequently, g1(·,vn)g1(·,v) in LVr(RN). Similarly, we can prove g2(·,vn)g2(·,v) in LVs. Since (53)g(·,vn)-g(·,v)rsg1(·,vn)-g1(·,v)LVr+g2(·,vn)-g2(·,v)LVs, it follows that g(·,vn)g(·,v) in LVr+LVs. This completes the proof.

Lemma 3.

Let (V), (h1), and (h2) hold. Then every bounded sequence {vn}E with Iθ(vn)0 possesses a convergent subsequence.

Proof.

Since {vn}E is bounded, then, by the compactness of the embedding ELs(RN) (2s<2*), passing to a subsequence, one has vnv in E, vnv in Ls(RN) for all 2s<2*, and vn(x)v(x) for a.e. xRN. By (25) (54)|RNh(x,f(vn))f(vn)(v-vn)dx|RNC(|vn|+|vn|(p/2)-1)|vn-v|dxC(vn2vn-v2+vnp/2(p/2)-1vn-vp/2)C(vnEvn-v2+vnE(p/2)-1vn-vp/2)0 as n. Similarly, RNh(x,f(v))f(v)(v-vn)dx0 as n. Hence, by the property of (f8), we have (55)Iθ(vn)-Iθ(v),vn-v=RN|(vn-v)|2dx+θRNV(x)|vn-v|2dx+RNV(x)[f(vn)f(vn)-f(v)f(v)]×(vn-v)dx-RN[h(x,f(vn))f(vn)-h(x,f(v))f(v)](vn-v)dxθvn-vE2-on(1), where on(1)0 as n. This shows that vn-vE20 as n. This completes the proof.

The following Lemma 4 has been proved in  (see Proposition 2.1(3) in ).

Lemma 4.

If vn(x)v(x) a.e. in RN and limnRNV(x)f2(vn)dx=RNV(x)f2(v)dx, then RNV(x)f2(vn-v)dx0 as n.

3. Main Results Theorem 5.

Assume conditions (V), (h1)(h3) hold. Let {θn}(0,1] be such that θn0. Let vnE be a critical point of Iθn with Iθn(vn)c for some constant c independent of n. Then, up to subsequence, one has vnv in E, Iθn(vn)I(v) and v is a critical point of I.

Proof.

By (h2), for 0<ε0<(1/4)(1/2-1/μ)a0, there exists δ0>0 such that (56)|1μsh(x,s)-H(x,s)|ε0s2,s[-δ0,δ0]. By (h1), for δ0|s|r (r is the constant appearing in condition (h3)), we have (57)|1μsh(x,s)-H(x,s)|2C(1δ02+rp-2)s2, where C is the constant appearing in condition (h1). Hence (58)|1μsh(x,s)-H(x,s)|ε0s2+2C(1δ02+rp-2)s2,s[-r,r]. Since lim|x|V(x)=+, there exists ρ0>0 such that (59)14(12-1μ)V(x)>2C(1δ02+rp-2) for all |x|ρ0. Hence (60)(12-1μ)RNV(x)f2(v)dx+{x:|f(v)|r}[1μf(v)h(x,f(v))-H(x,f(v))1μ]dx12(12-1μ)RNV(x)f2(v)dx-2C(1δ02+rp-2)r2|Bρ0|. Since vn is a critical point of Iθn, (61)Iθn(vn),ϕ=RNvnϕdx+RNV(x)f(vn)f(vn)ϕdx+θnRNV(x)vnϕdx-RNh(x,f(vn))f(vn)ϕdx=0 for all ϕE. Consequently, taking ϕ=f(vn)/f(vn)E, by (h3) and (f6) we have (62)cIθn(vn)=Iθn(vn)-1μIθn(vn),f(vn)f(vn)=RN[12-1μ(1+2f2(vn)1+2f2(vn))]|vn|2dx+(12-1μ)RNV(x)f2(vn)dx+θn2RNV(x)vn2dx-θnμRNV(x)vnf(vn)f(vn)dx+RN[1μh(x,f(vn))f(vn)-H(x,f(vn))]dx(12-2μ)RN|vn|2dx+(12-1μ)RNV(x)f2(vn)dx+(12-2μ)θnRNV(x)vn2dx+{x:|f(vn)|r}[1μh(x,f(vn))f(vn)-H(x,f(vn))1μ]dx(12-2μ)RN|vn|2dx+12(12-1μ)RNV(x)f2(vn)dx+(12-2μ)θnRNV(x)vn2dx-2C(1δ02+rp-2)r2|Bρ0|(12-2μ)RN|vn|2dx+(14-12μ)RNV(x)f2(vn)dx+(12-2μ)θnRNV(x)vn2dx-C1, and hence (63)RN|vn|2dx+RNV(x)f2(vn)dx+θnRNV(x)vn2dxC for some constant C independent of n. By the boundedness of RN|vn|2dx, there exists C2>0 such that (64)2Nf2(vn)|f(vn)|2dxN[1+2f2(vn)]|f(vn)|2dx=N|vn|2dxC2 for all n. Hence, by the Sobolev embedding theorem, one has (65)f(vn)2(2*)4=f2(vn)2*2C3f2(vn)22C.

Next, we prove that f(vn)L(RN) and f(vn)LC, where the positive constant C is independent of n. Setting T>2, r>0, define v~nT=b(vn), where b:RR is a smooth function satisfying b(s)=s for |s|T-1, b(-s)=-b(s); b(s)=0 for sT, and b(s) is decreasing in [T-1,T].

This means that v~nT=vn, for |vn|T-1; |v~nT|=|b(vn)||vn|, for T-1|vn|T; |v~nT|=CT>0, for |vn|T, where T-1CTT.

Let ϕ=(f(vn)/f(vn))|f(v~nT)|2r; then ϕE. By (61) I(vn),ϕ=0. Hence (66)I1+I2+I3+I4+I5=Nh(x,f(vn))f(vn)|f(v~nT)|2rdx, where (67)I5:=RNV(x)f2(vn)|f(v~nT)|2rdx+θnRNV(x)vnf(vn)f(vn)|f(v~nT)|2rdxRNV(x)f2(vn)|f(v~nT)|2rdx,I1{x:|vn|T}[1+2f2(vn)1+2f2(vn)]|f(v~nT)|2r|vn|2dx{x:|vn|T}|f(v~nT)|2r|vn|2dx={x:|vn|T}[1+2f2(vn)]×|f(vn)|2|f(v~nT)|2rdx2{x:|vn|T}f2(vn)|f(vn)|2|f(v~nT)|2rdx=12{x:|vn|T}|[f2(vn)fr(v~nT)]|2dx,(68)I2{x:|vn|T-1}[2r+1+2f2(vn)1+2f2(vn)]×|f(v~nT)|2r|vn|2dx{x:|vn|T-1}2f2(vn)1+2f2(vn)|f(v~nT)|2r|vn|2dx{x:|vn|T-1}|f(vn)|2r+2|f(vn)|2dx=1(r+2)2{x:|vn|T-1}|fr+2(vn)|2dx=1(r+2)2{x:|vn|T-1}|[f2(vn)fr(v~nT)]|2dx,I3:={x:T-1|vn|T}[1+2f2(vn)1+2f2(vn)]f2r(v~nT)|vn|2dx{x:T-1|vn|T}2f2(vn)1+2f2(vn)f2r(v~nT)|vn|2dx=12{x:T-1|vn|T}[fr(v~nT)f2(vn)]2dx,I4:=2r{x:T-1|vn|T}f2r-1(v~nT)f(v~nT)×b(vn)f(vn)f(vn)|vn|2dx. For T-1|vn|T, |v~nT|=|b(vn)||vn|. By the properties of f and b, the mean value theorem implies (69)|f(b(vn))|f(b(vn))b(vn)|vn|12f(b(vn))b(vn)f2(vn). Hence (70)I4=2r{x:T-1|vn|T}f2r-1(v~nT)f(v~nT)×b(vn)f(vn)f(vn)|vn|2dx=2r{x:T-1|vn|T}f2r-1(b(vn))f(b(vn))×b(vn)f(vn)×1+2f2(vn)|vn|2dx2r{x:T-1|vn|T}[fr-1(b(vn))f(b(vn))b(vn)]2×f(vn)vn2f2(vn)|vn|2dx2r{x:T-1|vn|T}[fr-1(b(vn))f(b(vn))b(vn)]2×f4(vn)|vn|2dx=2r{x:T-1|vn|T}f4(vn)×[fr-1(b(vn))f(b(vn))]2dx=2r{x:T-1|vn|T}[f2(vn)fr(v~nT)]2dx. Consequently, (71)I3+I4=12{x:T-1|vn|T}[fr(v~nT)f2(vn)]2dx+2r{x:T-1|vn|T}[f2(vn)fr(v~nT)]2dx1(r+2)2{x:T-1|vn|T}2[fr(v~nT)f2(vn)]2+2[f2(vn)fr(v~nT)]2dx1(r+2)2{x:T-1|vn|T}|[f2(vn)fr(v~nT)]|2dx. Combining (67) and (68), we have (72)I1+I2+I3+I41(r+2)2N|[f2(vn)fr(v~nT)]|2dx. For any ε>0, by (h1) and (h2), there exists C(ε)>0 such that (73)|h(x,s)|ε|s|+C(ε)|s|p-1. Combining (66), (72), and (73), one has (74)1(r+2)2RN|[f2(vn)fr(v~nT)]|2dxC(ε)RN|f(vn)|p|f(v~nT)|2rdx. By the Hölder inequality and (65), (75)RN|f(vn)|p|f(v~nT)|2rdx=RN|f(vn)|p-4|f(v~nT)|2rf4(vn)dx(RN|f(vn)|(p-4)·(4N/(p-4)(N-2))dx)(p-4)(N-2)/4N·(RN[|f(v~nT)|2r×f4(vn)]4N/(4N-(p-4)(N-2))dxRN)(4N-(p-4)(N-2))/4N=(RN|f(vn)|22*dx)((p-4)(N-2))/4N·(RN[|f(v~nT)|r×f2(vn)]8N/(4N-(p-4)(N-2))dxRN)(4N-(p-4)(N-2))/4NC(RN[|f(v~nT)|r×f2(vn)]8N/(4N-(p-4)(N-2))dxRN)(4N-(p-4)(N-2))/4N. Moreover, (76)1(r+2)2RN|[f2(vn)fr(v~nT)]|2dxC(r+2)2(RN[f2(vn)|f(v~nT)|r]2*dx)2/2*. Hence(77)(RN[f2(vn)|f(v~nT)|r]2*dx)2/2*C(r+2)2×(RN[|f(v~nT)|r×f2(vn)|f(v~nT)|r]8N/(4N-(p-4)(N-2))dxRN)(4N-(p-4)(N-2))/4N.Since 4<p<2(2*), d=2*/(8N/(4N-(p-4)(N-2)))=2*/2-p/4+1>1. Set q=8N/(4N-(p-4)(N-2)). Then (78)(RN[f2(vn)|f(v~nT)|r]qddx)1/qd(r+2)[C(r+2)2]1/2(r+2)×(RN[|f(v~nT)|rf2(vn)]qdx)1/q(r+2). Take r=r0 such that (2+r0)q=2(2*). Since |v~nT|=|b(vn)||vn|, |f(v~nT)||f(vn)|. Hence, from (65), we have (79)RN[|f(v~nT)|r0f2(vn)]qdxRN|f(vn)|(2+r0)qdx<C. Since f(v~nT)f(vn) as T+, taking T+ in (78) with r=r0, we have (80)(RN|f(vn)|(2+r0)qddx)1/qd(r0+2)[C(r0+2)2]1/2(r0+2)×(RN|f(vn)|(2+r0)qdx)1/q(r0+2). Set 2+r1=(2+r0)d. Then (81)(RN|f(vn)|(2+r1)qdx)1/q(r1+2)[C(r0+2)2]1/2(r0+2)×(RN|f(vn)|(2+r0)qdx)1/q(r0+2). Inductively, we have (82)(RN|f(vn)|(2+rk+1)qdx)1/q(rk+1+2)[C(rk+2)2]1/2(rk+2)×(RN|f(vn)|(2+rk)qdx)1/q(rk+2)i=0k[C(ri+2)2]1/2(ri+2)×(RN|f(vn)|(2+r0)qdx)1/q(r0+2), where (2+ri)=di(2+r0)  (i=0,1,,k), and (83)i=0k[C(ri+2)2]12(ri+2)=exp{i=0klnCdi(r0+2)di(r0+2)}=exp{i=0k[lnC(r0+2)di(r0+2)+ilnddi(r0+2)]} is convergent as k. Let Ck=i=0k[C(ri+2)2]1/2(ri+2). Then CkC>0 as k. Hence (84)f(vn)L(2+r0)qdk+1Ckf(vn)L2(2*). Let k; by (65), we have (85)f(vn)LCf(vn)L2(2*)C,f(v)LC. Hence, by (f9) and (85), we have (86)RNV(x)vn2dx={x:|vn(x)|1}V(x)vn2dx+{x:|vn(x)|>1}V(x)vn2dx1C{x:|vn(x)|1}V(x)f2(vn)dx+1C{x:|vn(x)|>1}V(x)f4(vn)dx1C{x:|vn(x)|1}V(x)f2(vn)dx+C{x:|vn(x)|1}V(x)f2(vn)dxCRNV(x)f2(vn)dx. By (63) we know that RNV(x)vn2dx is bounded, and hence {vn} is bounded in E. Up to subsequence, one has vnv in E, vnv in Ls(RN) for s[2,2*), and vn(x)v(x) a.e. xRN.

Now, we show that v is a critical point of I. For any ψC0(RN) with ψ0, by (85), we know that ϕ=ψexp(-f(vn))E. Take ϕ=ψexp(-f(vn)) as the test function in (61); we have (87)0=RNexp(-f(vn))vn·ψdx-RN|vn|2ψexp(-f(vn))f(vn)dx+θnRNV(x)vnψexp(-f(vn))dx+RNV(x)f(vn)f(vn)ψexp(-f(vn))dx-RNh(x,f(vn))f(vn)ψexp(-f(vn))dx. By |(vn-v)|2ψexp(-f(vn))f(vn)0, one has (88)limsupnRN-|vn|2ψexp(-f(vn))f(vn)dx-RN|v|2ψexp(-f(v))f(v)dx. Since θn0, by (63) (89)θnRNV(x)vnψexp(-f(vn))dx0 as n. Moreover, notice that vnv in E, vnv in Ls(RN) for s[2,2*), and vn(x)v(x) a.e. xRN; by Hölder inequality and Lebesgue theorem, we have (90)RNexp(-f(vn))vn·ψdxRNexp(-f(v))v·ψdx,RNV(x)f(vn)f(vn)ψexp(-f(vn))dxRNV(x)f(v)f(v)ψexp(-f(v))dx,RNh(x,f(vn))f(vn)ψexp(-f(vn))dxRNh(x,f(v))f(v)ψexp(-f(v))dx. Hence, from (87), we have (91)0RNexp(-f(v))v·ψdx-RN|v|2ψexp(-f(v))f(v)dx+θnRNV(x)vψexp(-f(v))dx+RNV(x)f(v)f(v)ψexp(-f(v))dx-RNh(x,f(v))f(v)ψexp(-f(v))dx=RNv·(ψexp(-f(v)))dx+RNV(x)f(v)f(v)ψexp(-f(v))dx-RNh(x,f(v))f(v)ψexp(-f(v))dx. For any φE with φ0, by (85) we know that ζ:=φexp(f(v))E. By Theorem 2.8 in , there exists a sequence {ψn}C0(RN) such that ψn0 and ψnζ and ψn(x)ζ(x) for a.e. xRN. Take ψ=ψn in (91), and let n; we have (92)0RNv·φdx+RNV(x)f(v)f(v)φdx-RNh(x,f(v))f(v)φdx. The opposite inequality can be obtained by taking ϕ=ψexp(f(vn)) and ζ=φexp(-f(v)). Consequently, (93)RNv·φdx+RNV(x)f(v)f(v)φdx-RNh(x,f(v))f(v)φdx=0,φE. This shows that vE is a critical point of I, and by taking φ=f(v)/f(v)E, one has (94)RN[1+2f2(v)1+2f2(v)]|v|2dx+RNV(x)f2(v)dx-RNh(x,f(v))f(v)dx=0.

Finally, taking ϕ=f(vn)/f(vn) as the test function in (61), we have (95)RN[1+2f2(vn)1+2f2(vn)]|vn|2dx+θnRNV(x)vnf(vn)f(vn)dx+RNV(x)f2(vn)dx-RNh(x,f(vn))f(vn)dx=0. Since (96)RNh(x,f(vn))f(vn)dxRNh(x,f(v))f(v)dx,RN[1+2f2(vn)1+2f2(vn)]|(vn-v)|2dx0, by Fatou’s Lemma, (63), (94), (95), up to subsequence, one has (97)θnRNV(x)vnf(vn)f(vn)dx0,RN|vn|2dxRN|v|2dx,RN2f2(vn)1+2f2(vn)|vn|2dxRN2f2(v)1+2f2(v)|v|2dx,(98)RNV(x)f2(vn)dxRNV(x)f2(v)dx. Hence Iθn(vn)I(v) as n. Set wn:=vn-vE. By (f8), (f12), and (85), one has (99)f2(wn)=f2(2·wn2)C[12f2(vn)+12f2(v)]C[f2(vn)+f2(v)]C. Consequently, by (f9), (98), and Lemma 4, one has (100)RNV(x)|wn|2dx={x:|wn|1}V(x)|wn|2dx+{x:|wn|1}V(x)|wn|2dxC{x:|wn|1}V(x)f2(wn)dx+C{x:|wn|1}V(x)f4(wn)dxC{x:|wn|1}V(x)f2(wn)dx+C{x:|wn|1}V(x)f2(wn)dx=CRNV(x)f2(wn)dx0 as n. Therefore, vnv in E. This completes the proof.

Theorem 6.

Assume conditions (V), (h1)(h3) hold; then (1) has a weak solution.

Proof.

First, we prove that, for each θ(0,1], Iθ satisfy the Cerami condition. Indeed, let {vn}E be an arbitrary Cerami sequence of Iθ. Set ϕ=f(vn)/f(vn). Then ϕECvnE. Similar to the proof of (63), we can prove that {vn} is bounded in E. Hence, by Lemma 3, the sequence {vn} possesses a convergent subsequence in E. This shows that Iθ satisfy the Cerami condition.

Next, for any ε>0, by (h1), (h2), (f3), and (f7), there exists C(ε)>0 such that (101)H(x,f(t))εt2+C(ε)|t|p/2 for all (x,t)RN×R. For small 0<ρ1, set (102)Sρ={vE:vE=ρ}. Then, from (101), for vSρ, (103)Iθ(v)=12RN[|v|2+V(x)f2(v)]dx+θ2RNV(x)v2dx-RNH(x,f(v))dxθ2RN[|v|2+V(x)v2]dx-εRNv2dx-C(ε)RN|v|p/2dxθ2vE2-εa22vE2-C(ε)ap/2p/2vEp/2ρ2(θ4-Cρ(p-4)/2)δ>0 for small ε>0 and ρ>0. Moreover, by (h3), for any (x,z)RN×R with |z|r, one has (104)H(x,z)c0|z|μ. Since μ>4, there exists a constant 4<α<min{μ,2(2*)}. Hence, by (f5), we have (105)lim|t|H(x,f(t))|t|α/2=lim|t|H(x,f(t))|f(t)|α·|f(t)|α|t|α/2=+ uniformly in xRN. Consequently, there exist constants τ>1 such that (106)H(x,f(t))|t|α/2,|t|τ, for all xRN. For any finite-dimensional subspace E~E, by the equivalency of all norms in the finite-dimensional space, there is a constant a>0 such that (107)vα/2avE,vE~. By (h1), (h2), and (106), there exists a positive constant C>0 such that (108)H(x,f(t))|t|α/2-Ct2,  (x,t)RN×R. Since 4<α<2(2*), by (f3), (107), and (108), we have (109)Iθ(v)=12RN[|v|2+V(x)f2(v)]dx+θ2RNV(x)v2dx-RNH(x,f(v))dxvE2-vα/2α/2+Cv22CvE2-aα/2vEα/2 for all vE~. Hence there exists a large R>0 such that Iθ<0 on E~BR. Set a fixed eE~ with eE=1. For any fixed T>ρ, define the path hT:[0,1]E~E by hT(t)=tTe. Then for large T>0, by (109), one has (110)Iθ(hT(1))CT2-aα/2Tα/2<0,hT(1)E=T>ρ,supt[0,1]Iθ(hT(t))CT2<+. Hence by Theorem 2.2 with the Cerami condition in , Iθ possesses a critical value (111)cθinfγΓmaxt[0,1]Iθ(γ(t))δ>0,cθsupt[0,1]Iθ(hT(t))CT2, where (112)Γ={γC([0,1],E):γ(0)=0,γ(1)=hT(1)}. Consequently, by Theorem 5, we know that (1) has a weak solution. This completes the proof of Theorem 6.

Remark 7.

Let v+=max{v,0} and v-=max{-v,0}. Set (113)I±(u)=12RN[|v|2+V(x)f2(v)]dx-RNH(x,f(v±))dx,Iθ±(v)=12θRNV(x)v2dx+I±(v) instead of I(u) and Iθ(u), respectively. Then, under the conditions of Theorem 6, we can obtain the existence of a positive solution and a negative solution for (1).

Theorem 8.

Assume conditions (V), (h1)(h3) hold. If h(x,s) is odd in s, then (1) has a sequence {vm} of solutions such that I(vm)+.

Proof.

Consider the eigenvalue of the operate L=-Δ+V. By assumption (V) and the compactness of the embedding EL2(RN), we know that the spectrum σ(L)={λ1,λ2,,λn} of L with (114)0<λ1<λ2<<λn< and λn+ as n+ (see page 3820 in ). Let φn be the eigenfunction corresponding to λn. By regularity argument we know that φnE. Set En=span{φ1,φ2,,φn}. Then we can decompose the space E as E=EnWn for n=1,2,, where Wn is orthogonal to En in E. For ρ>0, set (115)Qρ={vE:RN[|v|2+V(x)f2(v)]dxρ2}. By (109) there exists rn>0 independent of θ such that (116)Iθ(v)<0,vEnQrn-. Set (117)Dn=EnQrn,Gn={φC(Dn,E):φis  oddandφ|QrnEn=id},Γj={φ(DnQrn¯):φGn,nj,A=-AEnQrnis  closed  and  γ(A)n-j}, where γ(·) is the genus. Let (118)cj(θ)=infBΓjsupvBIθ,j=1,2. We have the following three facts (we refer the reader to  for their proofs).

Fact (1). For each BΓj, if 0<ρ<rn for all nj, then BQρWj-1.

Fact (2). There exist constants αjβj such that cj(θ)[αj,βj] and αj+ as j+.

Fact (3).  cj(θ), j=1,2, are critical values of Iθ.

Consequently, Theorem 8 follows from Theorem 5 and the above Facts (2)-(3). This completes the proof.

Corollary 9.

If the following conditions (h4) and (h5) are used in place of (h3); then the conclusions of Theorem 5, Theorem 6, and Theorem 8 hold:

lim|s|+infH(x,s)>0 uniformly in xN,

there exist μ>4 and τ>0 such that (119)μH(x,s)h(x,s)s

for all xN and |s|τ.

Proof.

By (h4), there are constants λ>0 and r1>0 such that whenever |s|r1, one has (120)H(x,s)>λ,xN. Set r=max{τ,r1}. Then, by (h5), (121)c0:=infxN,|s|=rH(x,s)λ>0,μH(x,s)h(x,s)s for all xN and |s|r. Therefore, condition (h3) holds. This completes the proof.

Acknowledgment

This work was supported partially by the National Natural Science Foundation of China (11261070).

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