JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 627570 10.1155/2013/627570 627570 Research Article Sign-Changing Solutions for a Fourth-Order Elliptic Equation with Hardy Singular Terms Pei Ruichang 1,2 Zhang Jihui 1 Li Wan-Tong 1 Institute of Mathematics School of Mathematics and Computer Sciences Nanjing Normal University Nanjing 210097 China nnu.cn 2 School of Mathematics and Statistics, Tianshui Normal University Tianshui 741001 China 2013 20 11 2013 2013 08 08 2013 28 10 2013 2013 Copyright © 2013 Ruichang Pei and Jihui Zhang. 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.

The existence and multiplicity of sign-changing solutions for a class of fourth elliptic equations with Hardy singular terms are established by using the minimax methods.

1. Introduction

Consider the following Navier boundary value problem: (1)2u(x)-N2(N-4)216u|x|4=f(x,u),in  Ω,u=u=0on  Ω, where Ω is a bounded smooth domain in N(N5), 0Ω.

The conditions imposed on f(x,t) are as follows:

there exists C>0 such that (2)|f(x,t)|C(1+|t|p),t,xΩ,

where 1<p<(N+4)/(N-4);

fC(Ω¯×,),f(x,t)  t0 for all xΩ, t;

lim|t|0f(x,t)/t=f0, lim|t|f(x,t)/t=l uniformly for xΩ, where f0 and l are constants;

lim|t|[f(x,t)t-2F(x,t)]=- uniformly for xΩ, where F(x,t)=0tf(x,s)ds;

there exist μ>2 and R>0 such that (3)0<μF(x,t)f(x,t)t,xΩ,|t|R;

f(x,t) is odd in t.

In recent years, this fourth-order semilinear elliptic problem: (4)2u(x)+cu=f(x,u),in  Ω,u=u=0on  Ω, can be considered as an analogue of a class of second-order problems which have been studied by many authors. In , there was a survey of results obtained in this direction. In , Micheletti and Pistoia showed that (4) admits at least two solutions by a variation of linking if f(x,u) is sublinear. And in , the authors proved that the problem (4) has at least three solutions by a variational reduction method and a degree argument. In , Zhang and Li showed that (4) admits at least two nontrivial solutions by Morse theory and local linking if f(x,u) is superlinear and subcritical on u.

To the authors’ knowledge, there seem few results about the sign-changing solutions on problem (1) with hardy singular terms. In this paper, motivated by , the existence and multiplicity of sign-changing solutions for problem (1) are obtained by introducing a compact embedding theorem and a maximum principle. Our results are new.

2. Preliminaries and Auxiliary Lemmas

We introduce the new working space E which is obtained by the completion of C0(Ω) with respect to the norm (see ) (5)u=(Ω(|u|2-N2(N-4)216|u|2|x|4)dx)1/2 associated with the inner product (6)u,v=Ω(uv-N2(N-4)216uv|x|4)dx.

Throughout this paper, we denoted by ·p the Lp(Ω) norm.

At first, we here give two important lemmas.

Lemma 1.

E L 2 ( Ω ) (see ).

Lemma 2 (see [<xref ref-type="bibr" rid="B6">6</xref>, Corollary 4.1]).

Assume N5,  VL(Ω), and V0. Let us suppose that the operator 2-(V/|x|4) is coercive on H2(Ω)H01(Ω). Let fL2(Ω) such that f0. Let uH2(Ω) be a solution of (7)2u(x)-V|x|4u=f,in  Ω,      u=u=0onΩ. Then u0 in Ω.

Now, we consider the following eigenvalue problem: (8)2u(x)-N2(N-4)216|x|4u=λu,in  Ω,(9)u=u=0on  Ω.

The first eigenvalue of this problem is given by (10)λ1=inf{u2:uE,u2=1}.

By Lemma 1, EW1,p(Ω)L2(Ω) for p2-. The minimizing sequence is compact in L2(Ω). By standard argument, we may assume that the first eigenfunction ϕ1 is positive in Ω (see [9, page 167]). The second eigenvalue is given by (11)λ2=inf{u2:uE,Ωuϕ1=0,u2=1} which possesses a sign-changing eigenfunction ϕ2. Similarly, we can characterize the nth eigenvalue λn with a sign-changing eigenfunction. By standard elliptic theory, λn as n.

It follows from (H1) that the functional (12)I(u)=12Ω|u|2dx-N2(N-4)232×Ωu2|x|4-ΩF(x,u)dx is of C1 on the space E. Under the condition (H1), the critical points of I are solutions of problem (1).

If l in the above condition (H3) is an eigenvalue of (2-(N2(N-4)2/16)(1/|x|4),E), then the problem (1) is called resonance at infinity. Otherwise, we call it nonresonance.

For looking for sign-changing solutions of problem (1), we recall a very useful result.

Proposition 3 (see [<xref ref-type="bibr" rid="B9">10</xref>, Theorem 3.2]).

Let X be a Hilbert space and f be a C1 functional defined on X. Assume that f satisfies the (PS) condition on X and f(u) has the expression f(u)=u-Au for uX. Assume that D1 and D2 are open convex subset of X with the properties that D1D2, A(D1)D1, and A(D2)D2. If there exists a path h:[0,1]X such that (13)h(0)D1D2,h(1)D2D1,infuD1¯D2¯f(u)>supt[0,1]f(h(t)), then f has at least four critical points, one in D1D2, one in D1D2¯, one in D2D1¯, and one in X(D1¯D2¯).

Remark 4.

If f satisfies the (C)c condition, then this proposition still holds (see ).

3. Main Results

Let us now state the main results.

Theorem 5.

Assume conditions (H2) and (H3) hold. If f0<λ1 and l(λk,λk+1) for some k>2, then problem (1) has a positive solution, a negative solution, and a sign-changing solution.

Remark 6.

This result is similar to [7, Theorem1.1]. As far as verifying the (PS) condition is concerned, our method is more simple than that in [12, 13].

Theorem 7.

Assume conditions (H2)(H4) hold. If f0<λ1 and l=λk for some k>2, then problem (1) has a positive solution, a negative solution, and a sign-changing solution.

Remark 8.

When l=λk(k>2), the case is called resonance and not considered by . This result is completely new.

Theorem 9.

Assume conditions (H1), (H5), and (H6) hold. If f0=0, then problem (1) has infinitely many sign-changing solutions.

Lemma 10.

Under the assumptions of Theorem 5, if λk<l<λk+1, then I satisfies the (PS) condition.

Proof.

Let {un}E be a sequence such that |I(un)|c,  <I(un), and   ϕ>0. Since(14)I(un),ϕ=Ω(unϕ-N2(N-4)216unϕ|x|4)dx-Ωf(x,un)ϕdx=o(ϕ) for all ϕE. If un2 is bounded, we can take ϕ=un. By (H3), there exists a constant c>0 such that |f(x,un(x))|c|un(x)|, a.e. xΩ. So un is bounded in E. If un2+, as n, set vn=un/un2, then vn2=1. Taking ϕ=vn in (14), it follows that vn is bounded. Without loss of generality, we assume vnv in E, and then vnv in L2(Ω). Hence, vnv a.e. in Ω and |vn|q(x) (q(x)L2(Ω)). Dividing both sides of (14) by un2, for all ϕE, we get (15)Ω(vnϕ-N2(N-4)216vnϕ|x|4)dx-Ωf(x,un)un2ϕdx=o(ϕun2).

Then for a.e. xΩ, we have f(x,un)/un2lv as n. In fact, if v(x)0, by (H3), we have (16)|un(x)|=|vn(x)|un2+,f(x,un)un2=f(x,un)unvnlv.

If v(x)=0, we have (17)|f(x,un)|un2c|vn|0.

Since |f(x,un)|/un2c|vn|cq(x), by (15) and the Lebesgue dominated convergence theorem, we arrive at (18)Ωvϕdx-N2(N-4)216vϕ|x|4-Ωlvϕdx=0,ϕE.

It is easy to see that v0. In fact, if v0, then v2=0 contradicts limnvn2=v2=1. Hence, l is an eigenvalue of (2-(N2(N-4)2/16)(1/|x|4),E). This contradicts our assumption. Thus {un} is bounded. By standard argument (see the proof of our Lemma 12 below), {un}u in E. The lemma is proved.

Lemma 11.

Under the assumptions of Theorem 7, if l=λk, then the functional I satisfies the (C) condition which is stated in .

Proof.

Suppose I satisfies(19)I(un)c,(1+un)I(un)0asn.

In view of (H3), it suffices to prove that un is bounded in E. Similar to the proof of Lemma 10, we have (20)Ω(vϕ-N2(N-4)216vϕ|x|4)dx-Ωlvϕdx=0,ϕE.

Therefore v0 is an eigenfunction of λk, and then |un(x)| for a.e. xΩ. It follows from (H4) that (21)limn+[f(x,un(x))un(x)-2F(x,un(x))]=- holds uniformly in xΩ, which implies that (22)Ω(f(x,un)un-2F(x,un))dx-as  n.

On the other hand, (19) implies that (23)2I(un)-I(un),un2cas  n.

Thus (24)Ω(f(x,un)un-2F(x,un))dx2c    as  n, which contradicts (22). Hence un is bounded.

Lemma 12.

Assume (H1) and (H5) hold. Then I satisfies the (PS) condition.

Proof.

Assume that {un} is a (PS) sequence; I(un)0 and {I(un)} is bounded. A routine argument implies that {un} is bounded. By [6, Theorem A.2], we have (25)EW01,q(Ω), where 1q<2. For p given in (H1),  p<(n+4)/(n-4), and we may choose q such that (p+1)<qN/(N-q),q<2. By the Sobolev embedding theorem, we have (26)W1,q(Ω)Lt(Ω),t<NqN-q. We infer from (26) that {un} is compact in Lp+1(Ω). By (H1), (27)un-um2=Ω|f(x,un)-f(x,um)||un-um|dx+o(1)C(Ω|un-um|p+1dx)1/(p+1)+o(1)0.

This completes the proof of this lemma.

For the aim of using Proposition 3 that proves our main results, we prove an important lemma below.

From previous Section 1, we know that I is C1 functional and its gradient at u is given by (28)I(u)=u-A(u),A:EE,  A(u)=(2-N2(N-4)2161|x|4)-1f(x,u).

Then A(u),ϕ=Ωf(x,u)ϕdx for all ϕE. We consider the convex cones P={uH:u0} and -P={uE:u0}; moreover, for ϵ>0, assume (29)Pϵ={uE:dist(u,P)<ϵ},-Pϵ={uE:dist(u,-P)<ϵ}.

Note that Pϵ and -Pϵ are open convex subsets of E  and E(Pϵ¯(-Pϵ¯)) contains only sign-changing functions.

Lemma 13.

Assume (H2) and (H3) hold. Then, there exists ϵ0>0 such that for 0<ϵϵ0 there holds (30)A((±Pϵ))±Pϵ.

Moreover, if u±Pϵ is a nontrivial solution of problem (1), then u is positive (negative) in the sense that u>0    (u<0) in Ω.

Proof.

Indeed, if uE and u+=max{u,0}, u-=min{u,0}, then (31)dist(A(u),P)infwPA(u)-w=infwPA(u)++A(u)--wA(u)-.

For every s(2,2N/(N-4)), there exists Cs>0 such that (32)u±sinfwPu-wsCsdist(u,P).

Choose ϵ>0 such that (f0+ϵ)<λ1. Using (32), the Hölder inequality, the Poincaré inequality, and the Sobolev embedding theorem, we have (33)dist(A(u),P)A(u)-A(u)-2=Ωf(x,u)A(u)-dxΩf(x,u-)A(u)-dxΩ((f0+ϵ)|u-|+Cϵ|u-|p)A(u)-dx(f0+ϵ)u-2A(u)-2+Cϵu-p+1pA(u)-p+1(f0+ϵ)infwPu-w2A(u)-2+CinfwPu-wp+1pA(u)-p+1f0+ϵλ1dist(u,P)A(u)-+Cdist(u,P)pA(u)-, where Cϵ,C>0 are constants. Hence (34)dist(A(u),P)(δ+Cdist(u,P)p-1)dist(u,P), where δ=(f0+ϵ)/λ1<1. Take ϵ0 such that δ1=δ+Cϵ0p-1<1. Now if dist(u,P)<ϵ<ϵ0, then we have (35)dist(A(u),P)δ1dist(u,P).

Thus for every uPϵ, by (35) we have (36)dist(A(u),P)δ1ϵ; thus A(u)Pϵ. Hence A(Pϵ)Pϵ. In a similar way, A((-Pϵ))(-Pϵ). If 0<ϵϵ0, and uPϵ (resp.,—Pϵ) is a nontrivial solution of problem (1), then I(u)=0. By (35) we have dist(u,P)=0; that is, uP (resp., u-P). By Lemma 2, we imply that u>0    (u<0) in Ω.

Lemma 14.

Assume (H1),   (H2),     and   (H5) hold. Then, there exists ϵ0>0 such that for 0<ϵϵ0 there holds (37)A((±Pϵ))±Pϵ.

Proof.

The proof is quite similar to that of Lemma 4.2 in . We omit it here.

Lemma 15.

Assume (H5) holds. Then (38)I(u)-,uEk, where the definition of Ek introduced in our proof of Theorem 9.

Proof.

Because dimEk<, then by (H5), (39)I(u)u212-ΩF(x,u)u2dx- as u,  uEk. This lemma follows immediately.

Lemma 16.

Assume (H2) and (H3) hold. Let 0<ϵϵ0, and then there exists C0>- such that infPϵ¯(-Pϵ¯)I(u)=C0.

Proof.

By the conditions (H2) and (H3), we know that, for any ϵ>0, there exists C>0, such that (40)|f(x,t)|(f0+ϵ)|t|+C|t|p(1<p<N+4N-4).

Using (40) and the Sobolev embedding theorem, we have (41)I(u)=12u2-ΩF(x,u)dx12u2-12(f0+ϵ)Ωu2dx-Cup+1p+1-12(f0+ϵ)u22-Cup+1p+1.

By (32) we have u±sCsϵ0 for every uPϵ(-Pϵ). So there exists C0>- such that (42)infPϵ¯(-Pϵ¯)I(u)=C0.

Hence this lemma is proved.

4. Proof of the Main Results Proof of Theorem <xref ref-type="statement" rid="thm3.1">5</xref> and Theorem <xref ref-type="statement" rid="thm3.2">7</xref>.

Motivated by the Proof of Theorem 4.2 in , we still define a path hR:[0,1]E as (43)hR(t)=Rϕ1cosπt+Rϕ2sinπt,0t1.

Obviously, hR(0)Pϵ(-Pϵ) and  hR(1)(-Pϵ)Pϵ. By the Fatou’s lemma, the condition (H3) with l>λ2 and a direct computation shows that (44)limR+supt[0,1]I(hR(t))=-.

So, it yields that there exists R0 such that I(hR0(t))<C0-c*(c*>0). Hence we obtain (45)infPϵ¯(-Pϵ¯)I(u)>supt[0,1]I(h(t)). By using Lemmas 10, 11, and 13, Proposition 3, and Lemma 16, we can find a critical point in Pϵ(-Pϵ¯) which is a positive solution, a critical point in (-Pϵ)Pϵ¯ which is a negative solution, and a critical point in E(Pϵ¯(-Pϵ¯)) which is a sign-changing solution.

Before beginning our proof of Theorem 9, we need the following important proposition.

Proposition 17 (see [<xref ref-type="bibr" rid="B13">9</xref>, Theorem 5.6]).

Assume E is a Hilbert space with inner product <, > and the corresponding norm ·, IC1(E,R) and I(u)=(1/2)u2-G(u), uE, where GC1(E,R). P denotes a positive closed convex cone of E.

Assume that  A(±D0)±D0, where D0:={uE:dist(u,P)<μ0}, μ0>0, and A=G.

Assume that, for any a,b>0, there is a constant C>0 such that(46)G(u)a,u*buC,

where ·* denotes another norm of E such that u*Cu for all uE.

Assume that limuY,uI(u)=-,supYI:=β.

If the even functional I satisfies (PS) condition at level c for each c[γ,β], then (47)𝒦[γ-ϵ,β+ϵ](E(-PP)) for all ϵ>0 small, where (supYIβ  (Y  andM are two subspaces of E with dimY<, dimY-codimM1),  infQ**I:=γ, and Q**:=Q*(ρ)Iβ    (Q*(ρ):={uM:(u*p/u2)+((uu*)/(u+D*u*))=ρ}, where ρ>0, D*>0,  and   p>2 are fixed constants.

Now, we give an outline proof for our Theorem 9.

Proof of Theorem <xref ref-type="statement" rid="thm3.3">9</xref>.

Let Nk denote the eigenspace of λk. We fix k and let Ek:=N1Nk. Consider another norm ·*=·p+1 of E, p(1,((N+4)/(N-4))). Write E=Ek-1Ek-1.

Let (48)Q*(ρ)={uEk-1:up+1p+1u2+uup+1u+D*up+1=ρ}, where ρ,D* are fixed constants. By our assumptions, we may find a constant C>0 such that (49)F(x,t)14λ1t2+C|t|p+1,xΩ,tR, where 1<p<(N+4)/(N-4). For any a,b>0, there is a constant C>0 such that (50)I(u)a,up+1buC. By Lemma 15, (51)limuY,uI(u)=-, where Y=Ek. Then (A1*) and (A2*) are satisfied. By Lemma 14, the condition (A) holds.

Now, we define (52)supYI=β. Let (53)Q**:=Q*(ρ)Iβ,      infQ**I=γ. By Lemma 12, I satisfies the (PS) condition. Thus, by Proposition 17 and the Proof of Theorem 5.7 in , we know that the functional I posses a sequence sign-changing solution {uk}.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant no. 10671156) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant no. 1301038C).

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