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),inΩ,u=△u=0on∂Ω,
where Ω is a bounded smooth domain in ℝN(N≥5), 0∈Ω.

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∈ℝ,∀x∈Ω,

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

f∈C(Ω¯×ℝ,ℝ),f(x,t)t≥0 for all x∈Ω, t∈ℝ;

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

lim|t|→∞[f(x,t)t-2F(x,t)]=-∞ uniformly for x∈Ω, 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,x∈Ω,|t|≥R;

f(x,t) is odd in t.

In recent years, this fourth-order semilinear elliptic problem:
(4)△2u(x)+c△u=f(x,u),inΩ,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 [1], there was a survey of results obtained in this direction. In [2], Micheletti and Pistoia showed that (4) admits at least two solutions by a variation of linking if f(x,u) is sublinear. And in [3], the authors proved that the problem (4) has at least three solutions by a variational reduction method and a degree argument. In [4], 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 [5–8], 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])
(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(△u△v-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↪↪L2(Ω) (see [5]).

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

Assume N≥5,V∈L∞(Ω), and V≥0. Let us suppose that the operator △2-(V/|x|4) is coercive on H2(Ω)∩H01(Ω). Let f∈L2(Ω) such that f≥0. Let u∈H2(Ω) be a solution of
(7)△2u(x)-V|x|4u=f,inΩ,u=△u=0on∂Ω.
Then u≥0 in Ω.

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

The first eigenvalue of this problem is given by
(10)λ1=inf{∥u∥2:u∈E,∥u∥2=1}.

By Lemma 1, E↪W1,p(Ω)↪↪L2(Ω) for p→2-. 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{∥u∥2:u∈E,∫Ωuϕ1=0,∥u∥2=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)=12∫Ω|△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 u∈X. Assume that D1 and D2 are open convex subset of X with the properties that D1∩D2≠∅, A(∂D1)⊂D1, and A(∂D2)⊂D2. If there exists a path h:[0,1]→X such that
(13)h(0)∈D1∖D2,h(1)∈D2∖D1,infu∈D1¯∩D2¯f(u)>supt∈[0,1]f(h(t)),
then f has at least four critical points, one in D1∩D2, one in D1∖D2¯, one in D2∖D1¯, and one in X∖(D1¯∪D2¯).

Remark 4.

If f satisfies the (C)c condition, then this proposition still holds (see [11]).

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 [7]. 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(△un△ϕ-N2(N-4)216unϕ|x|4)dx-∫Ωf(x,un)ϕdx=o(∥ϕ∥)
for all ϕ∈E. If ∥un∥2 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. x∈Ω. So un is bounded in E. If ∥un∥2→+∞, as n→∞, set vn=un/∥un∥2, then ∥vn∥2=1. Taking ϕ=vn in (14), it follows that ∥vn∥ is bounded. Without loss of generality, we assume vn⇀v in E, and then vn→v in L2(Ω). Hence, vn→v a.e. in Ω and |vn|≤q(x) (q(x)∈L2(Ω)). Dividing both sides of (14) by ∥un∥2, for all ϕ∈E, we get
(15)∫Ω(△vn△ϕ-N2(N-4)216vnϕ|x|4)dx-∫Ωf(x,un)∥un∥2ϕdx=o(∥ϕ∥∥un∥2).

Then for a.e. x∈Ω, we have f(x,un)/∥un∥2→lv as n→∞. In fact, if v(x)≠0, by (H3), we have
(16)|un(x)|=|vn(x)|∥un∥2⟶+∞,f(x,un)∥un∥2=f(x,un)unvn⟶lv.

If v(x)=0, we have
(17)|f(x,un)|∥un∥2≤c|vn|⟶0.

Since |f(x,un)|/∥un∥2≤c|vn|≤cq(x), by (15) and the Lebesgue dominated convergence theorem, we arrive at
(18)∫Ω△v△ϕdx-N2(N-4)216vϕ|x|4-∫Ωlvϕdx=0,∀ϕ∈E.

It is easy to see that v≢0. In fact, if v≡0, then ∥v∥2=0 contradicts limn→∞∥vn∥2=∥v∥2=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 [11].

Proof.

Suppose I satisfies(19)I(un)⟶c∈ℝ,(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(△v△ϕ-N2(N-4)216vϕ|x|4)dx-∫Ωlvϕdx=0,∀ϕ∈E.

Therefore v≢0 is an eigenfunction of λk, and then |un(x)|→∞ for a.e. x∈Ω. It follows from (H4) that
(21)limn→+∞[f(x,un(x))un(x)-2F(x,un(x))]=-∞
holds uniformly in x∈Ω, which implies that
(22)∫Ω(f(x,un)un-2F(x,un))dx⟶-∞asn⟶∞.

On the other hand, (19) implies that
(23)2I(un)-〈I′(un),un〉⟶2casn⟶∞.

Thus
(24)∫Ω(f(x,un)un-2F(x,un))dx⟶2casn⟶∞,
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)E↪W01,q(Ω),
where 1≤q<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-um∥2=∫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:E⟶E,A(u)=(△2-N2(N-4)2161|x|4)-1f(x,u).

Then 〈A(u),ϕ〉=∫Ωf(x,u)ϕdx for all ϕ∈E. We consider the convex cones P={u∈H:u≥0} and -P={u∈E:u≤0}; moreover, for ϵ>0, assume
(29)Pϵ={u∈E:dist(u,P)<ϵ},-Pϵ={u∈E: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<ϵ≤ϵ0 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 u∈E and u+=max{u,0}, u-=min{u,0}, then
(31)dist(A(u),P)≤infw∈P∥A(u)-w∥=infw∈P∥A(u)++A(u)--w∥≤∥A(u)-∥.

For every s∈(2,2N/(N-4)), there exists Cs>0 such that
(32)∥u±∥s≤infw∈∓P∥u-w∥s≤Csdist(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)-dx≤∫Ωf(x,u-)A(u)-dx≤∫Ω((f0+ϵ′)|u-|+Cϵ′|u-|p)A(u)-dx≤(f0+ϵ′)∥u-∥2∥A(u)-∥2+Cϵ′∥u-∥p+1p∥A(u)-∥p+1≤(f0+ϵ′)infw∈P∥u-w∥2∥A(u)-∥2+Cinfw∈P∥u-w∥p+1p∥A(u)-∥p+1≤f0+ϵ′λ1dist(u,P)∥A(u)-∥+Cdist(u,P)p∥A(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 u∈∂Pϵ, 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<ϵ≤ϵ0, and u∈Pϵ (resp.,—Pϵ) is a nontrivial solution of problem (1), then I′(u)=0. By (35) we have dist(u,P)=0; that is, u∈P (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<ϵ≤ϵ0 there holds
(37)A(∂(±Pϵ))⊂±Pϵ.

Proof.

The proof is quite similar to that of Lemma 4.2 in [8]. We omit it here.

Lemma 15.

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

Proof.

Because dimEk<∞, then by (H5),
(39)I(u)∥u∥2≤12-∫ΩF(x,u)∥u∥2dx⟶-∞
as ∥u∥→∞,u∈Ek. This lemma follows immediately.

Lemma 16.

Assume (H2) and (H3) hold. Let 0<ϵ≤ϵ0, 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)=12∥u∥2-∫ΩF(x,u)dx≥12∥u∥2-12(f0+ϵ′)∫Ωu2dx-C∥u∥p+1p+1≥-12(f0+ϵ′)∥u∥22-C∥u∥p+1p+1.

By (32) we have ∥u±∥s≤Csϵ0 for every u∈Pϵ∩(-Pϵ). So there exists C0>-∞ such that
(42)infPϵ¯∩(-Pϵ¯)I(u)=C0.

Hence this lemma is proved.

4. Proof of the Main ResultsProof 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 [10], we still define a path hR:[0,1]→E as
(43)hR(t)=Rϕ1cosπt+Rϕ2sinπt,0≤t≤1.

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 ∥·∥, I∈C1(E,R) and I(u)=(1/2)∥u∥2-G(u), u∈E, where G∈C1(E,R). P denotes a positive closed convex cone of E.

Assume that A(±D0)⊂±D0, where D0:={u∈E: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∥*≤b⟹∥u∥≤C,

where ∥·∥* denotes another norm of E such that ∥u∥*≤C∥u∥ for all u∈E.

Assume that limu∈Y,∥u∥→∞I(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∖(-P∪P))≠∅
for all ϵ>0 small, where (supYI≔β(YandM are two subspaces of E with dimY<∞, dimY-codimM≥1), infQ**I:=γ, and Q**:=Q*(ρ)∩Iβ(Q*(ρ):={u∈M:(∥u∥*p/∥u∥2)+((∥u∥∥u∥*)/(∥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:=N1⊕⋯⊕Nk. Consider another norm ∥·∥*∶=∥·∥p+1 of E, p∈(1,((N+4)/(N-4))). Write E=Ek-1⊕Ek-1⊥.

Let
(48)Q*(ρ)∶={u∈Ek-1⊥:∥u∥p+1p+1∥u∥2+∥u∥∥u∥p+1∥u∥+D*∥u∥p+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,∀x∈Ω,t∈R,
where 1<p<(N+4)/(N-4). For any a,b>0, there is a constant C>0 such that
(50)I(u)≤a,∥u∥p+1≤b⟹∥u∥≤C.
By Lemma 15,
(51)limu∈Y,∥u∥→∞I(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 [9], 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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