We study the existence of multiple solutions for the following elliptic problem: -Δpu-μ|u|p-2u/|x|p=|u|p*(t)-2/|x|tu+λ|u|q-2/|x|su,u∈W01,p(Ω). We prove that if 1≤q<p<N, then there is a μ0, such that for any μ∈0,μ0, the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result (Azorero and Alonso, 1991).

1. Introduction and Main Results

In this paper, we study the existence of multiple solutions to the following elliptic problem:
(1)-Δpu-μ|u|p-2u|x|p=|u|p*(t)-2|x|tu+λ|u|q-2|x|su,x∈Ω,u(x)=0,x∈∂Ω,
where Ω⊂ℜN(N≥3) is a smooth bounded domain containing the origin 0, Δpu=div(|∇u|p-2∇u) is the p-Laplacian of u, 1<p<N, 0≤μ<μ-≡((N-p)/p)p, 0≤t, s<p, 1≤q<p, and p*(t)=p(N-t)/(N-p) is the critical Sobolev-Hardy exponent; note that p*(0)=p*≡pN/(N-p) is the critical Sobolev exponent.

Problem (1) is related to the well-known Sobolev-Hardy inequalities [1]:
(2)(∫ℜN|u|p*(t)|x|tdx)p/p*(t)≤C∫ℜN|∇u|pdx,hhhhhhhhhhhhhhhhhhhh∀u∈W01,p(Ω).
As t=p, p*(t)=p, then the well-known Hardy inequality holds [1, 2]:
(3)∫ℜN|u|p|x|pdx≤1μ-∫ℜN|∇u|pdx,∀u∈W01,p(Ω).

In this paper, we use Lq(Ω,|x|s) to denote the usual weighted Lq(Ω) space with the weight |x|s. In W01,p(Ω), for μ∈[0,μ-), we use the norm
(4)∥u∥=∥u∥W01,p(Ω)=(∫Ω(|∇u|p-μ|u|p|x|p)dx)1/p.
By (3), this norm is equivalent to the usual norm (∫Ω|∇u|pdx)1/p. By the Hardy inequality and the Sobolev-Hardy inequality, for 0≤μ<μ-, 0≤t<p, we can define the following best constants:
(5)Aμ,t(Ω)=infu∈W01,p(Ω)∖{0}∥u∥p(∫Ω(|u|p*(t)/|x|t)dx)p/p*(t).
Note that Aμ,0 is the best constant in the Sobolev inequality, that is,
(6)Aμ,0(Ω)=infu∈W01,p(Ω)∖{0}∫Ω(|∇u|p-μ(|u|p/|x|p))dx(∫Ω|u|p*dx)p/p*.
The energy functional of (1) is defined as follows:
(7)I(u)=1p∫Ω(|∇u|p-μ|u|p|x|p)dx-1p*(t)∫Ω|u|p*(t)|x|tdx-λq∫Ω|u|q|x|sdx.
Then, I(u) is well defined on W01,p(Ω) and belongs to C1(W01,p(Ω),ℜ). The solutions of problem (1) are then the critical points of the functional I.

In recent years, the quasilinear problems related to Hardy inequality and Sobolev-Hardy inequality have been studied by some authors [3–7]. Ghoussoub and Yuan [5] studied problem (1) with μ=0, s=0, and p≤q≤p* and proved the existence results of positive solutions and sign-changing solutions. Kang in [3, 4] studied (1,1) when p≤q≤p*(s) and verified the existence of positive solutions of (1) when the parameters p, q, s, λ, μ satisfy suitable conditions. To the best of our knowledge, there are few results of problem (1) involving the p-sublinear of 1≤q<p<N. We are only aware of the works [6–9] which studied the existence and multiplicity of solution of problem (1) involving weight functions. Azorero and Alonso [9] studied problem (1) with s=t=μ=0, 1<q<p and proved that there exists λ1, such that (1) has infinitely many solutions for λ∈(0,λ1). Hsu [7] studied problem (1) and proved that there exists Λ0>0 such that (1) has at least two positive solutions for λ∈(0,Λ0). In this paper, we study (1) and extend the results of [7, 9].

Throughout this paper, let R0 be the positive constant such that Ω⊂B(0,R0), where B(0,R0)={x∈ℜN:|x|<R0}. By Holder inequalities, for all u∈W01,p(Ω), we obtain
(8)∫Ω|u|q|x|sdx≤(∫B(0,R0)|x|(tq-sp*(t))/(p*(t)-q)dx)(p*(t)-q)/p*(t)×(∫Ω|u|p*(t)|x|tdx)q/p*(t)≤(NωN∫0R0r((tq-sp*(t))/(p*(t)-q))+N-1dr)(p*(t)-q)/p*(t)×(∫Ω|u|p*(t)|x|tdx)q/p*(t)=(NωNR0((tq-sp*(t))/(p*(t)-q))+N(p*(t)-q)q(t-N)+p*(t)(N-s))(p*(t)-q)/p*(t)×(∫Ω|u|p*(t)|x|tdx)q/p*(t)=c0(∫Ω|u|p*(t)|x|tdx)q/p*(t),
where ωN=2πN/2/NΓ(N/2) is the volume of the unit ball in ℜN. The following inequality comes from the paper [7]:
(9)∫Ω|u|q|x|sdx≤(NωNR0N-sN-s)(p*(s)-q)/p*(s)Aμ,s-q/p∥u∥q.

Now we are ready to state our main results.

Theorem 1.

If Ω⊂ℜN is a bounded domain in ℜN, and 1≤q<p<N, then there is a μ0>0 such that problem (1) possesses infinitely many weak solutions in W01,p(Ω) for any μ∈(0,μ0).

2. The Palais-Smale Condition

Let X be a Banach space and X-1 be the dual space of X. The functional I∈C1(X,ℜ) is said to satisfy the Palais-Smale condition at level c ((PS)c), if any sequence (un)⊂X satisfying
(10)I(un)⟶c,I′(un)⟶0stronglyinX-1asn⟶∞
contains a subsequence converging in X to a critical point of the functional I. In this paper, we will take X=W01,p(Ω).

Lemma 2.

Let {un}⊂W01,p(Ω) be a Palais-Smale sequence for I defined by (7), that is,
(11)I(un)⟶c,(12)I′(un)⟶0inW-1,p′(Ω),1p+1p′=1.
If 1≤q<p and c<(p-t)/p(N-t)(Aμ,t)(N-t)/(p-t)-Kλp*(t)/(p*(t)-q), and K depends on p, q, N, then, there exists a subsequence {unk}⊂{un}, strongly convergent in W01,p(Ω).

Proof.

By (11) and (12), it is easy to prove that the sequence un is bounded in W01,p(Ω). Passing to a subsequence if necessary, we may assume that, as n→∞,
(13)un⇀uweaklyinW01,p(Ω),un⇀uweaklyinLp*(t)(Ω,|x|-t),un⇀uweaklyinLp(Ω,|x|-p),un⟶ustronglyinLq(Ω,|x|-s),un⟶ustronglyinLp(Ω),un⟶ua.e.inΩ.
Then, u∈W01,p(Ω) is a solution of problem (1). By the concentration compactness principle (see [10, 11]), there exists a subsequence, still denoted by un, at the most countable set ȷ, a set of different points {xj}j∈ȷ⊂Ω∖{0}, sets of nonnegative real numbers {μ~j}j∈ȷ∪{0}, {υ~j}j∈ȷ∪{0}, and nonnegative real numbers τ~0 and γ~0, such that
(14)|∇un|p⇀dμ~≥|∇u|p+∑j∈ȷμ~jδxj+μ~0δ0,|un|p*⇀dν~=|u|p*+∑j∈ȷν~jδxj+ν~0δ0,|un|p*(t)|x|t⇀dτ~=|u|p*(t)|x|t+τ~0δ0,|un|p|x|p⇀dγ~=|u|p|x|p+γ~0δ0,
where δx is the Dirac mass at x.

Case 1 (t=s=0 and p*(t)=p*). We claim that ȷ is finite, and, for any j∈ȷ, either
(15)ν~j=0orν~j≥(Aμ,0)N/p.

In fact, let ε>0 be small enough such that 0∈¯B(xj,ε) and B(xi,ε)⋂B(xj,ε)=∅ for i≠j, i,j∈ȷ. We consider φj∈C0∞(ℜN), such that
(16)φj≡1onB(xj,ε2),φj≡0onB(xj,ε)c,hhh|∇φj|≤4ε,hhh0≤φj≤1.
It is clear that the sequence {φjun} is bounded in W01,p(Ω). Note that
(17)〈I′(un),unφj〉=∫Ω|∇un|pφjdx+∫Ωun|∇un|p-2∇un∇φjdx-μ∫Ω|un|p|x|pφjdx-∫Ω|un|p*φjdx-λ∫Ω|un|qφjdx.
By (13), (16), and the Holder inequality, we obtain
(18)0≤limε→0limn→∞|∫Ωun|∇un|p-2∇un∇φjdx|≤Climε→0(∫B(xj,ε)|u|p*dx)1/p*=0,limε→0limn→∞|∫Ω|un|p|x|pφjdx|≤limε→0limn→∞|∫Bε(xj)|un|p(|xj|-ε)pφjdx|=0.
From (12)~(18), we get that
(19)0=limε→0limn→∞〈I′(un),unφj〉≥μ~j-ν~j.
By the Sobolev inequality, A0,0νjp/p*≤μ~j, hence, we deduce that
(20)ν~j=0orν~j≥(A0,0)N/p,
which implies that ȷ is finite.

Now we consider the possibility of concentration at the origin. Let ε>0 be small enough such that xj∈¯B(0,ε), ∀j∈ȷ. Take φ0∈C0∞(ℜN) such that
(21)φ0≡1onB(0,ε2),φ0≡0onB(0,ε)c,|∇φ0|≤4ε.
By (13) and (14), we also get that
(22)0=limε→0limn→∞〈I′(un),unφ0〉=limε→0(∫Ωφ0dμ~-∫Ωφ0dγ~-∫Ωφ0dν~-λ∫Ωφ0|u|q|x|sdx)≥μ~0-μγ~0-ν~0.
By the definition of Aμ,0, we deduce that
(23)Aμ,0ν~0p/p*≤μ~0-μγ~0.
From (22) we have
(24)Aμ,0ν~0p/p*≤μ~0-μγ~0≤ν~0,
which implies that ν~0=0 or
(25)ν~0≥(Aμ,0)N/p.
We will prove that (25) and ν~j≥(A0,0)N/p are not possible. By (13) and (14),
(26)c=limn→∞I(un)=limn→∞{I(un)-1p〈I′(un),un〉}=limn→∞{1N∫Ω|un|p*dx+λ(1p-1q)∫Ω|un|qdx}≥1N(∫Ω|u|p*dx+∑j∈ȷν~j+ν~0)+λ(1p-1q)∫Ω|u|qdx≥1N∫Ω|u|p*dx+1Nmin{(Aμ,0)N/p,(A0,0)N/p}+λ(1p-1q)∫Ω|u|qdx≥1N∫Ω|u|p*dx+1N(Aμ,0)N/p+λ(1p-1q)∫Ω|u|qdx.
By applying the Holder inequality at (26), we have
(27)c≥1N(Aμ,0)N/p+1N∫Ω|u|p*dx-λ(1q-1p)|Ω|(p*-q)/p*(∫Ω|u|p*dx)q/p*.
Let f1(x)=c1xp*-λc2xq, c1=1/N, c2=(1/q)-(1/p). This function obtains its absolute minimum (for x>0) at point x0=(λc2q/p*c1)1/(p*-q). That is,
(28)f1(x)≥f1(x0)=-K1λp*/(p*-q),
where
(29)K1=c2p/(p*-q)c1-q/(p*-q)×((qp*)q/(p*-q)-(qp*)p*/(p*-q))>0,
because of 1<q<p<Np/(N-p). But this result contradicts the hypothesis. Then, ν~j=0∀j∈ȷ∪{0} and we conclude.

Case 2 (0<t<p, then p<p*(t)<p*). We only need to consider the possibility of concentration at the origin. Let ε>0 be small enough such that B(0,ε)⊂Ω. Take φ0 a smooth cut-off function centered at the origin such that 0≤φ0≤1, φ0=1 for |x|≤ε/2, φ0=0 for |x|≥ε, and |∇φ0|≤4/ε. By (13) and (14), we get that
(30)0=limε→0limn→∞〈I′(un),unφ0〉=limε→0(∫Ωφ0dμ~-μ∫Ωφ0dγ~-∫Ωφ0dτ~-λ∫Ωφ0|u|q|x|sdx)≥μ~0-μγ~0-τ~0.
By the definition of Aμ,t, we deduce that
(31)Aμ,tτ~0p/p*(t)≤μ~0-μγ~0
From (30), we have
(32)Aμ,tτ~0p/p*(t)≤τ~0,
which implies that τ~0=0 or
(33)τ~0≥(Aμ,t)(N-t)/(p-t).
We will prove (33) is not possible. From the above arguments and (8), we conclude that
(34)c=limn→∞J(un)=limn→∞{J(un)-1p〈J(un),un〉}≥p-tp(N-t)∫Ω|u|p*(t)|x|tdx+p-tp(N-t)(Aμ,t)(N-t)/(p-t)+λc0(1q-1p)(∫Ω|u|p*(t)|x|tdx)q/p*(t).
Let f2(x)=c3xp*(t)-λc4xq, c3=(p-t)/p(N-t), c4=c0((1/q)-(1/p)). This function obtains its absolute minimum at point x0=(λc4q/p*(t)c3)1/(p*-q). That is,
(35)f2(x)≥f2(x0)=-K2λp*(t)/(p*(t)-q).
But this result contradicts the hypothesis. Hence, up to a subsequence, we obtain that un→u strongly in W01,p(Ω).

Thus, the proof of the Lemma is completed.

3. Existence of Infinitely Many Solutions

In this section, we will prove our main result of Theorem 1. We first recall some concepts and results in minimax theory.

Let X be a Banach space, and Σ denote all closed subsets of X-{0} which are symmetric with respect to the origin. For A∈Σ, we define the genus γ(A) by
(36)γ(A)=min{k∈N:∃ϕ∈C(A;Rk∖{0}),ϕ(x)=-ϕ(-x)},
if the minimum exists, and if such a minimum does not exist, then we define γ(A)=∞. The main properties of the genus are contained in the following lemma (see [12] for the details).

Lemma 3.

Let A,B∈Σ. Then one has the following.

If there exists f∈C(A,B), odd, then γ(A)≤(B).

If A⊂B, then γ(A)≤γ(B).

If there exists an odd homeomorphism between A and B, then γ(A)=γ(B).

If SN-1 is the sphere in ℜN, then γ(SN-1)=N.

Consider γ(A∪B)≤γ(A)+γ(B).

If γ(B)<+∞, then γ(A-B¯)≥γ(A)-γ(B).

If A is compact, then γ(A)<+∞, and there exists δ>0 such that γ(Nδ(A))=γ(A), where Nδ(A)={x∈X:d(x,A)≤δ}.

If X0 is a subspace of X with codimension k, and γ(A)<k, then A∩X0≠∅.

Let X be a Banach space and E be a C1 functional on X. Denote Ec={u∈X∣E(u)≤c}, Σk={C⊂W01,p(Ω)-{0},Cisclosed,C=-C,γ(C)≥k}.

Given the functional I, under the hypothesis 1<q<p<n, using Sobolev’s equality and (9), we obtain
(37)I(u)≥1p∥u∥p-1p*(t)Aμ,tp*(t)/p∥u∥p*(t)-λc0q∥u∥q.
If we define for x≥0(38)h(x)=1pxp-1p*(t)Aμ,tp*(t)/pxp*(t)-λc0qxq,
then
(39)I(u)≥h(∥u∥).
Because p*(t)>p and h(x)→-∞, as x→+∞, it is easy to see that there exists 0<μ0≤1 such that, if 0<μ≤μ0, h attains its positive maximum.

From the structure of h(x), we see that there are constants 0<R0<R1, such that h(R0)=h(R1)=0, h(R)≤0 if R<R0, h(R)>0 if R0<R<R1, and h(R)<0 if R>R1. Following [9], let τ:R+→[0,1]∈C∞ be nonincreasing, such that
(40)τ(x)={1,if0≤x≤R0,0,ifx≥R1,
and let φ(u)=τ(∥u∥); we consider the truncated functional
(41)J(u)=1p∫Ω(|∇u|p-μup|x|p)dx-1p*(t)∫Ω|u|p*(t)φ(u)|x|tdx-λq∫Ω|u|q|x|sdx.
Similar to (39), we have J(u)≥h-(∥u∥), where
(42)h-(x)=1pxp-τ(x)p*(t)Aμ,tp*(t)/pxp*(t)-λc0qxq.
Clearly, h-(x)≥h(x) for x≥0 and h-(x)=h(x) if 0≤x≤R0, h-(x)≥h(x)≥0, if R0<R<R1, and if x>R1, h-(x)=(1/p)xp-(λc0/q)xq is strictly increasing, and so h-(x)>0, if x>R1. Consequently, h-(x)≥0 for x≥R0.

Lemma 4.

(1) Consider J∈C1(W01,p(Ω),R).

(2) If J(u)≤0, then ∥u∥≤R0 and I(v)=J(v) for all v in a small enough neighborhood of u.

(3) There exists λ0>0, such that if 0<λ<λ0, then J verifies a local Palais-Smale condition for c≤0.

Proof.

(1) and (2) are immediate. To prove (3), observe that all Palais-Smale sequences for J with c≤0 must be bounded; then, by Lemma 2, if λ verifies ((p-t)/p(N-t))(Aμ,t)(N-t)/(p-t)-Kλp*(t)/(p*(t)-q)≥0, there exists a convergent subsequence.

Now, we use the idea in [9] to construct negative critical values of J via genus.

Lemma 5.

Given n∈N, there is an ε(n)>0, such that
(43)γ({u∈W01,p(Ω):J(u)≤-ε(n)})≥n.

Proof.

Fix n; let En be an n-dimensional subspace of W01,p(Ω); we take un∈En with norm ∥un∥=1 for 0<ρ<R0; we have
(44)J(ρun)=I(ρun)=1pρp-1p*(t)ρp*(t)×∫Ω|un|p*(t)|x|tdx-λqρq∫Ω|un|q|x|sdx.
Since En is a space of finite dimension, all the norms in En are equivalent. If we define
(45)αn=inf{∫Ω|u|p*(t)|x|tdx:u∈En,∥u∥=1}>0,βn=inf{∫Ω|u|q|x|sdx:u∈En,∥u∥=1}>0,
we have
(46)J(ρun)≤1pρp-αnp*(t)ρp*(t)-λβnqρq,
and we can choose ε (which depends on n), and η<R0, such that J(ηu)≤-ε if u∈En and ∥u∥=1.

Let Sη={u∈W01,p(Ω):∥u∥=η}. Consider Sη∩En⊂{u∈W01,p(Ω):J(u)≤-ε}; therefore, by Lemma 3, we see that
(47)γ({u∈W01,p(Ω):J(u)≤-ε(n)})≥γ(Sη∩En)=n.
We are now in a position to prove our main result.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

Let Σk={C⊂W01,p(Ω)-{0},Cisclosed,C=-C,γ(C)≥k}, ck=infC∈Σksupu∈CJ(u), Kc={u∈W01,p(Ω),J′(u)=0,J(u)=c}, and suppose that 0<λ<λ0 where λ0 is the constant given by Lemma 4. We claim that if k,r∈N are such that c=ck=ck+1=···=ck+r, then γ(Kc)≥r+1.

In fact, denote J-ε={u∈W01,p(Ω):J(u)≤-ε}; by Lemma 5, we see that for any k∈N, there is a ε(k)>0, such that γ(J-ε(k))≥k. Since J is continuous and even, J-ε(k)∈Σn and ck≤-ε(n)<0. As J is bounded from below, we see that ck>-∞ for all k∈N.

Suppose that c=ck=ck+1=···=ck+r<0; then J satisfies (PS)c condition by Lemma 2, and it is easy to see that Kc is a compact set.

If γ(Kc)≤r, then there is a closed and symmetric set U with Kc⊂U and γ(U)≤r by Lemma 3. Since c<0, we can also assume that the closed set U⊂J0. Since J satisfies (PS)c condition for c<0, by the Deformation Lemma, there is an odd homeomorphism,
(48)η:W01,p(Ω)⟶W01,p(Ω)
such that η(Jc+δ-U)⊂Jc-δ for some δ with 0<δ<-c.

Since c=ck+r=infC∈Σksupu∈CJ(u), there exists an A∈Σk+r, such that
(49)supu∈CJ(u)<c+δ,i.e.,A⊂Jc+δ,(50)η(A-U)⊂η(Jc+δ-U)⊂Jc-δ.

But by Lemma 3 and γ(U)≤r, we have
(51)γ(A-U¯)≥γ(A)-γ(U)≥n,γ(ηA-U¯)γ(A-U¯)≥n.
Hence, ηA-U¯∈Σk and supu∈ηA-U¯≥ck=c, which contradicts to (50). So we have proved that γ(Kc)≥r+1.

Now if for all k∈N, we have Σk+1⊂Σk, ck≤ck+1<0. If all ck are distinct, then γ(Kck)≥1, and we see that {ck} is a sequence of distinct critical values of J; if for some k0, there is a r≥1 such that
(52)c=ck0=ck0+1=···=ck0+r,
then
(53)γ(Kck0)≥r+1,
which shows that Kck0 contains infinitely many distinct elements.

Since J(u)=I(u) if J(u)<0, we see that there are infinitely many critical points of I(u). The theorem is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by Ningbo Scientific Research Foundation (2009B21003), K. C. Wong Magna Fund in Ningbo University, NSF of Hebei Province (A2013209278), and National Natural Science Foundation of China (nos. 61271398, 11220248 and 11175092).

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