Sharp Remainder Terms of the Rellich Inequality and its Application

MALAYSIANMATHEMATICAL

SCIENCESSOCIETY http://math.usm.my/bulletin

Sharp Remainder Terms of the Rellich Inequality and its Application

1ALNARDETALLA,²TOSHIOHORIUCHI AND3HIROSHIANDO

1Department of Mathematics, Central Mindanao University, University Town, 8710 Musuan, Bukidnon, Philippines

2,3Department of Mathematical Sciences, Ibaraki University Mito, Ibaraki, 310, Japan

1al [email protected],²[email protected]

Abstract. In this article we shall study the improvement of the Rellich inequality by adding terms with a singular weight of the type(log(1/|x|))⁻²in the right hand side. We show that this weight function is optimal in the sense that the improved inequality fails for any other weight more singular than this one. As an application, we use our improved inequality to an- alyze the behaviour of the first eigenvalue of the weighted eigenvalue problem for the opera- torL_µu=∆ |∆u|^p−2∆u

− µ/|x|^2p

|u|^p−2uasµincreases to((n−2p)/p)^p((np−n)/p)^p for 1<p<n/2.

2010 Mathematics Subject Classification: Primary: 35J70; Secondary: 35J60 Keywords and phrases: Rellich inequality, eigenvalue.

1. Introduction

In this paper we shall study the Rellich inequality (1.1)

Z

Ω

|∆u|^pdx≥

n−2p p

p np−n

p pZ

Ω

|u(x)|^p

|x|^2p dx

for anyu∈W₀^2,p(Ω), whereΩis a bounded domain inRⁿwith 0∈Ω,n≥3, and 1<p<

n/2. Here the best constantΛ_n,p= ((n−2p)/p)^p((np−n)/p)^pis given by the infimum of I(u) =

R

Ω|∆u|^pdx R

Ω

|u(x)|^p

|x|^2p dx .

Moreover there exists no extremal function inW₀^2,p(Ω)which attains the infimum of these problem. Roughly speaking, the candidates of the extremals are singular at the origin which are not in the classW₀^2,p(Ω). Hence it is natural to consider that there exist “missing terms” in the right hand side of (1.1). In view of this, we shall investigate the Rellich inequalities (1.1) and improve them by finding out missing terms.

Received:June 22, 2009;Revised:May 31, 2010.

For the case of gradient, such improved Hardy inequalities are known. For example, Adimurthi, Chaudhuri and Ramaswamy [1] have proved that there exists a constantC>0, depending onn≥2, 1<p<nandR>sup_Ω(|x|e^2/p)such that foru∈W₀^1,p(Ω)

(1.2) Z

Ω

|∇u|^pdx≥ n−p

p pZ

Ω

|u(x)|^p

|x|^p dx+C Z

Ω

|u(x)|^p

|x|^p

log R

|x|

−γ

dx

Our aim in this article is to achieve an optimal improvement of the inequality (1.1) by adding a second term involving the singular weight(log(1/|x|))⁻², in the sense that the improved inequality holds for this weight but fails if the weight is more singular than this one.

As an application, we use our improved inequality to determine exactly when the first eigenvalue of the weighted eigenvalue problem for the operator

(1.3) L_µu=∆ |∆u|^p−2∆u

− µ

|x|^2p|u|^p−2u will tend to 0 asµincreases toΛ_n,p.

This paper is organized in the following way. In§2 we shall describe our main results on the improvement of Rellich inequality. In§3 we shall prepare lemmas which are needed in the proof of the main theorem stated in§2. In§4 we shall prove the main results (Theorem 2.1 and Corollary 2.1). In§5 we shall apply our results to study the weighted eigenvalue problem.

2. Main results

Theorem 2.1. Let n≥3,0∈ΩandΩis a bounded domain inRⁿ. (1) Noncritical case(1<p<n/2)

Assumeγ≥2, then there exists K=K(n)>0and C=C(n)>0such that if R>

Ksup_Ω|x|then Z

Ω

|∆u|^pdx≥

n−2p p

p np−n

p pZ

Ω

|u(x)|^p

|x|^2p dx +C

Z

Ω

|u(x)|^p

|x|^2p

log R

|x|

−γ

dx (2.1)

for any u∈W₀^2,p(Ω).

(2) Critical case(p=n/2)

Assume thatγ≥n/2. Then there exists K^∗=K^∗(n)>0and C^∗=C^∗(n)>0such that if R>K^∗sup_Ω|x|, then

Z

Ω

|∆u|ⁿ²dx≥ n−2

√n nZ

Ω

|u(x)|ⁿ²

|x|ⁿ

log R

|x|

−γ

dx

+C^∗ Z

Ω

|u(x)|ⁿ²

|x|ⁿ

log R

|x|

−γ−1

dx (2.2)

for any u∈W₀^2,n/2(Ω).

Remark 2.1. In (2.1),γ≥2 is sharp. In (2.2),γ≥n/2 is also sharp, and((n−2)/√ n)ⁿis best constant.

Remark 2.2. The functionγ→(log(R/r))^−γis monotonically decreasing on[2,∞) ([n/2,∞)) provided thatR>sup_Ω|x|. Hence it suffices to assumeγ=2(γ=n/2)in noncritical case (critical case).

Remark 2.3. In the proof of the noncritical case, we will use decreasing rearrangement argument. Hence, the functiong(r) =r^−2p(log(R/r))⁻²should be monotone decreasing andR≥re^1/p. ThenK=e^1/p.

Remark 2.4. In the proof of the critical case, we will also use decreasing rearrangement argument. Hence, the function g^∗(r) =r⁻ⁿ(log(R/r))^−(n/2)−1 should also be monotone decreasing andR≥re(1/2)+(1/n). Moreover we need the condition to absorb the error terms in the right hand side of (2.2) withC^∗>0. HenceK^∗≥e(1/2)+(1/n).

Remark 2.5. CandC^∗may depend onRandγin a weak sense. Sinceg(r)andg^∗(r)tends to zero asγ→∞orR→∞, we can takeCandC^∗to be bigger.

Corollary 2.1. Let 1<p<n/2, and let F_p=

(

f:Ω→R⁺

f ∈L^∞_loc(Ω\ {0})with lim sup

|x|→0

|x|^2pf(x)

log 1

|x|

2

<∞ )

.

If f ∈F_p, then there existsλ(f)>0 such that foru∈W₀^2,p(Ω) (2.3)

Z

Ω

|∆u|^pdx≥Λn,p Z

Ω

|u(x)|^p

|x|^2p dx+λ(f) Z

Ω

|u(x)|^pf(x)dx

Iff ∈/F_pand if|x|^2pf(x) (log(1/|x|))²tends to∞as|x| →0, then no inequality of type (2.3) can hold.

3. Preliminary lemmas

In this section we shall prepare fundamental lemmas which are needed to prove our main results.

Lemma 3.1. For any R>1, q≤0,ν∈(0,1)satisfying2ν−1+q=0, (3.1)

Z 1 0

|h⁰(r)|²

logR r

q

r dr≥ν² Z 1

0

|h(r)|²

logR r

q−2

dr r holds for any h∈C([0,1])∩C¹(0,1), with h(0) =h(1) =0.

Proof. Leth(r) = (log(R/r))^νw(r). Thenw(0) =w(1) =0.

|h⁰(r)|²=

logR r

(ν−1)2

−ν

rw(r) +w⁰(r)logR r

2

≥ν r

2

|h(r)|²

logR r

−2

−ν r

d

drw²(r) logR r

2ν−1

. Hence for 2ν−1+q=0, we have

Z 1 0

|h⁰(r)|²

logR r

q

r dr≥ν² Z 1

0

|h(r)|²

logR r

q−2

dr r .

In the next lemma, we assume−∆u>0 andu>0 since we will use symmetric decreasing rearrangement in the proof of the main theorem. See also Lemma 3.4.

Lemma 3.2. Assume that f ∈C²(B₁) and u∈C₀²(B₁) are radial satisfying f(r)>0,

∆f(r)≤0, u(r)>0, and−∆u>0, where r=|x|. Set u(r) =f(r)v(r). Then Z

B₁

|∆u|ⁿ²dx≥ωn Z 1

0

rⁿ⁻¹|v(r)∆f(r)|ⁿ²dr

+n(n−2) 4 ω_n

Z 1 0

(v⁰(r))²vⁿ⁻⁴² (r)rⁿ⁻¹|∆f(r)|ⁿ⁻²² f(r)dr

+ω_n Z ₁

0

vⁿ²(r)∂_r rⁿ⁻¹

|∆f(r)|ⁿ⁻²² f⁰(r)−∂_r|∆f(r)|ⁿ⁻²² f(r) dr (3.2)

Proof. Note that∆u= f∆v+2v⁰f⁰+v∆f <0 andv(r)∆f(r)≤0. By the inequality(1+ x)^p≥1+px(x>−1), we have

|∆u|ⁿ² =|v(r)∆f(r)|ⁿ²

1+ f(r)∆v(r) +2f⁰(r)v⁰(r) v(r)∆f(r)

n 2

≥ |v(r)∆f(r)|ⁿ²

1+n 2

f(r)∆v(r) +2f⁰(r)v⁰(r) v(r)∆f(r)

and Z

B₁

|∆u|ⁿ²dx≥ω_n Z 1

0

rⁿ⁻¹|v∆f|ⁿ²dr

−n 2ω_n

Z ₁ 0

rⁿ⁻¹|v(r)∆f(r)|ⁿ⁻²² f(r)∆v(r) +2f⁰(r)v⁰(r) dr.

Then by using the integration by parts, we get the desired result.

The next lemma is the critical case of Theorem 2.1, whereΩ=B₁anduis radial.

Lemma 3.3. Consider positive radially nonincreasing function u∈C²₀(B₁). Then there exist K^∗=K^∗(n)>0and C^∗=C^∗(n)>0such that if R>K^∗, then

Z

B1

|∆u|ⁿ²dx≥ n−2

√n nZ

B1

|u(x)|ⁿ²

|x|ⁿ

log R

|x|

−ⁿ

2

dx

+C^∗ Z

B₁

|u(x)|ⁿ²

|x|ⁿ

log R

|x|

−ⁿ₂−1

dx.

(3.3)

A sketch of the proof of Lemma 3.3. Foru(r) =f(r)v(r), we setf(r) = (logR/r)^a, 0<a<

1. Then we use Lemma 3.1 and Lemma 3.2 to prove inequality (3.3).

We recall the rearrangement of domains and functions. For a domainΩ, we define a ballΩ^∗such that|Ω^∗|=|Ω|with center at the origin. We denote by|u|^∗ the symmetric decreasing rearrangement of functionu. It is well-known that the symmetric rearrangement does not change theL^p-norm and increases the integral^R_Ω(|u|^p)/(|x|^2p)dx. The following lemma is due to G. Talenti. As for the proof refer to [5] and [7].

Lemma 3.4. [7]LetΩbe a domain onRⁿ, n≥3and f∈C^∞₀(Ω). If u is the weak solution of Dirichlet problem−∆u= f inΩ, u|_∂Ω=0; v is the weak solution of Dirichlet problem

−∆v=|f|^∗inΩ^∗, v|_∂Ω^∗ =0; then v≥ |u|∗in pointwise.

From this lemma we see thatu^∗≤vinΩ^∗and Z

Ω

|∆u|^pdx= Z

Ω

|f|²dx= Z

Ω^∗

|f|^∗2dx Z

Ω^∗

|∆v|^pdx.

Further we see that Z

Ω

|u|^p

|x|^2pdx≤ Z

Ω^∗

|u^∗|^p

|x|^2pdx≤ Z

Ω^∗

|v|^p

|x|^2pdx.

Hence we have

R

Ω|∆u|^pdx R

Ω

|u|^p

|x|^2pdx

≥ R

Ω^∗|∆v|^pdx R

Ω^∗

|v|^p

|x|^2pdx .

Using this lemma we can assume thatuis radial andΩis a ball in the proof of the main results.

4. Proof of main results

We are now ready to give the proof of Theorem 2.1. We organize the proof in the following way: First for noncritical case (1<p<n/2), we prove inequality (2.1). Then we show the sharpness ofΛ_n,p, and then we show the optimality ofγ. Secondly for critical case (p=ⁿ₂), we first prove Inequality (2.2). Then we show the sharpness of((n−2)/√

n)ⁿ, and then we show the optimality ofγ.

Proof of Theorem 2.1. Case 1 (1<p<n/2): Let 1<p<n/2 andγ≥2. It suffices to prove (2.1) for smooth positive radially nonincreasing functionu defined on a ballB₁and from Remark 3.2 we can further assumeγ=2. Foru∈C₀^∞(B₁),u>0, radially nonincreasing, we define

(4.1) v(r) =u(r)r^(n/p)−2, r=|x|.

From Lemma 3.4, we may assume−∆u>0 and

∆(u(r)) =∆

v(r)r^2−np

=r^2−np

δ_β(v(r))−αv(r) r²

where

δ_β(v(r)) =∆_β(v(r)) =v”(r) +β−1 r v⁰(r), β=n+4−2n

p and α=(n−2p)(np−n)

p² .

Then by using(1+x)^p≥1+px (x>−1) and Lemma 3.1 (ν=1/2,q=0), we get Inequality (2.1) whereC= ((Λ_n,p)/α)((p−1)/p).

We construct a family of functions inW₀^2,p(B₁)which we will use to show the sharpness ofΛn,pand optimality ofγ. Forε>0 sufficiently small, let us define

(4.2) u_ε=







0, 0<r<ε²

r^1−np log¹

ε

⁻¹ log ^r

ε², ε²<r<ε r^1−np log¹

ε

⁻¹

log¹_r, ε<r<1

Letw_ε=^R_r¹u_ε(ρ)dρ. Then we get (4.3)

Z

B₁

|∆w_ε|^pdx= 2 p+1

n(p−1) p

p

ω_nlog1 ε+O

log1

ε −1!

,

(4.4)

Z

B1

|w_ε|^p

|x|^2p dx≥ 2 p+1

p n−2p

p

ωnlog1 ε+O

log1

ε −1!

.

Also we get (4.5)

Z

B₁

|w_ε|^p

|x|^2p

log R

|x|

−γ

≥O

log1 ε

1−γ! .

SHARPNESS OFΛn,p.The sharpness ofΛn,pwill follow if we can show that

(4.6) inf

u∈W₀^2,p(B1)\{0}

I(u):=

R

B1|∆u|^pdx R

B₁

|u|^p

|x|^2pdx

=Λn,p

Using the family of functionswε∈W₀^2,p(B₁)and from (4.3) and (4.4), we get

ε→0limI(w_ε)≤Λn,p

Also by Rellich inequality, we get lim

ε→0I(w_ε)≥Λn,p, hence lim

ε→0I(w_ε) =Λn,p. Thus sharp- ness follow.

OPTIMALITY OFγ. Suppose that 0≤γ<2. SinceΛn,pis the best constant for inequality (1.1), inequality (2.1) follows for the caseγ=0. So we assume 0<γ<2. Optimality will follow if we can prove that

(4.7) inf

u∈W₀^2,p(B1)\{0}

I_γ(u) = R

B₁|∆u|^pdx−Λn,pR B₁

|u|^p

|x|^2pdx R

B1

|u|^p

|x|^2p

log_|x|^R−γ

dx

=0.

Since 0<γ<2, from (4.3), (4.4) and (4.5),I_γ(w_ε)→0 asε→0 and hence optimality follows.

Case p=n/2: Let us assume p=n/2 andγ ≥n/2. From Remark 2.2, we can further assumeγ=n/2. In this case, inequality (2.2) follows immediately from Lemma 3.3 together with the rearrangement of function and domain arguments.

SHARPNESS. To show the sharpness of((n−2)/√

n)ⁿ, we consider the test function z_ε=

log R

r+ε ⁿ⁻²_n

−

log R 1+ε

ⁿ⁻²_n .

Then it is easy to verify by similar calculation as in the previous case(1<p<n/2)that

ε→0lim R

B₁|∆z_ε|ⁿ²dx R

B₁

|z_ε|ⁿ2

|x|ⁿ

log_|x|^R−γ

dx

= n−2

√n n

.

Hence sharpness follow.

OPTIMALITY.To show the optimality, we use the same test functionu_ε defined in (4.2) withp=n/2. Then forw_ε=^R_r¹u_ε(ρ)dρ, it is easy to verify by similar calculation as in the previous case(1<p<n/2)that

(4.8)

Z

B₁

|∆w_ε|ⁿ²dx= 4

n+2(n−2)ⁿ²ω_nlog1 ε+O

log1

ε −1!

and (4.9)

Z

B1

|w_ε|ⁿ²

|x|ⁿ

log R

|x|

−γ

dx≥O

log1 ε

ⁿ₂+1−γ!

Suppose that 0<γ<n/2. Optimality will follow if we can prove that inf

u∈W^2,

n 0 2(B₁)\{0}

I_γ(u) = R

B₁|∆u|ⁿ²dx R

B₁

|u|ⁿ2

|x|ⁿ

log_|x|^R−γ

dx

=0

Since 0<γ<n/2, from (4.8) and (4.9),I_γ(w_ε)→0 asε→0 and hence optimality follows.

Proof of Corollary 2.1. If f ∈F_p, then

ε→0limsup

x∈B_ε

f(x)|x|^2p

log 1

|x|

2

<∞

and hence for sufficiently smallε, inB_ε, f(x)<C|x|^−2p

log 1

|x|

−2

.

OutsideBε, both are bounded functions and henceCcan be chosen so that this inequality holds inΩ. Then (2.3) will follow from (2.1).

If f ∈/F_pand if|x|^2pf(x) (log(1/|x|))²tends to∞as|x| →0, then we can write f(x) = h(x)/

|x|^2p

log_|x|¹2

, whereh(x)tends to infinity asxtends to 0. Then from the calcu- lation of Theorem (2.1) forε>0 sufficiently small, we get

(4.10)

Z

B1

|w_ε|^ph(x)

|x|^2p

log_|x|¹2dx≥O

log1 ε

−1! m_ε

wherem_ε =min{inf_B

ε2h(x),inf_Bε\B

ε2h(x),inf_B1\Bεh(x)}. Sincem_ε tends to∞asε→0, we conclude thatI_f(w_ε)→0 asε→0 and Inequality (2.3) cannot hold for such f∈/F_p. 5. Application

Consider the weighted eigenvalue problem with a singular weight

∆ |∆u|^p−2∆u

− µ

|x|^2p|u|^p−2u=λ|u|^p−2u f inΩ u=∆u=0 on∂Ω (5.1)

wheref ∈Fpand Fp:=

f :Ω→R⁺

lim

|x|→0|x|^2pf(x) =0 with f∈L^∞_loc Ω¯\ {0}

,

1<p<n/2, 0≤µ<Λn,pandλ ∈R. We look for a weak solutionu∈W =W^2,p(Ω)∩ W₀^1,p(Ω)of this problem and study the asymptotic behaviour of the first eigenvalues for different singular weights as µincreases to Λn,p, after which the operatorL_µ is no more bounded from below. Here we define weak solution in the following way.

Definition 5.1. u∈W is said to be a weak solution of (5.1) iff for anyφ∈C²(Ω)¯ withφ=0 on∂Ω

Z

Ω

|∆u|^p−2∆u∆φ− µ

|x|^2p|u|^p−2uφ

dx=λ Z

Ω

|u|^p−2u fφdx.

Lemma 5.1. Forµ^∗<Λn,pand u∈W , there is a v∈W such that v>0and satisfies R

Ω|∆u|^pdx−µ^∗R_Ω ^|u|

p

|x|^2pdx R

Ω|u|^pf dx ≥ R

Ω|∆v|^pdx−µ^∗R_Ω ^|v|

p

|x|^2pdx R

Ω|v|^pf dx .

Remark 5.1. Sinceλ is first eigenvalue anduis the corresponding eigenfunction, by using Lemma 5.1, we can assumeu>0 inΩ. Then by the elliptic regularity theory,uis smooth near the boundary. From the definition of weak solution, one can derive the boundary condition of (5.1) by using integration by parts.

Theorem 5.1. The above problem admits a positive weak solution u∈W for all1<p<n/2 and0≤µ<Λn,p, corresponding to the first eigenvalueλ =λ_µ¹(f)>0. Moreover, asµ increases toΛn,p,λ_µ¹(f)→λ(f)≥0for all f ∈Fpand the limitλ(f)>0if f ∈F_p. If

f ∈/F_pand if|x|^2pf(x) (log(1/|x|))²tends to∞as|x| →0, then the limitλ(f) =0.

In order to prove the theorem we need the following results; the first one is a standard result from measure theory (see [4], Chapter 1, section 4) and the second one is due to Boccardo and Murat [4].

Lemma 5.2. Let(g_m)_m∈N⊂L^p(Ω),1≤p<∞, be such that, as m→∞, (i) gm*g weakly in L^p(Ω)and (ii) g_m(x)→g(x)a.e. inΩ. Then

m→∞lim kg_mk^p_p− kg_m−gk^p_p

=kgk^p_p.

Lemma 5.3(see Remark 2.7 in [2]).

|∆u_m|^p−2∆u_m− |∆u|^p−2∆u

∆(u_m−u)≥

(s|∆(u_m−u)|^p if p≥2;

s|∆(u_m−u)|²

(|∆u_m|+|∆u|)^2−p if p≤2, for some s>0.

Remark 5.2. In the proof of Theorem 5.1,uwill be characterized as a solution of variational problem defined in (5.2) and (5.1) becomes Euler-Lagrange equation of this variational problem.

Proof of Theorem 5.1. We define the functional

(5.2) J_µ(u) =

Z

Ω

|∆u|^p−µ|u|^p

|x|^2p

dx

By the Hardy inequality, one can check thatJµ is continuous, Gateaux differentiable and coercive onW. We minimize this functional over the manifold

M=

u∈W Z

Ω

|u(x)|^pf(x)dx=1

,

and letλ_µ¹>0 be the infimum. By standard arguments, we can choose (u_m)m∈N⊂M a minimizing sequence ofJµ such thatJµ(u_m)→λ_µ¹andJ_µ⁰(u_m)→0 strongly inW⁰. The coercivity ofJ_µ implies that(u_m)_m∈Nis a bounded sequence and hence we have for a sub- sequence, ask→∞,u_mk*uweakly inW. Moreover, we can also assume the following:

∆u_mk*∆uweakly inL^p(Ω), u_mk*uweakly inL^p(Ω,|x|^−2p), u_mk→ustrongly inL^p(Ω,f), um_k→ustrongly inL^p(Ω),

∇u_mk→∇ustrongly inL^p(Ω).

(5.3)

SinceW is compactly embedded inL^p(Ω,f), it follows thatMis weakly closed and hence u∈M. Alsou_mksatisfies inD⁰(Ω)that

∆ |∆u_mk|^p−2∆u_mk

= µ

|x|^2p|u_mk|^p−2u_mk+λ_mk|u_mk|^p−2u_mkf+f_mk

where f_mk→0 strongly inW⁰andλm_k→λ asm_k→∞. Letφ∈C^∞(Ω)such that for any ballB_ρ,B_ρ∩suppφ=/0 and

(5.4) φ=

(0, |x|<ρ; 1, |x|>2ρ.

Using the test functionv_mk=φ(u_mk−u)∈W, we obtain Z

Ω

|∆u_mk|^p−2∆um_k− |∆u|^p−2∆u

∆vm_kdx

=− Z

Ω

|∆u|^p−2∆u∆v_mkdx+µ Z

Ω

|u_mk|^p−2u_mk

|x|^2p v_mkdx +λ_mk

Z

Ω

|u_mk|^p−2u_mkf v_mkdx+ Z

Ω

f_mkv_mkdx

From (5.3) one deduces that v_mk →0 weakly in W and L^p(Ω,|x|^−2p), and strongly in L^p(Ω,f)andL^p(Ω). Hence form_k→∞and by expanding∆vm_k, we get

(5.5) lim

m_k→∞

Z

Ω

e_mkφ=0, wheree_mk= |∆u_mk|^p−2∆u_mk− |∆u|^p−2∆u

∆(u_mk−u).

Fixθwith 0<θ<1. Then Z

Ω

e^θ_m

k=

Z

Ω

e^θ_m

kφ+ Z

Ω

e^θ_m

k(1−φ)

≤ Z

Ω

e^θ_m

kφ+ _Z

Ω

e_mk θ_Z

Ω

(1−φ)^1−θ1 1−θ

. From (5.4) and (5.5), we get

mlim_k→∞

Z

Ω

e^θ_m

k≤C_θρ^n(1−θ). which by lettingρtends to 0 implies that for any subsequencee⁰_m

kofe_mk,

(5.6) e⁰_m

k→0 a.e. inΩ We apply Lemma 5.3 on (5.5) and get

(5.7) ∆u_m⁰

k(x)→∆u(x) a.e.x∈Ω.

We apply Lemma 5.2 onu_mk and also on∆u_mkto obtain λ_µ¹=k∆(u_mk−u)k^p_p−µku_mk−uk^p

L^p(Ω,|x|^−2p)+k∆uk^p_p−µkuk^p

L^p(Ω,|x|^−2p)+o(1)

≥(Λ_n,p−µ)ku_mk−uk^p

L^p(Ω,|x|^−2p)+λ_µ¹+o(1),

whereo(1)→0 asm_k→∞. Asµ<Λn,p,we conclude thatku_mk−uk^p

L^p(Ω,|x|^−2p)→0 as m_k→∞and alsok∆(u_mk−u)k_L^pp(Ω)→0 asm_k→∞and hence we haveJµ(u) =λ_µ¹ and λ=λ_µ¹. Sinceuis the first eigenfunction, it is not difficult to show by using Lemma 5.1 that u>0 inΩ. Then by the Euler-Lagrange equation,uis a distribution solution of (5.1) and sinceu∈W, it is a weak solution to the eigenvalue problem (5.1), corresponding toλ=λ_µ¹. The remaining part of the proof follows from the corollary of the main theorem.

References

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