A Class of --Laplacian Type Equation with Potentials Eigenvalue Problem in

Volume 2009, Article ID 185319,19pages doi:10.1155/2009/185319

Research Article

A Class of

p

-

q

-Laplacian Type Equation with

Potentials Eigenvalue Problem in

R

N

Mingzhu Wu

_{and Zuodong Yang}

1_{School of Mathematics Science, Institute of Mathematics, Nanjing Normal University, Jiangsu,} Nanjing 210097, China

2_{College of Zhongbei, Nanjing Normal University, Jiangsu, Nanjing 210046, China}

Correspondence should be addressed to Zuodong Yang,zdyang [email protected]

Received 20 October 2009; Accepted 6 December 2009

Recommended by Wenming Zou

The nonlinear elliptic eigenvalue problem −div|∇u|p−2_∇

u −div|∇u|q−2_∇

u λax|u|p−2

u

λbx|u|q−2ufx, u, u∈W1,p_∩W1,q_RN_{, where 2≤q≤p < Nandax∈L}N/p_RN_{, bx∈}

LN/q_RN_{, ax, bx > 0 are studied. The key ingredient is a special constrained minimization}

method.

Copyrightq2009 M. Wu and Z. Yang. 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.

1. Introduction

In this paper, we are interested in finding nontrivial weak solutions for the nonlinear eigenvalue problem

−div|∇u|p−2∇u−div|∇u|q−2∇uax|u|p−2ubx|u|q−2ufx, u,

u∈W1,p_∩W1,q_RN_{, u /0,}

1.1

where 2 ≤ q ≤ p < N and ax ∈ LN/p_RN_{, bx ∈ L}N/q_RN_{, ax, bx > 0,}

infax,infbx/0,fx, usatisfy the following conditions:

Af ∈CRN_{×R,R, lim}

t→0fx, t/|t|p−1 0,and lim|t| → ∞fx, t/|t|p−1p2/N 0

uniformly inx∈RN_,

Remark 1.1. We can see ifax∈LN/p_RN_{, bx∈L}N/q_RN_{, then}

RN

ax|u|pdx <

RN

axN/p

1−p/p∗

RN

up∗

p/p∗

<∞,

RNbx|u| q

dx <

RNbx

N/q1−q/q

∗

RNu q∗

q/q∗

<∞,

1.2

wherep∗Np/N−pandq∗Nq/N−q.

Problem1.1comes, for example, from a general reaction-diffusion system:

utdivDu∇u cx, u, 1.3

whereDu |∇u|p−2|∇u|q−2. This system has a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. In such applications, the functionudescribes a concentration, the first term on the right-hand side of 1.3corresponds to the diffusion with a diffusion coefficientDu; whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological applications, the reaction termcx, uis a polynomial ofuwith variable coefficients.

When p q 2, problem1.1is a normal Schrodinger equation which has been extensively studied, for example,1–8. The authors used many different methods to study the equation. In8, the authors established some embedding results of weighted Sobolev spaces of radially symmetric functions which are used to obtain ground state solutions. In 6, the authors studied the equation depending upon the local behavior ofV near its global minimum. In3, the authors used spectral properties of the Schrodinger operator to study nonlinear Schrodinger equations with steep potential well. In9, the author imposed on functionskandKconditions ensuring that this problem can be written in a variational form. We know thatW1,p_RN_{is not a Hilbert space for 1 < p < N, except for p 2. The space}

W1,p_RN_{withp /2 does not satisfy the Lieb lemmae.g., see9. AndR}N_{results in the loss}

of compactness. So there are many difficulties to study equation1.1ofpq /2 by the usual methods. There seems to be little work on the casep q /2 for problem1.1, to the best of our knowledge. In this paper, we overcome these difficulties and study1.1ofp≥q≥2.

Recently, whenp q, ax bx,andfx, u 0 then the problem is the following eigenvalue problem has been studied by many authors:

−div|∇u|p−2∇uVx|u|p−2u,

u∈D1₀,pΩ, u /0,

1.4

whereΩ ⊆RN_{. We can see10–13. In13, Szulkin and Willem generalized several earlier}

Whenp qandax, bxis constant then the problem is the following quasilinear elliptic equation:

−div|∇u|p−2∇uλ|u|p−2ufx, u, inΩ,

u∈W₀1,pΩ, u /0,

1.5

where 1< p < N, N≥3,λis a parameter,Ωis an unbounded domain inRN_{. There are many}

results about it we can see14–18. Because of the unboundedness of the domain, the Sobolev compact embedding does not hold. There are many methods to overcome the difficulty. In 15, the authors used the concentration-compactness principle posed by P. L. Lions and the mountain pass lemma to solve problem1.5. In17,18, the authors studied the problem in symmetric Sobolev spaces which possess Sobolev compact embedding. By the result and a min-max procedure formulated by Bahri and Li16, they considered the existence of positive solutions of

−div|∇u|p−2∇uup−1qxuα inRN, 1.6

where qxsatisfies some conditions. We can see if λ is function, then it cannot easily be proved by the above methods.

Whenax, bxis positive constant, He and Li used the mountain pass theorem and concentration-compactness principle to study the following elliptic problem in19:

−div|∇u|p−2∇u−div|∇u|q−2∇um|u|p−2un|u|q−2ufx, u inRN_,

u∈W1,p∩W1,qRN, 1.7

wherem, n > 0, N ≥ 3,and 1 < q < p < N,fx, u/up−1 _{tends to a positive constantl as}

u → ∞. The authors prove in this paper that the problem possesses a nontrivial solution even if the nonlinearityfx, tdoes not satisfy the Ambrosetti-Rabinowitz condition.

In20, Li and Liang used the mountain pass theorem to study the following elliptic problem:

−div|∇u|p−2∇u−div|∇u|q−2∇u|u|p−2u|u|q−2ufx, u inRN_,

u∈W1,p_∩W1,q_RN_, 1.8

where 1< q < p < N. They generalized a similar result forp-Laplacian type equation in15. It is our purpose in this paper to study the existence of ground state to the problem 1.1inRN_{. We call any minimizer a ground state for1.1. We inspired by9,16,21try to}

use constrained minimization method to study problem1.1. Let us point out that although the idea was used before for other problems, the adaptation to the procedure to our problem is not trivial at all. But since bothp- andq-Laplacian operators are involved, careful analysis is needed. A typical difficulty for problem1.1inRN_{is the lack of compactness of the Sobolev}

imbedding due to the invariance ofRN _{under the translations and rotations. However, our}

results, we have to overcome two main difficulties; one is that RN _{results in the loss of}

compactness; the other is thatW1,p_RN _{is not a Hilbert space for 1 < p < N and it does}

not satisfy the Lieb lemma, except forp2.

The paper is organized as follows. InSection 2, we state some condition and many lemmas which we need in the proof of the main theorem. InSection 3, we give the proof of the main result of the paper.

2. Preliminaries

Let

Fx, t

t

0

fx, sds, Ft

t

0

fsds 2.1

and we define variational functionalsI:W1,p_∩W1,q_RN _{→ RandI}∞_:W1,p_∩W1,q_RN _{→ R}

by

Iu 1 p

RN|∇u| p

dx1 q

RN|∇u| q

dx−

RNF

x, udx,

I∞u 1 p

RN

|∇u|pdx1 q

RN

|∇u|qdx−

RN

Fudx.

2.2

Solutions to problem1.1will be found as minimizers of the variational problem

Iλinf Iu;u∈W1,p

RN_,

RNax|u| p

bx|u|qdxλ

, λ >0. Iλ

To find a solution of problemIλwe introduce thelimitvariational problem

I_λ∞inf I∞u;u∈W1,pRN,

RNax|u| p

bx|u|qdxλ

, λ >0. I_λ∞

Lemma 2.1. Letun⊆W₀1,pΩa bounded sequence andp≥2. Going if necessary to a subsequence,

one may assume thatun uinW₀1,pΩ,un → ualmost everywhere, whereΩ ⊆ RNis an open

subset. Then,

lim

n→ ∞

Ω|∇un|

p

dx≥ lim

n→ ∞

Ω|∇un− ∇u|

p

dx lim

n→ ∞

Ω|∇u|

p

Proof. Whenp2 from Brezis-Lieb lemmasee21, Lemma 1.32we have

lim

n→ ∞

Ω|∇un|

2

dx lim

n→ ∞

Ω|∇un− ∇u|

2

dx lim

n→ ∞

Ω|∇u|

2

dx, 2.4

when 3 ≥ p > 2, using the lower semicontinuity of theLp_{-norm with respect to the weak}

convergence andun uinW1,pΩ, we deduce

|∇un|p−2∇un,∇un

≥|∇u|p−2∇u,∇uo1,

lim

n→ ∞

|∇un− ∇u|p−2∇un,∇un

≥ lim

n→ ∞

|∇un− ∇u|p−2∇un,∇u

lim

n→ ∞

|∇un− ∇u|p−2∇u,∇un

lim

n→ ∞

|∇un− ∇u|p−2∇u,∇u

.

2.5

Then,

lim

n→ ∞

Ω

|∇un|p− |∇u|p

dx

lim

n→ ∞

Ω|∇un|

p−2_|∇

un|2− |∇u|2

dx lim

n→ ∞

Ω

|∇un|p−2− |∇u|p−2

|∇u|2dx

lim

n→ ∞

Ω

|∇un|p−2|∇u|p−2

|∇un|2− |∇u|2

dx

lim

n→ ∞

Ω

|∇un|p−2|∇u|2− |∇u|p−2|∇un|2

dx.

2.6

Fromun uinW1,pΩ,

lim

n→ ∞

Ω

|∇un|p−2|∇u|2− |∇u|p−2|∇un|2

dx0. 2.7

So

lim

n→ ∞

Ω

|∇un|p− |∇u|p

dx lim

n→ ∞

Ω

|∇un|p−2|∇u|p−2

|∇un|2− |∇u|2

dx

≥ lim

n→ ∞

Ω|∇un− ∇u|

p−2_|∇

un|2− |∇u|2

.

So we have

|∇un|p−2∇un,∇un

|∇un− ∇u|p−2∇u,∇un

|∇un− ∇u|p−2∇un,∇u

≥|∇un− ∇u|p−2∇un,∇un

|∇un− ∇u|p−2∇u,∇u

|∇u|p−2∇u,∇uo1.

2.9

Then,

|∇un|p−2∇un,∇un

≥|∇un− ∇u|p−2∇un− ∇u,∇un− ∇u

|∇u|p−2∇u,∇uo1

lim

n→ ∞

Ω|∇un|

p_{dx≥ lim} n→ ∞

Ω|∇un− ∇u|

p_{dx lim} n→ ∞

Ω|∇u|

p_dx,

2.10

whenp >3, there exists ak∈Nthat 0< p−k≤1. Then, we only need to prove the following inequality:

lim

n→ ∞

Ω

|∇un|p− |∇u|p

dx≥ lim

n→ ∞

Ω|∇un− ∇u|

p−k_|∇u

n|k− |∇u|k

. 2.11

The proof of it is similar to the above, so we omit it here. So, the lemma is proved.

Lemma 2.2. Let{un}be a bounded sequence inW1,pRNsuch that

lim

n→ ∞_y∈sup_RN

By,R

uqndx0, p≤q < p∗ 2.12

for someR >0. Thenun → 0inLsRNforp < s < p∗, wherep∗Np/N−p.

Proof. We consider the caseN ≥ 3. Letq < s < p∗ andu ∈ W1,p_RN_{. Holder and Sobolev}

inequalities imply that

|u|Ls_By,R≤ |u|1_L−λq_By,R|u|λ_Lp∗_By,R

≤C|u|1L−λq_By,R

By,R

|u|p|∇u|p

λ/p

,

2.13

whereλ s−q/p∗−qp∗/s. Choosingλp/s, we obtain

By,R|u|

s_≤

Cs|u|_L1q−λ_Bs_y,R

By,R

Now, coveringRN_{by balls of radiusr, in such a way that each point ofR}N_{is contained}

in at mostN1 balls, we find

RN|u| s_≤

N1Cssup

y∈RN

By,R|u|

q

1−λs/q

By,R

|u|p|∇u|p. 2.15

Under the assumption of the lemma,un → 0 inLsRN, p < s < p∗. The proof is

complete.

Corollary 2.3. Let{um}be a sequence inW1,pRNsatisfying0 < ρ

RN|um|pdxand such that

um 0inW1,pRN. Then there exist a sequence{ym} ⊂ RN and a function0/u∈ W1,pRN

such that up to a subsequenceum·ym uinW1,pRN.

Lemma 2.4. Letf ∈CRN_{×Rand suppose that}

lim

|s| → ∞

fx, s

|s|p∗−1 0 2.16

uniformly inx∈RN_and

_{fx, s≤C|s|}p−1_|

s|p∗−1 2.17

for allx∈RNandt∈R. Ifum u0inW1,pRNandum → u0a.e. onRN, then

lim

m→ ∞

RNFx, umdx−

RNFx, u0dx−

RNFx, um−u0dx

0, 2.18

whereFx, u u₀fx, tdt.

Proof. LetR >0. Applying the mean value theorem we have

RNFx, umdx

|x|≤RFx, umdx

|x|≥RFx, u0 um−u0dx

|x|≤RFx, umdx

|x|≥R

Fx, um−u0 fx, θu0 um−u0u0

dx,

whereθdepends onxandRand satisfies 0< θ <1. We now write

RNFx, umdx−

RNFx, u0dx−

RNFx, um−u0dx

≤

|x|≤RFx, um−Fx, u0dx

|x|≥RFx, u0dx

|x|≤RFx, um−u0dx

|x|≥Rfx, θu0 um−u0u0dx

.

2.20

For each fixedR >0

lim

m→ ∞

|x|≤RFx, um−Fx, u0dx0,

lim

m→ ∞

|x|≤RFx, um−u0dx0.

2.21

Applying2.20and the Holder inequality we get that

|x|≥Rfx, θu0 um−u0u0dx

≤C

|x|≥R

|θu0 um−u0|p−1|u0||θu0 um−u0|p ∗₋₁

|u0|dx

≤C

|x|≥R|u0| p

1/p

|x|≥R|θu0 um−u0| p

p−1/p

C

|x|≥R|u0| p∗

1/p∗

|x|≥R|θu0 um−u0| p∗

p∗−1/p∗

.

2.22

Since{um}is bounded inW1,pRNwe see that

lim

R→ ∞

|x|≥Rfx, θu0 um−u0u0dx

0. 2.23

The result follows from2.21and2.23.

Lemma 2.5. FunctionsIλandI_λ∞are continuous on0,∞and minimizing sequences for problems

Proof. From conditionA, we observe that for eachε >0 there existsCε>0 such that

Fu,|Fx, u| ≤ε

RN|u| p

dxε

RN|u| pp2_/N

dxCε

RN|u| α

dx, 2.24

wherep < α < pp2_{/Nandε >0.}

By the Holder and Sobolev inequalities we have

RN|u| pp2_/N

dx

RN|u|

pp∗−p−p2_/N/p∗_−pp∗_p2_/N/p∗_−p

dx

≤

RN|u| p

p∗−p−p2_/N/p∗_−p

RN|u| p∗

p2_/N/p∗_−p

≤S−1

RN|u| pp/N

RN|∇u| p

dx,

2.25

where|u|p_p∗≤S−1|∇u|pp.

Similarly we have

RN|u| α_dx

RN|u|

pp∗−α/p∗−pp∗α−p/p∗−p_dx

≤

RN|u| p

dx

p∗−α/p∗−p

RN|u| p∗

dx

α−p/p∗−p

≤S−p∗α−p/pp∗−p

RN|u| p

dx

p∗−α/p∗−p

RN|∇u| p

dx

p∗α−p/pp∗−p

.

2.26

Consequently by the Young inequality we have

RN|u| α

dx≤η

RN|∇u| p

dxKη RN|u|

α

dx

pp∗−α/p2_p∗_−p2_−p∗_α

2.27

forη >0, whereKη>0 is a constant.

Becauseu∈W1,p∩W1,q_RN_{so we can by Sobolev embedding andλ}

RNax|u|p

bx|u|qdxlettingλ_RN|u|pdx <∞, we derive the following estimates forIuandI∞u:

Iu, I∞u≥

1

p−εS

−1_λp/N_−C εη

RN

|∇u|pdx

1

q

RN|∇u|

q_{dx−ελ−KηC} ελpp

∗_−α/p2_p∗_−p2_−p∗_α

.

Choosingε >0 andη >0 so that

1

p −εS

−1_λp/N_−C

εη >0, 2.29

we see thatIλandI_λ∞ are finite and moreover minimizing sequences for problemsIλand I_λ∞are bounded. It is easy to check thatIλandI_λ∞are continuous on0,∞.

We observe thatIμ∞≤0 for allμ >0. Indeed, letu∈C∞0RNand

RNax

u_σx/σN/q

pdx

RNbx

u_σx/σN/q

qdxμ, 2.30

then for eachσ >0 we have

Iμ∞≤

1

pσpp/q−1N

RN|∇u| p

dx 1 qσq

RN|∇u| q

dx−σN

RNF

σ−N/qudx−→0 2.31

asσ → ∞.

Lemma 2.6. Suppose that I_λ∞ < 0 for someλ > 0, then Iμ∞/μ is nonincreasing on 0,∞ and

limμ→0I_μ∞/μ 0. Moreover there existsλ∗≤λsuch that

Iμ∞

μ > I_λ∞

λ forμ∈0, λ

∗_{. 2.32}

Proof. We observe that

inf I

∞_u

RNax|u|pbx|u|qdx

inf I

∞_ux/σ1/N

RNa

x/σ1/N_ux/σ1/Np_dxbx/σ1/N_ux/σ1/Nq_dx.

2.33

So if _RNax|u|

p

bx|u|qdx k and _RNax/σ1/N|ux/σ1/N|pdx bx/σ1/N|ux/

σ1/N|qdxkthenI∞ux I∞ux/σ1/N I_k∞.

We have that ifσ >0 andα >0 with_RNax|u|pbx|u|qdxα, then

RN

a

x σ1/N

u

x σ1/N

pdxb

x σ1/N

u

x σ1/N

qdxσα, I∞

u

x σ1/N

Iσα∞.

Consequently, for allα1>0 andα2>0 we have

Iα∞1 inf 1 p α1 α2

_N−p/N

RN

|∇u|pdx1 q

α1

α2

_N−q/N

RN

|∇u|qdx−α1 α2

RN

Fudx;

RN

ax|u|pbx|u|qdxα2

.

2.35

If 0< α1 < α2, then for eachε >0 there existsu∈W1,p∩W1,qRNwith

RNax|u| p

bx|u|qdxα2such that

Iα∞1ε > 1

p

α1

α2

_N−p/N

RN

|∇u|pdx1 q

α1

α2

_N−q/N

RN

|∇u|qdx−α1 α2

RN

Fudx

≥ α1

α2

1

p

RN|∇u| p_dx1

q

RN|∇u| q_dx−

RNFudx

≥ α1

α2I ∞ α2.

2.36

This inequality yields

Iα∞1

α1 >

Iα∞2

α2.

2.37

SinceIμ∞≤0 for allμ >0, we see that

lim

μ→0

Iμ∞

μ c≤0. 2.38

We claim thatc0. Indeed, it follows from2.36and from the estimate obtained in the

Lemma 2.1that for every 0< μ < λthere exists anuμ∈W1,p∩W1,qRN, with

RNax|uμ|p

bx|uμ|qdxλsuch that

Iμ∞μ2>

1

p

_μ

λ

N−p/N

RN

_∇uμp

dx1 q

_μ

λ

N−q/N

RN

_∇uμq

dx− μ λ

RNF

uμ dx ≥ μ λ 1 p RN

_∇uμp

dx1 q

RN

_∇uμq

dx−

RNF

uμ dx ≥ μ λ

C1λ

RN

_∇uμp

dxC2λ

RN

_∇uμq

dx−C3λ

,

2.39

whereC1λ>0, C2λ>0,andC3λ>0 are constants. Hence

μ2≥ μ λ

C1λ

RN

_∇uμp

dxC2λ

RN

_∇uμq

dx−C3λ

that is, _RN|∇uμ|pdx ≤ C4λ,

RN|∇uμ|pdx ≤ C5λ for some constant C4λ, C5λ > 0

independent ofμ. We see that there existsε0>0 and a sequenceμn → 0 such that

RN

_∇uμnp

dx≥ε0,

RN

_∇uμnq

dx≥ε0. 2.41

If_RN|∇uμn|pdx≥ε0then

RN|∇uμn|qdx≥η≥0.

Then, using the fact that_RNFuμndx≤Cfor some constantC >0, we get

Iμn∞

μn μn≥

1

pλ

−N−p/N_μ

n−p/Nε0

1

qλ

−N−q/N_μ

n−q/Nη−C

λ −→ ∞ 2.42

asμn → 0 and this contradicts the fact that limμ→0Iμ∞/μ c≤0. Therefore

lim

μ→0

RN

_∇uμp

dx0, lim

μ→0I ∞

μ 0, 2.43

when_RN|∇uμn|pdx≥ε0>0 we can use the same method to obtain that limμ→0

RN|∇uμ|qdx

0.

So

lim

μ→0

RN

_∇uμp

dx lim

μ→0

RN

_∇uμq

dx0, lim

μ→0I ∞

μ 0, 2.44

this implies that limμ→0

RNFuμdx0 and consequently

Iμ∞

μ μ≥ −

1

λ

RNF

uμ

dx−→0. 2.45

This shows that limμ→0I_μ∞/μ 0. Finally, we observe that lim_μ→0I_μ∞/μ 0 > I_λ∞/λ which obtain2.32.

3. Proof of Main Theorems

Theorem 3.1. Suppose thatI_λ∞ <0for someλ >0, then there exists0< α0 ≤ λsuch that problem

Iα∞0has a minimizer. Moreover each minimizing sequence forI

∞

α0up to a translation is relatively compact inW1,p_∩W1,q_RN_.

Proof. According toLemma 2.6the set

α1; I ∞ α

α > I_λ∞

λ for eachα∈0, α1

is nonempty. We define

α0sup α1; I ∞ α

α > I_λ∞

λ for eachα∈0, α1

. 3.2

It follows from the continuity ofI_λ∞that

0< α0≤λ ,

Iα∞0

α0

λ I

∞ λ <0,

Iα∞>

α λI

∞ λ ,

3.3

for all 0< α < α0. This yields

Iα∞0

α0

λ I

∞

λ

α0−α

λ I

∞

λ

α λI

∞

λ < Iα∞0−αI

∞

α 3.4

for eachα∈0, α0.

Let{um} ⊂W1,p∩W1,qRNbe a minimizing sequence forIα∞0. Since{um}is bounded we may assume thatum uinW1,p∩W1,qRN,um → ua.e. onRN. First we consider the

caseu ≡ 0. In this case byLemma 2.2um → 0 forq < α < p∗ or Corollary there exists a

sequence{um} ⊂RNsuch thatum·ym v /0 inW1,p∩W1,qRN.

In the first case limm→ ∞_RNFumdx0 and consequently

Iα∞0 _mlim_{→ ∞}I

∞_u

m _{m→ ∞}lim

1

p

RN

|∇um|pdx

1

q

RN

|∇um|qdx−

RN

Fumdx

≥0, 3.5

which is impossible. Henceum·ym v /0 inW1,p∩W1,qRNholds and lettingvmx

umxymfrom Brezis-Lieb lemmasee21, Lemma 1.32we have

RN

ax|um|pbx|um|qdx

RN

axym

|vm|pb

xym

|vm|qdx

RN

axym

|v|pbxym

|v|qdx

RNa

xym

|vm−v|p

bxym

|vm−v|qdxo1.

3.6

We now show that

RNa

xym

|v|pbxym

In the contrary case fromLemma 2.1we have

0<

RNa

xym

|v|pbxym

|v|qdx < α0. 3.8

By3.21we have

lim

m→ ∞

RNa

xym

|vm−v|pb

xym

|vm−v|qdx−→α0−λ,

λ

RNa

xym

|v|pbxym

|v|qdx.

3.9

On the other hand, by Lemmas2.1and2.4we have

RNFvmdx

RNFvdx

RNFvm−vdxo

1,

RN

|∇vm|p|∇vm|qdx≥

RN

|∇v|p|∇v|qdx

RN

|∇vm−v|p|∇vm−v|qdxo1,

3.10

and this implies that

Iα∞0≥I

∞_{v I}∞_v

m−v o1≥Iλ∞Iα∞0−λ0o1. 3.11

Lettingm → ∞we getIα∞0 ≥I

∞

λ Iα∞0−λ0which contradicts3.4. Therefore

RNaxym|v|p

bxym|v|qdxα0. It then follows from3.6thatvm → vinLp∩LqRN. By the

Gagliardo-Nirenberg inequalityvm → vinLsRN,q≤s <∞. Obviously this implies thatIα∞0 I

∞_v

I∞v·−ymand

RNax|v·−ym|pbx|v·−ym|qdxα0. To complete the proof we show

thatvm → vinW1,p∩W1,qRN. Indeed, we have

Iα∞0 1

p

RN|∇vm| p

dx 1 q

RN|∇vm| q

dx−

RNFvmdx

o1

≥ 1

p

RN

|∇v|pdx1 q

RN

|∇v|qdx1 p

RN

|∇vm−v|pdx

1

q

RN

|∇vm−v|qdx

−

RNFvdx

RN

Fv−Fvm

dxo1.

3.12

Since limm→ ∞_RNFv−Fvmdx0, we deduce from3.12that∇vm → ∇vinLp∩LqRN

and hencevm → vinW1,p∩W1,qRN.

If_{u /}0, we repeat the previous argument to show thatIα∞0is attained.

Theorem 3.2. Suppose thatFx, t ≥ FtonRN _{×Rand thatI}

λ < 0for someλ > 0, then the

Proof. SinceFx, t≥FtonRN_{×Rwe haveI}

μ≤Iμ∞forμ≥0. We distinguish two cases:i

IλI_λ∞<0,iiIλ< I_λ∞.

Casei. ByTheorem 3.1there existsλ0∈0, λsuch that

I_λ∞₀ I∞u,

RN

ax|u|pbx|u|qdxλ0 for someu∈W1,p∩W1,q

RN_{. 3.13}

Thus

Iλ0 ≤Iu 1

p

RN|∇u| p

dx 1 q

RN|∇u| q

dx−

RNFx, udx

≤ 1

p

RN|∇u| p

dx 1 q

RN|∇u| q

dx−

RNFudxI

∞_{u I}∞ λ0.

3.14

IfIλ0 I_λ∞₀, thenI also attains its infimumIλ0 atu. Therefore it remains to consider the case

Iλ0< Iλ∞0. Consequently we need to prove the following claim.

IfIλ < I_λ∞for someλ >0, then there existsα0 ∈ 0, λsuch that problemIα0has a solution. This obviously completes the proof of caseiand also provides the proof of case ii.

By virtue ofLemma 2.5,IβI_λ−β∞ is continuous forβ ∈0, λand alsoI0 I₀∞ 0. If

Iλ< I_λ∞for someλ >0, then there existsγ >0 such that

Iλ< IβI_λ−β∞ 3.15

for allβ∈0, γ. Let

α0sup

γ; Iλ< IβI_λ−β∞ , for 0≤β < γ

. 3.16

Then we have

IλIα0Iλ−α∞ ₀,

Iλ< IαI_λ−α∞

3.17

for 0≤α < α0. This implies that

Iα0Iλ−α∞ ₀ Iλ< Iλ∞≤0, 3.18

and hence

Iα0 < Iλ∞−Iλ−α∞ 0≤I

∞

α0 ≤0, 3.19

we show thatIα0 is attained by au ∈ W1,p∩W1,qRNand every minimizing sequence for

Iα0is relatively compact inW

1,p_∩W1,q_RN_{. Let{u}

{um}is bounded, we may assume that um uin W1,p ∩W1,qRN,um u a.e. on RN.

Arguing indirectly we assume thatu≡0 onRN_{. We see that}

lim

m→ ∞

B0,R|Fx, um|dxm→ ∞lim

B0,R

Fumdx0 3.20

for eachR >0. We now write

Ium

1

p

RN

|∇um|pdx

1

q

RN

|∇um|qdx−

RN

Fx, umdx

I∞um

RN

Fum−Fx, um

dx.

3.21

We show that

lim

m→ ∞

RN

Fum−Fx, um

dx0. 3.22

Towards this end we write

RN

Fum−Fx, umdx

≤

B0,R|Fx, um|dx

B0,R

Fumdx

|x|≥R,|um|≤δ

|x|≥R,δ≤|um|≤1/δ

|x|≥R,|um|>1/δ

Fum−Fx, um.

3.23

We now define the following quantities:

1δ sup 0<|t|<δ,x∈RN

Ft−Fx, t

|t|p

R sup

δ≤|t|≤1/δ,|x|≥R

Ft−Fx, t,

2δ sup |t|≥1/δ

Ft−Fx, t

|t|pN/N−p .

It follows from assumption A that limδ→01δ limδ→02δ 0 and by

BlimR→ ∞R 0 for each fixedδ > 0. Inserting these quantities into3.23 we derive

the following estimate:

RN

Fum−Fx, umdx

≤1δ

RN|um| p

dxR δp

RN|um| p

dx

2δ

RN|um|

pN/N−p_dx

B0,R|Fx, um|dx

B0,R

Fumdx.

3.25

First lettingm → ∞,R → ∞,and thenδ → 0, relation3.22readily follows. Combining 3.21and3.22we obtain

Ium I∞um o1, 3.26

which implies Ium ≥ Iα∞0 o1 and consequently Iα0 ≥ Iα∞0 and this contradicts 3.19. Therefore 0<_RNax|u|pbx|u|qdx≤α0. Suppose thatλ

RNax|u|pbx|u|qdx < α0.

Writing

RN|∇um| p_|∇

um|qdx≥

RN|∇u| p_|∇

u|qdx

RN|∇um−u| p_|∇

um−u|qdxo1

RNax|um| p

bx|um|qdx

RNax|u| p

bx|u|qdx

RNax|um−u| p

bx|um−u|qdxo1,

3.27

RN

Fx, umdx

RN

Fx, udx

RN

Fx, um−udxo1, 3.28

we deduce that

Iα0 ≥Iu Ium−u o1. 3.29

By a similar method used to obtain3.22we also have

Ium−u I∞um−u o1. 3.30

Hence the last two relations yield

and this contradicts 3.19. Consequently _RNax|u|pbx|u|qdx α0 and 3.27 yields

um → uinLp∩LqRN. By the Gagliardo-Nirenberg inequality we haveum → uinLsRN,

q≤s < p∗. This obviously show thatIα0Iuand

RNax|u|pbx|u|qdxα0; that is,uis

a solution of problemIα0. Finally, writing

Iα0≥ 1

p

RN|∇u| p

dx 1 q

RN|∇u| q

dx 1 p

RN|∇un

−u|pdx 1 q

RN|∇un

−u|qdx

−

RN

Fudx−

RN

Fx, um−Fum

dx

RN

Fu−Fum

dxo1,

3.32

and using3.22we deduce from this that∇um → ∇uinLp∩LqRNand henceum → uin

W1,p_∩W1,q_RN_.

Theorem 3.3. Suppose thatFx, t ≥ FtonRN _{×Rand thatFζ > 0for someζ ∈ R, then}

problemIλhas a minimizer for someλ >0.

Proof. The conditionFζ > 0 for some ζ > 0 implies that_RNFuxdx > 0 for some u ∈

W1,p_∩W1,q_RN_{. Lettingvx ux/σ,σ >0, we have}

I∞v σN

1

pσp

RN|∇u| p

dx 1 qσq

RN|∇u| q

dx−

RNFuxdx

<0 3.33

forσ > 0 sufficiently large. Hence there existsλ > 0 such that Iλ ≤ I_λ∞ < 0 and the result

follows fromTheorem 3.2.

Remark 3.4. It is a standard argument that minimizers ofIμcorrespond to weak solutions of

problem1.1withλappearing as a Lagrange multiplier. Such aλis then called the principal eigenvalue for problem1.1.

Remark 3.5. Ifa∈LN/p_RN_,b∈LN/q_RN_{,a, b <0, we can use the similar method to study}

it, where Iλ inf{Iu;u ∈ W1,p ∩W1,qRN,

RNa−x|u|pb−x|u|qdx λ}, λ > 0, a− −a, b−−b.

Acknowledgments

This project was supported by the National Natural Science Foundation of China no. 10871060 and the Natural Science Foundation of Educational Department of Jiangsu Provinceno. 08KJB110005.

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