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

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

Mingzhu Wu

and Zuodong Yang

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

2College 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−2u λbx|u|^q−2ufx, u, u∈W^1,p∩W^1,qR^N, where 2≤q≤p < Nandax∈L^N/pR^N, bx∈ L^N/qR^N, 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∇u

ax|u|^p−2ubx|u|^q−2ufx, u, u∈W^1,p∩W^1,q

R^N

, u /0,

1.1

where 2 ≤ q ≤ p < N and ax ∈ L^N/pR^N, bx ∈ L^N/qR^N, ax, bx > 0, infax,infbx/0,fx, usatisfy the following conditions:

Af ∈CR^N×R,R, lim_t→0fx, t/|t|^p−1 0,and lim_{|t| → ∞}fx, t/|t|^p−1p2/N 0 uniformly inx∈R^N,

Blim_{|x| → ∞}fx, t ftuniformly fortin bounded subsets ofR.

Remark 1.1. We can see ifax∈L^N/pR^N, bx∈L^N/qR^N, then

R^Nax|u|^pdx <

R^Nax^N/p

_1−p/p^∗

R^Nu^p∗ _p/p^∗

<∞,

R^Nbx|u|^qdx <

R^Nbx^N/q

_1−q/q^∗

R^Nu^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 thatW^1,pR^Nis not a Hilbert space for 1 < p < N, except for p 2. The space W^1,pR^Nwithp /2 does not satisfy the Lieb lemmae.g., see9. AndR^Nresults 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∇u

Vx|u|^p−2u,

u∈D^1,p₀ Ω, u /0, 1.4

whereΩ ⊆R^N. We can see10–13. In13, Szulkin and Willem generalized several earlier results concerning the existence of an infinite sequence of eigenvalues.

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

−div

|∇u|^p−2∇u

λ|u|^p−2ufx, u, inΩ,

u∈W₀^1,pΩ, u /0, 1.5

where 1< p < N, N≥3,λis a parameter,Ωis an unbounded domain inR^N. 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∇u

u^p−1qxu^α inR^N, 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∇u

m|u|^p−2un|u|^q−2ufx, u inR^N, u∈W^1,p∩W^1,q

R^N

, 1.7

wherem, n > 0, N ≥ 3,and 1 < q < p < N,fx, u/u^p−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 inR^N, u∈W^1,p∩W^1,q

R^N

, 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 1.1inR^N. 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 problem1.1inR^Nis the lack of compactness of the Sobolev imbedding due to the invariance ofR^N under the translations and rotations. However, our method has essential difference with the methods used in 19,20. In order to obtain the

results, we have to overcome two main difficulties; one is that R^N results in the loss of compactness; the other is thatW^1,pR^N 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:W^1,p∩W^1,qR^N → RandI^∞:W^1,p∩W^1,qR^N → R by

Iu 1 p

R^N|∇u|^pdx1 q

R^N|∇u|^qdx−

R^NFx, udx, I^∞u 1

p

R^N|∇u|^pdx1 q

R^N|∇u|^qdx−

R^NFudx.

2.2

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

Iλinf Iu;u∈W^1,p R^N

,

R^Nax|u|^pbx|u|^qdxλ

, λ >0. Iλ

To find a solution of problemIλwe introduce thelimitvariational problem

I_λ^∞inf I^∞u;u∈W^1,p R^N

,

R^Nax|u|^pbx|u|^qdxλ

, λ >0. I_λ^∞

Lemma 2.1. Letun⊆W₀^1,pΩa bounded sequence andp≥2. Going if necessary to a subsequence, one may assume thatun uinW₀^1,pΩ,un → ualmost everywhere, whereΩ ⊆ R^Nis an open subset.

Then,

nlim→ ∞

Ω|∇un|^pdx≥ lim

n→ ∞

Ω|∇un− ∇u|^pdx lim

n→ ∞

Ω|∇u|^pdx. 2.3

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

n→ ∞lim

Ω|∇un|²dx lim

n→ ∞

Ω|∇un− ∇u|²dx lim

n→ ∞

Ω|∇u|²dx, 2.4 when 3 ≥ p > 2, using the lower semicontinuity of theL^p-norm with respect to the weak convergence andun uinW^1,pΩ, we deduce

|∇un|^p−2∇un,∇un

≥

|∇u|^p−2∇u,∇u o1,

nlim→ ∞

|∇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,

nlim→ ∞

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 uinW^1,pΩ,

nlim→ ∞

Ω

|∇un|^p−2|∇u|²− |∇u|^p−2|∇un|²

dx0. 2.7

So

nlim→ ∞

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 .

2.8

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,∇u

o1. 2.9

Then,

|∇un|^p−2∇un,∇un

≥

|∇un− ∇u|^p−2∇un− ∇u,∇un− ∇u

|∇u|^p−2∇u,∇u o1

nlim→ ∞

Ω|∇un|^pdx≥ lim

n→ ∞

Ω|∇un− ∇u|^pdx lim

n→ ∞

Ω|∇u|^pdx,

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

nlim→ ∞

Ω

|∇un|^p− |∇u|^p

dx≥ lim

n→ ∞

Ω|∇un− ∇u|^p−k

|∇un|^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 inW^1,pR^Nsuch that

nlim→ ∞sup

y∈R^N

B^y,Ru^qndx0, p≤q < p^∗ 2.12 for someR >0. Thenun → 0 inL^sR^Nforp < s < p^∗, wherep^∗Np/N−p.

Proof. We consider the caseN ≥ 3. Letq < s < p^∗ andu ∈ W^1,pR^N. Holder and Sobolev inequalities imply that

|u|_L^s_By,R≤ |u|^1−λ_L^q_By,R|u|^λ_Lp∗By,R

≤C|u|^1−λ_L^q_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≤C^s|u|^1−λs_LqBy,R

By,R

|u|^p|∇u|^p

. 2.14

Now, coveringR^Nby balls of radiusr, in such a way that each point ofR^Nis contained in at mostN1 balls, we find

R^N|u|^s≤N1C^ssup

y∈R^N

By,R|u|^q

_1−λs/q

By,R

|u|^p|∇u|^p

. 2.15

Under the assumption of the lemma,un → 0 inL^sR^N, p < s < p^∗. The proof is complete.

Corollary 2.3. Let{um}be a sequence inW^1,pR^Nsatisfying 0< ρ

R^N|um|^pdxand such that um 0 inW^1,pR^N. Then there exist a sequence{ym} ⊂ R^N and a function 0/u∈ W^1,pR^N such that up to a subsequenceum·ym uinW^1,pR^N.

Lemma 2.4. Letf ∈CR^N×Rand suppose that

|s| → ∞lim fx, s

|s|^p∗−1 0 2.16

uniformly inx∈R^Nand

fx, s≤C

|s|^p−1|s|^p∗−1

2.17

for allx∈R^Nandt∈R. Ifum u0inW^1,pR^Nandum → u0a.e. onR^N, then

mlim→ ∞

R^NFx, umdx−

R^NFx, u0dx−

R^NFx, um−u0dx

0, 2.18

whereFx, u _u

0fx, tdt.

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

R^NFx, umdx

|x|≤RFx, umdx

|x|≥RFx, u0 um−u0dx

|x|≤RFx, umdx

|x|≥R

Fx, um−u0 fx, θu0 um−u0u0

dx, 2.19

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

R^NFx, umdx−

R^NFx, u0dx−

R^NFx, 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

mlim→ ∞

|x|≤RFx, um−Fx, u0dx0,

m→ ∞lim

|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 inW^1,pR^Nwe see that

Rlim→ ∞

|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 IλandI_λ^∞are bounded inW^1,pR^N.

Proof. From conditionA, we observe that for eachε >0 there existsCε>0 such that Fu,|Fx, u| ≤ε

R^N|u|^pdxε

R^N|u|^pp2/NdxCε

R^N|u|^αdx, 2.24 wherep < α < pp²/Nandε >0.

By the Holder and Sobolev inequalities we have

R^N|u|^pp2/Ndx

R^N|u|^{pp∗−p−p2/N/p∗−pp∗p2/N/p∗−p}dx

≤

R^N|u|^p

_p^∗_−p−p²_/N/p^∗_−p

R^N|u|^p∗

_p²_/N/p^∗_−p

≤S⁻¹

R^N|u|^p p/N

R^N|∇u|^pdx,

2.25

where|u|^p_p∗≤S⁻¹|∇u|^pp. Similarly we have

R^N|u|^αdx

R^N|u|^{pp∗−α/p∗−pp∗α−p/p∗−p}dx

≤

R^N|u|^pdx

_p^∗_−α/p^∗_−p

R^N|u|^p∗dx

_α−p/p^∗_−p

≤S^{−p∗α−p/pp∗−p}

R^N|u|^pdx

_p^∗_−α/p^∗_−p

R^N|∇u|^pdx

p^{∗α−p/pp∗−p} .

2.26

Consequently by the Young inequality we have

R^N|u|^αdx≤η

R^N|∇u|^pdxK

η

R^N|u|^αdx

_pp^∗_−α/p²p^∗−p²−p^∗α

2.27

forη >0, whereKη>0 is a constant.

Becauseu∈W^1,p∩W^1,qR^Nso we can by Sobolev embedding andλ

R^Nax|u|^p bx|u|^qdxlettingλ

R^N|u|^pdx <∞, we derive the following estimates forIuandI^∞u:

Iu, I^∞u≥ 1

p−εS⁻¹λ^p/N−Cεη

R^N|∇u|^pdx 1

q

R^N|∇u|^qdx−ελ−K η

Cελ^{pp∗−α/p2p∗−p2−p∗α}.

2.28

Choosingε >0 andη >0 so that 1

p −εS⁻¹λ^p/N−Cεη >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 thatI_μ^∞≤0 for allμ >0. Indeed, letu∈C^∞₀R^Nand

R^Nax ux/σ

σ^N/q

^pdx

R^Nbx ux/σ

σ^N/q

^qdxμ, 2.30

then for eachσ >0 we have

I_μ^∞≤ 1 pσ^pp/q−1N

R^N|∇u|^pdx 1 qσ^q

R^N|∇u|^qdx−σ^N

R^NF

σ^−N/qu

dx−→0 2.31

asσ → ∞.

Lemma 2.6. Suppose that I_λ^∞ < 0 for some λ > 0, thenI_μ^∞/μ is nonincreasing on 0,∞ and limμ→0I_μ^∞/μ 0. Moreover there existsλ^∗≤λsuch that

I_μ^∞ μ > I_λ^∞

λ forμ∈0, λ^∗. 2.32

Proof. We observe that

inf I^∞u

R^Nax|u|^pbx|u|^qdx

inf I^∞ u

x/σ^1/N

R^Na

x/σ^1/Nu

x/σ^1/Npdxb

x/σ^1/Nu

x/σ^1/Nqdx.

2.33

So if

R^Nax|u|^p bx|u|^qdx k and

R^Nax/σ^1/N|ux/σ^1/N|^pdx bx/σ^1/N|ux/

σ^1/N|^qdxkthenI^∞ux I^∞ux/σ^1/N I_k^∞. We have that ifσ >0 andα >0 with

R^Nax|u|^pbx|u|^qdxα, then

R^Na x

σ^1/N u

x σ^1/N

^pdxb x

σ^1/N u

x σ^1/N

^qdxσα, I^∞

u x

σ^1/N

I_σα^∞. 2.34

Consequently, for allα1>0 andα2>0 we have

I_α^∞₁ inf 1

p α1

α2

_N−p/N

R^N|∇u|^pdx1 q

α1

α2

_N−q/N

R^N|∇u|^qdx−α1

α2

R^NFudx;

R^Nax|u|^pbx|u|^qdxα2

.

2.35

If 0< α1 < α2, then for eachε >0 there existsu∈W^1,p∩W^1,qR^Nwith

R^Nax|u|^p bx|u|^qdxα2such that

I_α^∞₁ε > 1 p

α1

α2

_N−p/N

R^N|∇u|^pdx1 q

α1

α2

_N−q/N

R^N|∇u|^qdx−α1

α2

R^NFudx

≥ α1

α2

1 p

R^N|∇u|^pdx1 q

R^N|∇u|^qdx−

R^NFudx

≥ α1

α2I_α^∞₂.

2.36

This inequality yields

I_α^∞₁ α1 > I_α^∞₂

α2. 2.37

SinceI_μ^∞≤0 for allμ >0, we see that

μlim→0

I_μ^∞

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∈W^1,p∩W^1,qR^N, with

R^Nax|uμ|^p bx|uμ|^qdxλsuch that

I_μ^∞μ²> 1 p

μ λ

N−p/N

R^N

∇uμ^pdx1 q

μ λ

N−q/N

R^N

∇uμ^qdx− μ λ

R^NF uμ

dx

≥ μ λ

1 p

R^N

∇uμ^pdx1 q

R^N

∇uμ^qdx−

R^NF uμ

dx

≥ μ λ

C1λ

R^N

∇uμ^pdxC2λ

R^N

∇uμ^qdx−C3λ

,

2.39

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

μ²≥ μ λ

C1λ

R^N

∇uμ^pdxC2λ

R^N

∇uμ^qdx−C3λ

, 2.40

that is,

R^N|∇uμ|^pdx ≤ C4λ,

R^N|∇uμ|^pdx ≤ C5λ for some constant C4λ, C5λ > 0 independent ofμ. We see that there existsε0>0 and a sequenceμn → 0 such that

R^N

∇uμn^pdx≥ε0,

R^N

∇uμn^qdx≥ε0. 2.41 If

R^N|∇uμn|^pdx≥ε0then

R^N|∇uμn|^qdx≥η≥0.

Then, using the fact that

R^NFuμndx≤Cfor some constantC >0, we get I_μ^∞_n

μn μn≥ 1

pλ^−N−p/Nμn−p/Nε01

qλ^−N−q/Nμn−q/Nη−C

λ −→ ∞ 2.42

asμn → 0 and this contradicts the fact that lim_μ→0I_μ^∞/μ c≤0. Therefore

μlim→0

R^N

∇uμ^pdx0, lim

μ→0I_μ^∞0, 2.43

when

R^N|∇uμn|^pdx≥ε0>0 we can use the same method to obtain that lim_μ→0

R^N|∇uμ|^qdx 0.

So

μ→lim0

R^N

∇uμ^pdx lim

μ→0

R^N

∇uμ^qdx0, lim

μ→0I_μ^∞0, 2.44 this implies that lim_μ→0

R^NFuμdx0 and consequently I_μ^∞

μ μ≥ −1 λ

R^N

F uμ

dx−→0. 2.45

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

3. Proof of Main Theorems

Theorem 3.1. Suppose thatI_λ^∞ <0 for someλ >0, then there exists 0< α0 ≤ λsuch that problem I_α^∞₀has a minimizer. Moreover each minimizing sequence forI_α^∞₀up to a translation is relatively compact inW^1,p∩W^1,qR^N.

Proof. According toLemma 2.6the set

α1; I_α^∞ α > I_λ^∞

λ for eachα∈0, α1

3.1

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 , I_α^∞₀ α0

λ I_λ^∞<0, I_α^∞> α

λI_λ^∞,

3.3

for all 0< α < α0. This yields I_α^∞₀ α0

λ I_λ^∞ α0−α λ I_λ^∞ α

λI_λ^∞< I_α^∞_0−αI_α^∞ 3.4 for eachα∈0, α0.

Let{um} ⊂W^1,p∩W^1,qR^Nbe a minimizing sequence forI_α^∞₀. Since{um}is bounded we may assume thatum uinW^1,p∩W^1,qR^N,um → ua.e. onR^N. First we consider the caseu ≡ 0. In this case byLemma 2.2um → 0 forq < α < p^∗ or Corollary there exists a sequence{um} ⊂R^Nsuch thatum·ym v /0 inW^1,p∩W^1,qR^N.

In the first case limm→ ∞

R^NFumdx0 and consequently I_α^∞₀ lim

m→ ∞I^∞um lim

m→ ∞

1 p

R^N|∇um|^pdx1 q

R^N|∇um|^qdx−

R^NFumdx

≥0, 3.5

which is impossible. Henceum·ym v /0 inW^1,p∩W^1,qR^Nholds and lettingvmx umxymfrom Brezis-Lieb lemmasee21, Lemma 1.32we have

R^Nax|um|^pbx|um|^qdx

R^Na xym

|vm|^pb xym

|vm|^qdx

R^Na xym

|v|^pb xym

|v|^qdx

R^Na xym

|vm−v|^p

b xym

|vm−v|^qdxo1.

3.6

We now show that

R^Na xym

|v|^pb xym

|v|^qdxα0. 3.7

In the contrary case fromLemma 2.1we have

0<

R^Na xym

|v|^pb xym

|v|^qdx < α0. 3.8

By3.21we have

m→ ∞lim

R^Na xym

|vm−v|^pb xym

|vm−v|^qdx−→α0−λ,

λ

R^Na xym

|v|^pb xym

|v|^qdx.

3.9

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

R^NFvmdx

R^NFvdx

R^NFvm−vdxo1,

R^N|∇vm|^p|∇vm|^qdx≥

R^N|∇v|^p|∇v|^qdx

R^N|∇vm−v|^p|∇vm−v|^qdxo1, 3.10

and this implies that

I_α^∞₀≥I^∞v I^∞vm−v o1≥I_λ^∞I_α^∞_0−λ0o1. 3.11 Lettingm → ∞we getI_α^∞₀ ≥I_λ^∞I_α^∞

0−λ0which contradicts3.4. Therefore

R^Naxym|v|^p bxym|v|^qdxα0. It then follows from3.6thatvm → vinL^p∩L^qR^N. By the Gagliardo- Nirenberg inequalityvm → vinL^sR^N,q≤s <∞. Obviously this implies thatI_α^∞₀ I^∞v I^∞v·−ymand

R^Nax|v·−ym|^pbx|v·−ym|^qdxα0. To complete the proof we show thatvm → vinW^1,p∩W^1,qR^N. Indeed, we have

I_α^∞₀ 1 p

R^N|∇vm|^pdx 1 q

R^N|∇vm|^qdx−

R^NFvmdxo1

≥ 1 p

R^N|∇v|^pdx1 q

R^N|∇v|^qdx1 p

R^N|∇vm−v|^pdx1 q

R^N|∇vm−v|^qdx

−

R^NFvdx

R^N

Fv−Fvm

dxo1.

3.12

Since lim_{m→ ∞}

R^NFv−Fvmdx0, we deduce from3.12that∇vm → ∇vinL^p∩L^qR^N and hencevm → vinW^1,p∩W^1,qR^N.

Ifu /0, we repeat the previous argument to show thatI_α^∞₀is attained.

Theorem 3.2. Suppose thatFx, t ≥ FtonR^N ×R and thatIλ < 0 for someλ > 0, then the infimumIλ0is attained for some 0< λ0≤λ.

Proof. SinceFx, t≥FtonR^N×R we haveIμ≤I_μ^∞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,

R^Nax|u|^pbx|u|^qdxλ0 for someu∈W^1,p∩W^1,q R^N

. 3.13

Thus

Iλ0 ≤Iu 1 p

R^N|∇u|^pdx 1 q

R^N|∇u|^qdx−

R^NFx, udx

≤ 1 p

R^N|∇u|^pdx 1 q

R^N|∇u|^qdx−

R^NFudxI^∞u I_λ^∞₀.

3.14

IfIλ0 I_λ^∞₀, thenI also attains its infimumIλ0 atu. Therefore it remains to consider the case Iλ0< I_λ^∞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

IλIα0I_λ−α^∞ ₀, Iλ< IαI_λ−α^∞

3.17

for 0≤α < α0. This implies that

Iα0I_λ−α^∞ ₀ Iλ< I_λ^∞≤0, 3.18 and hence

Iα0 < I_λ^∞−I_λ−α^∞ ₀≤I_α^∞₀ ≤0, 3.19 we show thatIα0 is attained by au ∈ W^1,p∩W^1,qR^Nand every minimizing sequence for Iα0is relatively compact inW^1,p∩W^1,qR^N. Let{um}be a minimizing sequence forIα0. Since

{um}is bounded, we may assume that um uin W^1,p ∩W^1,qR^N,um u a.e. on R^N. Arguing indirectly we assume thatu≡0 onR^N. We see that

mlim→ ∞

B0,R|Fx, um|dx lim

m→ ∞

B0,R

Fumdx0 3.20

for eachR >0. We now write

Ium

1 p

R^N|∇um|^pdx 1 q

R^N|∇um|^qdx−

R^NFx, umdx I^∞um

R^N

Fum−Fx, um dx.

3.21

We show that

mlim→ ∞

R^N

Fum−Fx, um

dx0. 3.22

Towards this end we write

R^N

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∈R^N

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 .

3.24