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Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝN

Huei-li Lin

Author Affiliations

Department of Natural Sciences in the Center for General Education, Chang Gung University, Tao-Yuan 333, Taiwan

Boundary Value Problems 2012, 2012:24  doi:10.1186/1687-2770-2012-24

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/24


Received:13 July 2011
Accepted:24 February 2012
Published:24 February 2012

© 2012 Lin; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we investigate the effect of the coefficient f(z) of the sub-critical nonlinearity. For sufficiently large λ > 0, there are at least k + 1 positive solutions of the semilinear elliptic equations

- Δ v + λ v = f ( z ) v p - 1 + h ( z ) v q - 1 in N ; v H 1 ( N ) ,

where 1 ≤ q < 2 < p < 2* = 2N/(N - 2) for N ≥ 3.

AMS (MOS) subject classification: 35J20; 35J25; 35J65.

Keywords:
semilinear elliptic equations; concave and convex; positive solutions

1 Introduction

For N ≥ 3, 1 ≤ q < 2 < p < 2* = 2N/(N - 2), we consider the semilinear elliptic equations

- Δ v + λ v = f ( z ) v p - 1 + h ( z ) v q - 1 in N ; v H 1 ( N ) , ( E λ )

where λ > 0.

Let f and h satisfy the following conditions:

(f 1) f is a positive continuous function in ℝN and lim|z| → ∞ f(z) = f> 0.

(f2) there exist k points a1, a2,..., ak in ℝN such that

f ( a i ) = f max = max z N f ( z ) for 1 i k ,

and f< fmax.

(h 1) h L p p - q ( N ) L ( N ) and h 0 .

Semilinear elliptic problems involving concave-convex nonlinearities in a bounded domain

- Δ u = c h ( z ) u q - 2 u + u p - 2 u in Ω ; u H 0 1 ( Ω ) , ( E c )

have been studied by Ambrosetti et al. [1] (h ≡ 1, and 1 < q < 2 < p ≤ 2* = 2N/(N- 2)) and Wu [2] h C ( Ω ̄ ) and changes sign, 1 < q < 2 < p < 2*). They proved that this equation has at least two positive solutions for sufficiently small c > 0. More general results of Equation (Ec) were done by Ambrosetti et al. [3], Brown and Zhang [4], and de Figueiredo et al. [5].

In this article, we consider the existence and multiplicity of positive solutions of Equation (Eλ) in ℝN. For the case q = λ = 1 and f(z) ≡ 1 for all z ∈ ℝN, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (Eλ) admits at least two positive solutions in ℝN. Without the condition of exponential decay, Cao and Zhou [7] and Hirano [8] proved that Equation (Eλ) admits at least two positive solutions in ℝN. For the case q = λ = 1, by using the idea of category and Bahri-Li's minimax argument, Adachi and Tanaka [9] asserted that Equation (Eλ) admits at least four positive solutions in ℝN, where f(z) ≢ 1, f(z) ≥ 1 - C exp((-(2 + δ) |z|) for some C, δ > 0, and sufficiently small h H - 1 > 0 . Similarly, in Hsu and Lin [10], they have studied that there are at least four positive solutions of the general case -Δu + u = f(z)vp-1 + λh(z) vq-1 in ℝN for sufficiently small λ > 0.

By the change of variables

ε = λ - 1 2 and u ( z )  =  ε 2 p - 2 v ( ε z ) ,

Equation (Eλ) is transformed to

- Δ u + u = f ( ε z ) u p - 1 + ε 2 ( p - q ) p - 2 h ( ε z ) u q - 1 in N ; u H 1 ( N ) , ( E ε )

Associated with Equation (Eε), we consider the C1-functional Jε, for u H1 (ℝN),

J ε ( u ) = 1 2 u H 2 - 1 p N f ( ε z ) u + p d z - 1 q N ε 2 ( p - q ) p - 2 h ( ε z ) u + q d z ,

where u H 2 = N Δ u 2 + u 2 d z is the norm in H1 (ℝN) and u+ = max{u, 0} ≥ 0. We know that the nonnegative weak solutions of Equation (Eε) are equivalent to the critical points of Jε. This article is organized as follows. First of all, we use the argument of Tarantello [11] to divide the Nehari manifold Mε into the two parts M ε + and M ε - . Next, we prove that the existence of a positive ground state solution u 0 M ε + of Equation (Eε). Finally, in Section 4, we show that the condition (f2) affects the number of positive solutions of Equation (Eε), that is, there are at least k critical points u 1 , . . . , u k M ε - of Jε such that J ε ( u i ) = β ε i ( ( PS ) - value ) for 1 ≤ i k.

Let

S = sup u H 1 ( N ) u H = 1 u L p ,

then

u L p S u H for any u H 1 ( N ) \ { 0 } . (1.1)

For the semilinear elliptic equations

- Δ u + u = f ( ε z ) u p - 1 in N ; u H 1 ( N ) , (E0)

we define the energy functional I ε ( u ) = 1 2 u H 2 - 1 p N f ( ε z ) u + p d z , and

γ ε = inf u N ε I ε ( u ) ,

where Nε = {u H1 (ℝN) \ {0} | u+ ≢ 0 and I ε ( u ) , u = 0 }. Note that

(i) if f f, we define I ( u ) = 1 2 u H 2 - 1 p N f u + p d z and

γ = inf u N I ( u ) ,

where N= {u H1 (ℝN) \ {0} | u+ ≢ 0 and I ( u ) , u = 0 };

(ii) if f fmax, we define I max ( u ) = 1 2 u H 2 - 1 p N f max u + p d z and

γ max = inf u N max I max ( u ) ,

where Nmax = {u H1 (ℝN) \ {0} | u+ ≢ 0 and I max ( u ) , u = 0 }.

Lemma 1.1

γ max = p - 2 2 p ( f max S p ) - 2 / ( p - 2 ) > 0 .

Proof. It is similar to Theorems 4.12 and 4.13 in Wang [[12], p. 31].

Our main results are as follows.

(I) Let Λ = ε2(p-q)/(p-2). Under assumptions (f 1) and (h1), if

0 < Λ < Λ 0 = ( p - 2 ) 2 - q f max 2 - q p - 2 ( p - q ) S 2 q - p p - 2 h # - 1 ,

where ∥h# is the norm in L p p - q ( N ) , then Equation (Eε) admits at least a positive ground state solution. (See Theorem 3.4)

(II) Under assumptions (f1) - (f2) and (h1), if λ is sufficiently large, then Equation (Eλ) admits at least k + 1 positive solutions. (See Theorem 4.8)

2 The Nehari manifold

First of all, we define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in H1(ℝN) for some functional J.

Definition 2.1 (i) For β ∈ ℝ, a sequence {un} is a (PS)β-sequence in H1(ℝN) for J if J(un) = β + on(1) and J'(un) = on(1) strongly in H-1 (ℝN) as n → ∞, where H-1 (ℝN) is the dual space of H1(ℝN);

(ii) J satisfies the (PS)β-condition in H1(ℝN) if every (PS)β-sequence in H1(ℝN) for J contains a convergent subsequence.

Next, since Jε is not bounded from below in H1 (ℝN), we consider the Nehari manifold

M ε = u H 1 ( N ) \ { 0 } u + 0 and J ε ( u ) , u = 0 , (2.1)

where

J ε ( u ) , u = u H 2 - N f ( ε z ) u + p d z - N ε 2 ( p - q ) p - 2 h ( ε z ) u + q d z .

Note that Mε contains all nonnegative solutions of Equation (Eε). From the lemma below, we have that Jε is bounded from below on Mε.

Lemma 2.2 The energy functional Jε is coercive and bounded from below on Mε.

Proof. For u Mε, by (2.1), the Hölder inequality p 1 = p p - q , p 2 = p q and the Sobolev embedding theorem (1.1), we get

J ε ( u ) = 1 2 - 1 p u H 2 - 1 q - 1 p N ε 2 ( p - q ) p - 2 h ( ε z ) u + q d z u H q p p - 2 2 u H 2 - q - p - q q ε 2 ( p - q ) p - 2 h # S q .

Hence, we have that Jε is coercive and bounded from below on Mε.

Define

ψ ε ( u ) = J ε ( u ) , u .

Then for u Mε, we get

ψ ε ( u ) , u = 2 u H 2 - p N f ( ε z ) u + p d z - q N ε 2 ( p - q ) p - 2 h ( ε z ) u + q d z = ( p - q ) N ε 2 ( p - q ) p - 2 h ( ε z ) u + q d z - ( p - 2 ) u H 2 (2.2)

= ( 2 - q ) u H 2 - ( p - q ) N f ( ε z ) u + p d z . (2.3)

We apply the method in Tarantello [11], let

M ε + = { u M ε ψ ε ( u ) , u > 0 } ; M ε 0 = { u M ε ψ ε ( u ) , u = 0 } ; M ε - = { u M ε ψ ε ( u ) , u < 0 } .

Lemma 2.3 Under assumptions (f1) and (h1), if 0 < Λ (= ε2(p-q)/(p-2)) < Λ0, then M ε 0 = .

Proof. See Hsu and Lin [[10], Lemma 5].

Lemma 2.4 Suppose that u is a local minimizer for Jε on Mε and u M ε 0 . Then J ε ( u ) = 0 in H-1 (ℝN).

Proof. See Brown and Zhang [[4], Theorem 2.3].

Lemma 2.5 We have the following inequalities.

(i) N h ( ε z ) u + q d z > 0 for each u M ε + ;

(ii) u H < p - q p - 2 Λ h # S q 1 / ( 2 - q ) for each u M ε + ;

(iii) u H > 2 - q ( p - q ) f max S p 1 / ( p - 2 ) for each u M ε - ;

(iv) If 0 < Λ = ε 2 ( p - q ) / ( p - 2 ) < q Λ 0 2 , then Jε(u) > 0 for each u M ε - .

Proof. (i) It can be proved by using (2.2).

(ii) For any u M ε + M ε , by (2.2), we apply the Hölder inequality ( p 1 = p p - q , p 2 = p q ) to obtain that

0 < ( p - q ) N Λ h ( ε z ) u + q d z - ( p - 2 ) u H 2 ( p - q ) Λ h # S q u H q - ( p - 2 ) u H 2 .

(iii) For any u M ε - , by (2.3), we have that

u H 2 < p - q 2 - q N f ( ε z ) u + p d z p - q 2 - q S p u H p f max .

(iv) For any u M ε - M ε , by (iii), we get that

J ε ( u ) = 1 2 - 1 p u H 2 - 1 q - 1 p N Λ h ( ε z ) u + q d z u H q p p - 2 2 u H 2 - q - p - q q Λ h # S q > 1 p 2 - q ( p - q ) f max S p q p - 2 p - 2 2 2 - q ( p - q ) f max S p 2 - q p - 2 - p - q q Λ h # S q .

Thus, if 0 < Λ < q 2 ( p - 2 ) 2 - q f max 2 - q p - 2 ( p - q ) S 2 q - p p - 2 h # - 1 , we get that Jε(u) ≥ d0 > 0 for some constant d0 = d0(ε, p, q, S, ∥h# , fmax).

For u H1 (ℝN) \ {0} and u+ ≢ 0, let

t ̄ = t ̄ ( u ) = ( 2 - q ) u H 2 ( p - q ) N f ( ε z ) u + p d z 1 / ( p - 2 ) > 0 .

Lemma 2.6 For each u H1 (ℝN)\ {0} and u+ ≢ 0, we have that

(i) if N h ( ε z ) u + q d z = 0 , then there exists a unique positive number t - = t - ( u ) > t ̄ such that t - u M ε - and Jε(t-u) = supt ≥ 0 Jε(tu);

(ii) if 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0 and N h ( ε z ) u + q d z > 0 , then there exist unique positive numbers t + = t + ( u ) < t ̄ < t - = t - ( u ) such that t + u M ε + , t - u M ε - and

J ε ( t + u ) = inf 0 t t ̄ J ε ( t u ) , J ε ( t - u ) = sup t t ̄ J ε ( t u ) .

Proof. See Hsu and Lin [[10], Lemma 7].

Applying Lemma 2.3 ( M ε 0 = for 0 < Λ < Λ 0 ) , we write M ε = M ε + M ε - , where

M ε + = u M ε | ( 2 - q ) u H 2 - ( p - q ) N f ( ε z ) u + p d z > 0 , M ε - = u M ε | ( 2 - q ) u H 2 - ( p - q ) N f ( ε z ) u + p d z < 0 .

Define

α ε = inf u M ε J ε ( u ) ; α ε + = inf u M ε + J ε ( u ) ; α ε - = inf u M ε - J ε ( u ) .

Lemma 2.7 (i) If 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0, then α ε α ε + < 0 ;

(ii) If 0 < Λ < qΛ0/2, then α ε - d 0 > 0 for some constant d0 = d0 (ε, p, q, S, ∥h#, fmax).

Proof. (i) Let u M ε + , by (2.2), we get

( p - 2 ) u H 2 < ( p - q ) N Λ h ( ε z ) u + q d z .

Then

J ε ( u ) = 1 2 - 1 p u H 2 - 1 q - 1 p N Λ h ( ε z ) u + q d z < 1 2 - 1 p - 1 q - 1 p p - 2 p - q u H 2 = - ( 2 - q ) ( p - 2 ) 2 p q u H 2 < 0 .

By the definitions of αε and α ε + , we deduce that α ε α ε + < 0 .

(ii) See the proof of Lemma 2.5 (iv).

Applying Ekeland's variational principle and using the same argument in Cao and Zhou [7] or Tarantello [11], we have the following lemma.

Lemma 2.8 (i) There exists a ( P S ) α ε -sequence {un} in Mε for Jε;

(ii) There exists a ( P S ) α ε + -sequence {un} in M ε + for Jε;

(iii) There exists a ( P S ) α ε - -sequence {un} in M ε - for Jε.

3 Existence of a ground state solution

In order to prove the existence of positive solutions, we claim that Jε satisfies the (PS)β-condition in H1(ℝN) for β - , γ - C 0 Λ 2 2 - q , where Λ = ε2(p-q)/(p-2) and C0 is defined in the following lemma.

Lemma 3.1 Assume that h satisfies (h1) and 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0. If {un} is a (PS)β-sequence in H1(ℝN) for Jε with un u weakly in H1 (ℝN), then J ε ( u ) = 0 in H-1 (ℝN) and J ε ( u ) - C 0 Λ 2 2 - q - C 0 , where

C 0 = ( 2 - q ) ( p - q ) h # S q 2 2 - q / 2 p q ( p - 2 ) q 2 - q ,

and

C 0 = ( p - 2 ) ( 2 - q ) p p - 2 / 2 p q f max ( p - q ) 2 p - 2 S 2 p p - 2 .

Proof. Since {un} is a (PS)β-sequence in H1(ℝN) for Jε with un u weakly in H1 (ℝN), it is easy to check that J ε ( u ) = 0 in H-1(ℝN) and u ≥ 0. Then we have J ε ( u ) , u = 0 , that is, N f ( ε z ) u p d z = u H 2 - N Λ h ( ε z ) u q d z . Hence, by the Young inequality p 1 = 2 q and p 2 = 2 2 - q

J ε ( u ) = 1 2 - 1 p u H 2 - 1 q - 1 p N Λ h ( ε z ) u q d z p - 2 2 p u H 2 - p - q p q Λ h # S q u H q p - 2 2 p u H 2 - p - 2 p q q u H 2 2 + p - q p - 2 Λ h # S q 2 2 - q 2 - q 2 - ( p - 2 ) ( 2 - q ) p p - 2 2 p q f max ( p - q ) 2 p - 2 S 2 p p - 2 .

Lemma 3.2 Assume that f and h satisfy (f1) and (h1). If 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0, then Jε satisfies the (PS)β-condition in H1(ℝN) for β - , γ - C 0 Λ 2 2 - q .

Proof. Let {un} be a (PS)β-sequence in H1(ℝN) for Jε such that Jε(un) = β + on(1) and J ε ( u n ) = o n (1) in H-1(ℝN). Then

β + c n + d n u n H p J ε ( u n ) - 1 p J ε ( u n ) , ( u n ) = 1 2 - 1 p u n H 2 - 1 q - 1 p N ε 2 ( p - q ) p - 2 h ( ε z ) ( u n ) + q d z p - 2 2 p u n H 2 - p - q p q Λ h # S q u n H q ,

where cn = on(1), dn = on(1) as n → ∞. It follows that {un} is bounded in H1(ℝN). Hence, there exist a subsequence {un} and a nonnegative u H1 (ℝN) such that J ε ( u ) = 0 in H-1 (ℝN), un u weakly in H1 (ℝN), un u a.e. in ℝN, un u strongly in L l o c s N for any 1 ≤ s < 2*. Using the Brézis-Lieb lemma to get (3.1) and (3.2) below.

N f ( ε z ) ( u n - u ) + p d z = N f ( ε z ) ( u n ) + p d z - N f ( ε z ) u p d z + o n ( 1 ) ; (3.1)

N h ( ε z ) ( u n - u ) + q d z = N h ( ε z ) ( u n ) + q d z - N h ( ε z ) u q d z + o n ( 1 ) . (3.2)

Next, claim that

N h ( ε z ) u n - u q d z 0 as n . (3.3)

For any σ > 0, there exists r > 0 such that [ B N ( 0 ; r ) ] c h ( ε z ) p p - q d z < σ . By the Hölder inequality and the Sobolev embedding theorem, we get

N h ( ε z ) u n - u q d z B N ( 0 ; r ) h ( ε z ) u n - u q d z + [ B N ( 0 ; r ) ] c h ( ε z ) u n - u q d z h # B N ( 0 ; r ) u n - u p d z q / p + S q [ B N ( 0 ; r ) ] c h ( ε z ) p p - q d z p - q p u n - u H q C σ + o n ( 1 ) . ( { u n } is bounded in H 1 ( N ) and u n u in L l o c p ( N ) )

Applying (f1) and un u in L l o c p ( N ) , we get that

N f ( ε z ) ( u n - u ) + p d z = N f ( u n - u ) + p d z + o n ( 1 ) . (3.4)

Let pn = un - u. Suppose pn ↛ 0 strongly in H1 (ℝN). By (3.1)-(3.4), we deduce that

p n H 2 = u n H 2 - u H 2 + o n ( 1 ) = N f ( ε z ) ( u n ) + p d z - N ε 2 ( p - q ) p - 2 h ( ε z ) ( u n ) + q d z - N f ( ε z ) u p d z + N ε 2 ( p - q ) p - 2 h ( ε z ) u q d z + o n ( 1 ) = N f ( ε z ) ( u n - u ) + p d z + o n ( 1 ) = N f ( p n ) + p d z + o n ( 1 ) .

Then

I ( p n ) = 1 2 p n H 2 - 1 p N f ( p n ) + p d z = 1 2 - 1 p p n H 2 + o n ( 1 ) > 0 .

By Theorem 4.3 in Wang [12], there exists a sequence {sn} ⊂ ℝ+ such that sn = 1 + on(1), {sn pn} ⊂ Nand I(sn pn) = I(pn) + on(1). It follows that

γ I ( s n p n ) = I ( p n ) + o n ( 1 ) = J ε ( u n ) - J ε ( u ) + o n ( 1 ) = β - J ε ( u ) + o n ( 1 ) < γ ,

which is a contradiction. Hence, un u strongly in H1(ℝN).

Remark 3.3 By Lemma 1.1, we obtain

C 0 = 2 - q q 2 - q p - q 2 p - 2 γ max < γ max < γ ,

and γ - C 0 Λ 2 2 - q > 0 for 0 < Λ < Λ0.

By Lemma 2.8 (i), there is a ( PS ) α ε -sequence {un} in Mε for Jε. Then we prove that Equation (Eε) admits a positive ground state solution u0 in ℝN.

Theorem 3.4 Under assumptions (f1), (h1), if 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0, then there exists at least one positive ground state solution u0 of Equation (Eε) in N. Moreover, we have that u 0 M ε + and

J ε ( u 0 ) = α ε = α ε + - C 0 Λ 2 2 - q . (3.5)

Proof. By Lemma 2.8 (i), there is a minimizing sequence {un} ⊂ Mε for Jε such that Jε(un) = αε + on(1) and J ε ( u n ) = o n ( 1 ) in H-1 (ℝN). Since α ε < 0 < γ - C 0 Λ 2 2 - q , by Lemma 3.2, there exist a subsequence {un} and u0 H1 (ℝN) such that un u0 strongly in H1 (ℝN). It is easy to see that u 0 0 is a solution of Equation (Eε) in ℝN and Jε(u0) = αε. Next, we claim that u 0 M ε + . On the contrary, assume that u 0 M ε - ( M ε 0 = for 0 < Λ ( = ε 2 ( p - q ) / ( p - 2 ) ) < Λ 0 ) .

We get that

N Λ h ( ε z ) ( u 0 ) + q d z > 0 .

Otherwise,

0 = N Λ h ( ε z ) ( u 0 ) + q d z = N Λ h ( ε z ) ( u n ) + q d z + o n ( 1 ) = u n H 2 - N f ( ε z ) ( u n ) + p d z + o n ( 1 ) .

It follows that

α ε + o n ( 1 ) = J ε ( u n ) = 1 2 - 1 p u n H 2 + o n ( 1 ) ,

which contradicts to αε < 0. By Lemma 2.6 (ii), there exist positive numbers t + < t ̄ < t - = 1 such that t + u 0 M ε + , t - u 0 M ε - and

J ε ( t + u 0 ) < J ε ( t - u 0 ) = J ε ( u 0 ) = α ε ,

which is a contradiction. Hence, u 0 M ε + and

- C 0 Λ 2 2 - q J ε ( u 0 ) = α ε = α ε + .

By Lemma 2.4 and the maximum principle, then u0 is a positive solution of Equation (Eε) in ℝN.

4 Existence of k + 1 solutions

From now, we assume that f and h satisfy (f1)-(f2) and (h1). Let w H1 (ℝN) be the unique, radially symmetric, and positive ground state solution of Equation (E0) in ℝN for f = fmax. Recall the facts (or see Bahri and Li [13], Bahri and Lions [14], Gidas et al. [15], and Kwong [16]).

(i) w L ( N ) C l o c 2 , θ ( N ) for some 0 < θ < 1 and lim z w ( z ) = 0 ;

(ii) for any ε > 0, there exist positive numbers C1, C 1 , C 2 ε , and C 3 ε such that for all z ∈ ℝN

C 2 ε exp - 1 - ε z w ( z ) C 1 exp - z

and

w ( z ) C 3 ε exp - ( 1 - ε ) z .

For 1 ≤ i k, we define

w ε i ( z ) = w z - a i ε , where f ( a i ) = f max .

Clearly, w ε i ( z ) H 1 N . By Lemma 2.6 (ii), there is a unique number t ε i - > 0 such that t ε i - w ε i M ε - M ε , where 1 ≤ i k.

We need to prove that

lim ε 0 + J ε t ε i - w ε i γ max uniformly in i .

Lemma 4.1 (i) There exists a number t0 > 0 such that for 0 t t0 and any ε > 0, we have that

J ε t w ε i < γ max u n i f o r m l y i n i ;

(ii) There exist positive numbers t1 and ε1 such that for any t > t1 and ε < ε1, we have that

J ε t w ε i < 0 u n i f o r m l y i n i .

Proof. (i) Since Jε is continuous in H 1 N , { w ε i } is uniformly bounded in H1 (ℝN) for any ε > 0, and γmax > 0, there is t0 > 0 such that for 0 ≤ t t0 and any ε > 0

J ε t w ε i < γ max .

(ii) There is an r0 > 0 such that f (z) ≥ fmax/2 for z BN (ai; r0) uniformly in i. Then there exists ε1 > 0 such that for ε < ε1

J ε t w ε i = t 2 2 w ε i H 2 - t p p N f ( ε z ) w ε i p d z - t q q N Λ h ( ε z ) w ε i q d z t 2 2 N w 2 w 2 d z - t p 2 p B N ( 0 ; 1 ) f max w p d z .

Thus, there is t1 >0 such that for any t > t1 and ε < ε1

J ε t w ε i < 0 uniformly in i .

Lemma 4.2 Under assumptions (f1), (f2), and (h1). If 0 < Λ ( = ε2(p-q)/(p-2)) < q Λ0/2, then

lim ε 0 + sup t 0 J ε t w ε i γ max u n i f o r m l y i n i .

Proof. By Lemma 4.1, we only need to show that

lim ε 0 + sup t 0 t t 1 J ε t w ε i γ max uniformly in i .

We know that supt ≥0 Imax (tw) = γmax. For t0 t t1, we get

J ε t w ε i = 1 2 t w ε i H 2 - 1 p N f ( ε z ) t w ε i p d z - 1 q N Λ h ( ε z ) t w ε i q d z = t 2 2 N w z - a i ε 2 + w z - a i ε d z - t p p N f ( ε z ) w z - a i ε p d z - t q q N Λ h ( ε z ) w z - a i ε q d z = t 2 p N w 2 + w 2 d z - t p p N f max w p d z + t p p N ( f max - f ( ε z ) ) w z - a i ε p d z - t q q Λ N h ( ε z ) w z - a i ε q d z γ max + t 1 p p N ( f max - f ( ε z ) ) w z - a i ε p d z - t 0 q q Λ N h ( ε z ) w z - a i ε d z .

Since

N ( f max - f ( ε z ) ) w z - a i ε p d z = N f max - f ( ε z + a i ) w p d z = o ( 1 ) as ε 0 + uniformly in i ,

and

Λ N h ( ε z ) w z - a i ε q d z ε 2 ( p - q ) p - 2 h # S q w H q = o ( 1 ) as ε 0 + ,

then lim ε 0 + sup t 0 t t 1 J ε t w ε i γ max , that is, lim ε 0 + sup t 0 J ε t w ε i γ max uniformly in i.

Applying the results of Lemmas 2.6, 2.7(ii), and 4.2, we can deduce that

0 < d 0 α ε - γ max + o ( 1 ) as ε 0 + .

Since γmax < γ, there exists ε0 > 0 such that

γ max < γ - C 0 Λ 2 2 - q for any ε < ε 0 . (4.1)

Choosing 0 < ρ0 < 1 such that

B ρ 0 N ( a i ) ¯ B ρ 0 N ( a j ) ¯ = for i j and 1 i , j k ,

where B ρ 0 N ( a i ) ¯ = z N | z - a i ρ 0 and f(ai) = fmax. Define K = {ai | 1 ≤ i k} and K ρ 0 / 2 = i = 1 k B ρ 0 / 2 N ( a i ) ¯ . Suppose i = 1 k B ρ 0 N ( a i ) ¯ B r 0 N ( 0 ) for some r0 > 0.

Let Qε : H1 (ℝN) \ {0} → ℝN be given by

Q ε ( u ) = N χ ( ε z ) u p d z N u p d z ,

where χ : ℝN → ℝN, χ (z) = z for |z| ≤ r0 and χ (z) = r0z/|z| for |z| > r0.

Lemma 4.3 There exists 0 < ε0 ε0 such that if ε < ε0, then Q ε t ε i - w ε i K ρ 0 / 2 for each 1 ≤ i k.

Proof. Since

Q ε t ε i - w ε i = N χ ( ε z ) w z - a i ε p d z N w z - a i ε p d z = N χ ( ε z + a i ) w z p d z N w z p d z a i a s ε 0 + ,

there exists ε0 > 0 such that

Q ε t ε i - w ε i K ρ 0 / 2 for any ε < ε 0 and each 1 i k .

Lemma 4.4 There exists a number δ ̄ > 0 such that if u Nε and I ε ( u ) γ max + δ ̄ , then Q ε ( u ) K ρ 0 / 2 for any 0 < ε < ε0.

Proof. On the contrary, there exist the sequences {εn} ⊂ ℝ+ and { u n } N ε n such that ε n 0 , I ε n ( u n ) = γ max ( > 0 ) + o n (1) as n → ∞ and Q ε n ( u n ) K ρ 0 / 2 for all n ∈ ℕ. It is easy to check that {un} is bounded in H1 (ℝN). Suppose un → 0 strongly in Lp (ℝN). Since

u n H 2 = N f ( ε n z ) ( u n ) + p d z for each n ,

and

I ε n ( u n ) = 1 2 u n H 2 - 1 p N f ( ε n z ) ( u n ) + p d z = γ max + o n ( 1 ) ,

then

γ max + o n ( 1 ) = I ε n ( u n ) = 1 2 - 1 p N f ( ε n z ) ( u n ) + p d z = o n ( 1 ) ,

which is a contradiction. Thus, un ↛ 0 strongly in Lp (ℝN). Applying the concentration-compactness principle (see Lions [17] or Wang [[12], Lemma 2.16]), then there exist a constant d0 > 0 and a sequence z n ̃ N such that

B N ( z n ̃ ; 1 ) u n ( z ) 2 d z d 0 > 0 . (4.2)

Let v n ( z ) = u n z + z n ̃ , there are a subsequence {vn} and v H1 (ℝN) such that vn v weakly in H1 (ℝN). Using the similar computation in Lemma 2.6, there is a sequence s max n + such that v n ̃ = s max n v n N max and

0 < γ max I max v n ̃ I ε n s max n u n I ε n ( u n ) = γ max + o n ( 1 ) as n .

We deduce that a convergent subsequence s max n satisfies s max n s 0 > 0 . Then there are subsequences v n ̃ and H 1 N such that n ( = s 0 v ) weakly in H1 (ℝN). By (4.2), then 0 . Moreover, we can obtain that v n ̃ strongly in H1 (ℝN) and I max ( ) = γ max . Now, we want to show that there exists a subsequence { z n } = ε n z n ̃ such that zn z0 K.

(i) Claim that the sequence {zn} is bounded in ℝN. On the contrary, assume that |zn| → ∞, then

γ max = I max ( ) < I ( ) lim inf n 1 2 v n ̃ H 2 - 1 p N f ( ε n z + z n ) ( v n ̃ ) + p d z = lim inf n ( s max n ) 2 2 u n H 2 - ( s max n ) p p N f ( ε n z ) ( u n ) + p d z = lim inf n I ε n ( s max n u n ) lim inf n I ε n ( u n ) = γ max ,

which is a contradiction.

(ii) Claim that z0 K. On the contrary, assume that z0 K, that is, f(z0) < fmax. Then using the above argument to obtain that

γ max = I max ( ) < 1 2 H 2 - 1 p N f ( z 0 ) ( ) + p d z lim inf n 1 2 v n ̃ H 2 - 1 p N f ( ε n z + z n ) ( v n ̃ ) + p d z = γ max ,

which is a contradiction. Since vn v ≠ 0 in H1 (ℝN), we have that

Q ε n ( u n ) = N χ ( ε n z ) v n ( z - z n ̃ ) p d z N v n ( z - z n ̃ ) p d z = N χ ( ε n z + ε n z n ̃ ) v n p d z N v n p d z z 0 K ρ 0 / 2 as n ,

which is a contradiction.

Hence, there exists a number δ ̄ > 0 such that if u Nε and I ε ( u ) γ max + δ ̄ , then Q ε ( u ) K ρ 0 / 2 for any 0 < ε < ε0.

From (4.1), choosing 0 < δ 0 < δ ̄ such that

γ max + δ 0 < γ - C 0 Λ 2 2 - q for any 0 < ε < ε 0 . (4.3)

For each 1 ≤ i k, define

O ε i = { u M ε - | Q ε ( u ) - a i < ρ 0 } , O ε i = { u M ε - | Q ε ( u ) - a i = ρ 0 } ,

β ε i = inf u O ε i J ε ( u ) and β ̃ ε i = inf u O ε i J ε ( u ) .

Lemma 4.5 If u M ε - and Jε (u) ≤ γmax + δ0/2, then there exists a number 0 < ε ̄ < ε 0 such that Q ε ( u ) K ρ 0 / 2 for any 0 < ε < ε ̄ .

Proof. We use the similar computation in Lemma 2.6 to get that there is a unique positive number

s ε u = u H 2 N f ( ε z ) u + p d z 1 / ( p - 2 )

such that s ε u u N ε . We want to show that s ε u < c for some constant c > 0 (independent of u). First, since u M ε - M ε ,

0 < d 0 α ε - J ε ( u ) γ max + δ 0 / 2 ,

and Jε is coercive on Mε, then 0 < c 2 < u H 2 < c 1 for some constants c1 and c2 (independent of u). Next, we claim that u L p p > c 3 > 0 for some constant c3 > 0 (independent of u). On the contrary, there exists a sequence { u n } M ε - such that

u n L p p = o n ( 1 ) as n .

By (2.3),

2 - q p - q < N f ( ε z ) ( u n ) + p d z u n H 2 f max u n L p p c 2 = o n ( 1 ) ,

which is a contradiction. Thus, s ε u < c for some constant c > 0 (independent of u). Now, we get that

γ max + δ 0 / 2 J ε ( u ) = sup t 0 J ε ( t u ) J ε ( s ε u u ) = 1 2 s ε u u H 2 - 1 p N f ( ε z ) ( s ε u u ) + p d z - 1 q N Λ h ( ε z ) ( s ε u u ) + q d z I ε ( s ε u u ) - 1 q N Λ h ( ε z ) ( s ε u u ) + q d z .

From the above inequality, we deduce that

I ε ( s ε u u ) γ max + δ 0 / 2 + 1 q N Λ h ( ε z ) ( s ε u u ) + q d z γ max + δ 0 / 2 + Λ h S q s ε u u H q < γ max + δ 0 / 2 + Λ c q ( c 1 ) q / 2