Open Access

On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions

Boundary Value Problems20142014:117

DOI: 10.1186/1687-2770-2014-117

Received: 1 March 2014

Accepted: 29 April 2014

Published: 14 May 2014

Abstract

In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is λ f ( x ) | u | q 1 u and the convex nonlinear term is h ( x ) | u | p 1 u with λ R + . By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as λ ( 0 , λ ) , here the explicit expression of λ = λ ( f , h , p , q , S ) is provided.

MSC:35J35, 35J40, 35J65.

Keywords

biharmonic equations concave-convex nonlinearities weight functions

1 Introduction

In recent years, there has been extensive attention on semilinear second-order elliptic equations,
{ Δ u = g λ ( x , u ) , in  Ω , u = 0 , on  Ω ,
(1.1)
here Ω is a bounded smooth domain in R N ( N 3 ), g λ : Ω × R R and λ is a positive parameter; see [18] and the references therein. As g λ is sublinear, say, g λ = λ u q , 0 < q < 1 , the monotone iteration scheme or the method of sub-solutions and super-solutions are effective; see [9]. As g λ is superlinear, for example, g λ = λ u + | u | p 1 u , 1 < p < N + 2 N 2 , variational methods are applicable; see [10]. In contrast with the pure sublinear case and the pure superlinear case, in [2] Ambrosetti et al. considered problem (1.1) when g λ is, roughly, the sum of a sublinear and a superlinear term. To be precise, they considered the following problem:
{ Δ u = λ u q + u p , in  Ω , 0 u H 0 1 ( Ω ) ,
(1.2)
with 0 < q < 1 < p N + 2 N 2 . They proved that problem (1.2) admits at least two positive solutions for λ sufficiently small. In [6], Sun and Li considered a similar problem:
{ Δ u = u q + λ u p , in  Ω , 0 u H 0 1 ( Ω ) ,

with 0 < q < 1 < p = N + 2 N 2 , the authors studied the value of Λ, the supremum of the set λ, related to the existence and multiplicity of positive solutions and established uniform lower bounds for Λ. In [8], Wu considered the subcritical case of problem (1.2) with λ u q replaced by λ f ( x ) u q , here f ( x ) C ( Ω ¯ ) is a sign-changing function, and he showed that problem (1.2) has at least two positive solutions as λ is small enough.

Some interesting generalizations of (1.2) have been provided in the framework of quasi-linear elliptic equations or systems, semilinear second-order elliptic systems or fourth-order elliptic equations. More recently, the semilinear fourth-order elliptic equations have been studied by many authors, we refer the reader to [1113] and the references therein. Motivated by some work in [6, 8, 13], we deal with the following semilinear biharmonic elliptic equation:
{ Δ 2 u = λ f ( x ) | u | q 1 u + h ( x ) | u | p 1 u , in  Ω , u = Δ u = 0 , on  Ω ,
(1.3)

where Ω is a bounded smooth domain in R N ( N 4 ), 0 < q < 1 < p < 2 ( 2 = N + 4 N 4 for N > 4 and 2 = for N = 4 ), λ > 0 is a parameter, f C ( Ω ¯ ) is a positive or sign-changing weight function and h C ( Ω ¯ ) is a positive weight function.

For convenience and simplicity, we introduce some notations. The norm of u in L r ( Ω ) is denoted by | u | r = ( Ω | u ( x ) | r ) 1 / r , the norm of u in C ( Ω ¯ ) is denoted by | u | = max x Ω ¯ | u ( x ) | ; H 0 1 ( Ω ) H 2 ( Ω ) is denoted by H ( Ω ) , endowed with the norm u = | Δ u | 2 ; S denotes the best Sobolev constant for the embedding of H ( Ω ) in L p + 1 ( Ω ) (see [14]); to be precise, | u | p + 1 S u for all u H ( Ω ) .

Now we define
J λ ( u ) = 1 2 u 2 λ q + 1 Ω f ( x ) | u | q + 1 d x 1 p + 1 Ω h ( x ) | u | p + 1 d x , u H ( Ω ) .

It is well known that the weak solutions of problem (1.3) are the critical points of the energy functional J λ (see Rabinowitz [15]).

Next, we consider the Nehari minimization problem: for λ > 0 ,
α λ ( Ω ) = inf { J λ ( u ) u M λ ( Ω ) } ,
where M λ ( Ω ) = { u H ( Ω ) { 0 } J λ ( u ) , u = 0 } . Define
ψ λ ( u ) = J λ ( u ) , u = u 2 λ Ω f ( x ) | u | q + 1 d x Ω h ( x ) | u | p + 1 d x .
Then for u M λ ( Ω ) ,
ψ λ ( u ) , u = 2 u 2 λ ( q + 1 ) Ω f ( x ) | u | q + 1 d x ( p + 1 ) Ω h ( x ) | u | p + 1 d x .
Similarly to the method used in Tarantello [16], we split M λ ( Ω ) into three parts:
M λ + ( Ω ) = { u M λ ( Ω ) ψ λ ( u ) , u > 0 } , M λ 0 ( Ω ) = { u M λ ( Ω ) ψ λ ( u ) , u = 0 } , M λ ( Ω ) = { u M λ ( Ω ) ψ λ ( u ) , u < 0 } .

Note that all solutions of (1.3) are clearly in the Nehari manifold, M λ ( Ω ) . Hence, our approach to solve problem (1.3) is to analyze the structure of M λ ( Ω ) , and then to deal with the minimization problems for J λ on M λ + ( Ω ) and M λ ( Ω ) applying the direct variational method.

The following is our main result.

Theorem 1.1 Let λ = p 1 p q [ 1 q ( p q ) | h | ] 1 q p 1 S 2 ( p q ) 1 p | f | p 1 with p = p + 1 p q , then problem (1.3) has at least two positive solutions for any λ ( 0 , λ ) .

The paper is organized as follows: in Section 2, we give some lemmas; in Section 3, we prove Theorem 1.1.

2 Preliminaries

In this section, we prove several lemmas.

Lemma 2.1 For λ ( 0 , λ ) (where λ is given in Theorem  1.1), we have M λ 0 ( Ω ) = ϕ .

Proof Suppose that M λ 0 ( Ω ) ϕ for all λ > 0 . If u M λ 0 ( Ω ) , then we have
u 2 = λ Ω f ( x ) | u | q + 1 d x + Ω h ( x ) | u | p + 1 d x
(2.1)
and
2 u 2 = λ ( q + 1 ) Ω f ( x ) | u | q + 1 d x + ( p + 1 ) Ω h ( x ) | u | p + 1 d x .
(2.2)
By (2.1)-(2.2), the Sobolev inequality, and the Hölder inequality, we get
u 2 = p q 1 q Ω h ( x ) | u | p + 1 d x p q 1 q | h | S p + 1 u p + 1
(2.3)
and
u 2 = λ p q p 1 Ω f ( x ) | u | q + 1 d x λ p q p 1 | f | p S q + 1 u q + 1 ,
(2.4)
where p = p + 1 p q . Thus, using (2.3) and (2.4), we have
λ p 1 p q | f | p 1 S ( q + 1 ) [ 1 q p q | h | 1 S ( p + 1 ) ] 1 q p 1 = p 1 p q [ 1 q ( p q ) | h | ] 1 q p 1 S 2 ( p q ) 1 p | f | p 1 = λ .
(2.5)

Hence, by (2.5) the desired conclusion yields. □

Lemma 2.2 If u M λ ( Ω ) , then
u > S 1 + p 1 p [ 1 q ( p q ) | h | ] 1 p 1 and Ω h ( x ) | u | p + 1 d x > | h | 2 1 p [ ( p q ) S 2 1 q ] 1 + p 1 p .
Proof From u M λ ( Ω ) , it is easy to see that
u 2 < p q 1 q Ω h ( x ) | u | p + 1 d x .
By the Sobolev inequality, we get
u > S 1 + p 1 p [ 1 q ( p q ) | h | ] 1 p 1 .
In addition,
Ω h ( x ) | u | p + 1 d x > | h | 2 1 p [ ( p q ) S 2 1 q ] 1 + p 1 p .

The proof is completed. □

By Lemma 2.1, for λ ( 0 , λ ) we write M λ ( Ω ) = M λ + ( Ω ) M λ ( Ω ) and define
α λ + ( Ω ) = inf u M λ + ( Ω ) J λ ( u ) , α λ ( Ω ) = inf u M λ ( Ω ) J λ ( u ) .

The following lemma shows that the minimizers on M λ ( Ω ) are ‘usually’ critical points for  J λ .

Lemma 2.3 For λ ( 0 , λ ) , if u 0 is a local minimizer for J λ on M λ ( Ω ) , then J λ ( u 0 ) = 0 in [ H ( Ω ) ] .

Proof If u 0 is a local minimizer for J λ on M λ ( Ω ) , then u 0 is a solution of the optimization problem
minimize J λ ( u ) subject to ψ λ ( u ) = 0 .
Hence, by the theory of Lagrange multipliers, there exists θ R such that
J λ ( u 0 ) = θ ψ λ ( u 0 ) in  [ H ( Ω ) ] .
(2.6)
Thus,
J λ ( u 0 ) , u 0 = θ ψ λ ( u 0 ) , u 0 .
(2.7)

From u 0 M λ ( Ω ) and Lemma 2.1, we have J λ ( u 0 ) , u 0 = 0 and ψ λ ( u 0 ) , u 0 0 . So, by (2.6)-(2.7) we get J λ ( u 0 ) = 0 in [ H ( Ω ) ] . □

For each u H ( Ω ) { 0 } , we write
t max = ( ( 1 q ) u 2 ( p q ) Ω h ( x ) | u | p + 1 d x ) 1 p 1 > 0 .

Then we have the following lemma.

Lemma 2.4 For each u H ( Ω ) { 0 } and λ ( 0 , λ ) , we have
  1. (i)

    there is a unique t = t ( u ) > t max > 0 such that t ( u ) u M λ ( Ω ) and J λ ( t ( u ) u ) = max t 0 J λ ( t u ) ;

     
  2. (ii)

    t ( u ) is a continuous function for nonzero u;

     
  3. (iii)

    M λ ( Ω ) = { u H ( Ω ) { 0 } 1 u t ( u u ) = 1 } ;

     
  4. (iv)

    if Ω f ( x ) | u | q + 1 d x > 0 , then there is a unique 0 < t + = t + ( u ) < t max such that t + ( u ) u M λ + ( Ω ) and J λ ( t + ( u ) u ) = min 0 t t J λ ( t u ) .

     
Proof (i) Fix u H ( Ω ) { 0 } . Let
s ( t ) = t 1 q u 2 t p q Ω h ( x ) | u | p + 1 d x , t 0 .
Then we have s ( 0 ) = 0 , s ( t ) as t , s ( t ) is concave and reaches its maximum at t max . Moreover,
s ( t max ) = t max 1 q u 2 t max p q Ω h ( x ) | u | p + 1 d x = u q + 1 [ ( 1 q p q ) 1 q p 1 ( 1 q p q ) p q p 1 ] ( u p + 1 Ω h ( x ) | u | p + 1 d x ) 1 q p 1 u q + 1 ( p 1 p q ) ( 1 q p q ) 1 q p 1 ( 1 | h | S p + 1 ) 1 q p 1 .
(2.8)

Case I. Ω f ( x ) | u | q + 1 d x 0 .

There is a unique t > t max such that s ( t ) = λ Ω f ( x ) | u | q + 1 d x and s ( t ) < 0 . Now,
J λ ( t u ) , t u = t u 2 λ Ω f ( x ) | t u | q + 1 d x Ω h ( x ) | t u | p + 1 d x = ( t ) q + 1 [ s ( t ) λ Ω f ( x ) | u | q + 1 d x ] = 0
and
ψ λ ( t u ) , t u = ( 1 q ) t u 2 ( p q ) Ω h ( x ) | t u | p + 1 d x = ( t ) 2 + q [ ( 1 q ) ( t ) q u 2 ( p q ) ( t ) p q 1 Ω h ( x ) | u | p + 1 d x ] = ( t ) 2 + q s ( t ) < 0 .
Thus, t u M λ ( Ω ) . In addition,
d J λ ( t u ) d t = t u 2 λ t q Ω f ( x ) | u | q + 1 d x t p Ω h ( x ) | u | p + 1 d x = t 1 J λ ( t u ) , t u = 0 if and only if t = t
and
d 2 J λ ( t u ) d t 2 | t = t = u 2 λ q ( t ) q 1 Ω f ( x ) | u | q + 1 d x p ( t ) p 1 Ω h ( x ) | u | p + 1 d x = ( t ) 2 ψ λ ( t u ) , t u < 0 .

Hence, J λ ( t u ) = max t 0 J λ ( t u ) .

Case II. Ω f ( x ) | u | q + 1 d x > 0 .

From (2.8) and
s ( 0 ) = 0 < λ Ω f ( x ) | u | q + 1 d x λ | f | p S q + 1 u q + 1 < u q + 1 ( p 1 p q ) ( 1 q p q ) 1 q p 1 ( 1 | h | S p + 1 ) 1 q p 1 s ( t max ) for  λ ( 0 , λ ) ,
there exist unique t + and t such that 0 < t + < t max < t ,
s ( t + ) = λ Ω f ( x ) | u | q + 1 d x = s ( t )
and
s ( t + ) > 0 > s ( t ) .
Similar to the argument in Case I above, we have t + u M λ + ( Ω ) , t u M λ ( Ω ) , and
J λ ( t u ) = max t 0 J λ ( t u ) , J λ ( t + u ) = min 0 t t J λ ( t u ) .
  1. (ii)

    By the uniqueness of t ( u ) and the external property of t ( u ) , we find that t ( u ) is continuous function of u 0 .

     
  2. (iii)
    For u M λ ( Ω ) , let v = u u . By item (i), there is a unique t ( v ) > 0 such that t ( v ) v M λ ( Ω ) , that is, t ( u u ) 1 u u M λ ( Ω ) . Since u M λ ( Ω ) , we have t ( u u ) 1 u = 1 , which implies
    M λ ( Ω ) { u H ( Ω ) { 0 } | 1 u t ( u u ) = 1 } .
     
Conversely, let u H ( Ω ) { 0 } such that 1 u t ( u u ) = 1 . Then t ( u u ) u u M λ ( Ω ) . Therefore,
M λ ( Ω ) = { u H ( Ω ) { 0 } | 1 u t ( u u ) = 1 } .
  1. (iv)

    By Case II of item (i). □

     
By f C ( Ω ¯ ) and changes sign in Ω, we have Θ = { x Ω f ( x ) > 0 } is an open set in R N . Without loss of generality, we may assume that Θ is a domain in R N . Consider the following biharmonic equation:
{ Δ 2 u = h ( x ) | u | p 1 u , in  Θ , u = Δ u = 0 , on  Θ .
(2.9)
Associated with (2.9), we consider the energy functional
K ( u ) = 1 2 u 2 1 p + 1 Θ h ( x ) | u | p + 1 d x , u H ( Θ )
and the minimization problem
β ( Θ ) = inf { K ( u ) u N ( Θ ) } ,

where N ( Θ ) = { u H ( Θ ) { 0 } K ( u ) , u = 0 } . Now we prove that problem (2.9) has a positive solution w 0 such that K ( w 0 ) = β ( Θ ) > 0 .

Lemma 2.5 For any u H ( Θ ) { 0 } , there exists a unique t ( u ) > 0 such that t ( u ) u N ( Θ ) . The maximum of K ( t u ) for t 0 is reached at t = t ( u ) , the map
t : H ( Θ ) { 0 } ( 0 , + ) ; u t ( u )

is continuous and the induced continuous map u t ( u ) u defines a homeomorphism of the unit sphere of H ( Θ ) with N ( Θ ) .

Proof For any given u H ( Θ ) { 0 } , consider the function g ( t ) = K ( t u ) , t 0 . Clearly,
g ( t ) = 0 t u N ( Θ ) u 2 = t p 1 Ω h ( x ) | u | p + 1 d x .
(2.10)

It is easy to verify that g ( 0 ) = 0 , g ( t ) > 0 for t > 0 small and g ( t ) < 0 for t > 0 large. Hence, max t 0 g ( t ) is reached at a unique t = t ( u ) such that g ( t ( u ) ) = 0 and t ( u ) u N ( Θ ) . To prove the continuity of t ( u ) , assume that u n u in H ( Θ ) { 0 } . It is easy to verify that { t ( u n ) } is bounded. If a subsequence of { t ( u n ) } converges to t 0 , it follows from (2.10) that t 0 = t ( u ) and then t ( u n ) t ( u ) . Finally the continuous map from the unit sphere of H ( Θ ) to N ( Θ ) , u t ( u ) u , is inverse to the retraction u u u . □

Define
c = inf u H ( Θ ) { 0 } max t 0 K ( t u ) , c = inf γ Γ max t [ 0 , 1 ] K ( γ ( t ) ) ,

where Γ = { γ C ( [ 0 , 1 ] , H ( Θ ) ) γ ( 0 ) = 0 , K ( γ ( 1 ) ) < 0 } .

Lemma 2.6 β ( Θ ) = c = c > 0 is a critical value of K.

Proof From Lemma 2.5, we know that β ( Θ ) = c . Since K ( t u ) < 0 for u H ( Θ ) { 0 } and t large, we obtain c c . The manifold N ( Θ ) separates H ( Θ ) into two components. The component containing the origin also contains a small ball around the origin. Moreover, K ( u ) 0 for all u in this component, because K ( t u ) , u 0 , t [ 0 , t ( u ) ] . Then each γ Γ has to cross N ( Θ ) and β ( Θ ) c . Since the embedding H ( Θ ) L p + 1 ( Θ ) is compact (see [14]), it is easy to prove that c > 0 is a critical value of K and w 0 a positive solution corresponding to c. □

With the help of Lemma 2.6, we have the following result.

Lemma 2.7 (i) For λ ( 0 , λ ) , there exists t λ > 0 such that
α λ ( Ω ) α λ + ( Ω ) < 1 q q + 1 t λ 2 β λ ( Θ ) < 0 ;
  1. (ii)

    J λ is coercive and bounded below on M λ ( Ω ) for all λ > 0 .

     
Proof (i) Let w 0 be a positive solution of problem (2.9) such that K ( w 0 ) = β ( Θ ) . Then
Ω f ( x ) w 0 q + 1 d x = Θ f ( x ) w 0 q + 1 d x > 0 .
Set t λ = t + ( w 0 ) as defined by Lemma 2.4(iv). Hence, t λ w 0 M λ + ( Ω ) and
J λ ( t λ w 0 ) = 1 2 t λ w 0 2 λ q + 1 Ω f ( x ) | t λ w 0 | q + 1 d x 1 p + 1 Ω h ( x ) | t λ w 0 | p + 1 d x = ( 1 2 1 q + 1 ) t λ w 0 2 + ( 1 q + 1 1 p + 1 ) Ω h ( x ) | t λ w 0 | p + 1 d x < 1 q q + 1 t λ 2 β ( Θ ) < 0 .
This implies
α λ ( Ω ) α λ + ( Ω ) < 1 q q + 1 t λ 2 β ( Θ ) < 0 .
  1. (ii)
    For u M λ ( Ω ) , we have u 2 = λ Ω f ( x ) | u | q + 1 d x + Ω h ( x ) | u | p + 1 d x . Then by the Hölder, Sobolev, and Young inequalities,
    J λ ( u ) = p 1 2 ( p + 1 ) u 2 λ ( p q ) ( p + 1 ) ( q + 1 ) Ω f ( x ) | u | q + 1 d x p 1 2 ( p + 1 ) u 2 λ ( p q ) ( p + 1 ) ( q + 1 ) | f | p S q + 1 u q + 1 p 1 4 ( p + 1 ) u 2 λ 2 1 q C ( p , q ) ( | f | p S q + 1 ) 2 1 q ,
     

here C ( p , q ) = [ p q ( p + 1 ) ( q + 1 ) ] 2 1 q [ 4 ( p + 1 ) p 1 ] 1 + q 1 q .

Thus, J λ is coercive on M λ ( Ω ) and
J λ ( u ) λ 2 1 q C ( p , q ) ( | f | p S q + 1 ) 2 1 q

for all λ > 0 . □

Next, we will use the idea of Tarantello [16] to get the following results.

Lemma 2.8 For λ ( 0 , λ ) and any given u M λ ( Ω ) , there exist ϵ > 0 and a differentiable functional ξ : B ( 0 ; ϵ ) H ( Ω ) R + such that ξ ( 0 ) = 1 , the function ξ ( v ) ( u + v ) M λ ( Ω ) and
ξ ( 0 ) , v = 2 Ω Δ u Δ v λ ( q + 1 ) Ω f | u | q 1 u v ( p + 1 ) Ω h | u | p 1 u v ( 1 q ) u 2 ( p q ) Ω h ( x ) | u | p + 1 d x
(2.11)

for all v H ( Ω ) .

Proof Define F : R × H ( Ω ) R as follows:
F ( ξ , w ) = ξ 2 u + w 2 λ ξ q + 1 Ω f ( x ) | u + w | q + 1 d x ξ p + 1 Ω h ( x ) | u + w | p + 1 d x .
Since F ( 1 , 0 ) = J λ ( u ) , u = 0 and by Lemma 2.1, we obtain
F ξ ( 1 , 0 ) = 2 u 2 λ ( q + 1 ) Ω f ( x ) | u | q + 1 d x ( p + 1 ) Ω h ( x ) | u | p + 1 d x = ψ λ ( u ) , u 0 ,

we can get the desired results applying the implicit function theorem at the point ( 1 , 0 ) . □

Lemma 2.9 For λ ( 0 , λ ) and any given u M λ ( Ω ) , there exist ϵ > 0 and a differentiable functional ξ : B ( 0 ; ϵ ) H ( Ω ) R + such that ξ ( 0 ) = 1 , the function ξ ( v ) ( u + v ) M λ ( Ω ) and
( ξ ) ( 0 ) , v = 2 Ω Δ u Δ v λ ( q + 1 ) Ω f | u | q 1 u v ( p + 1 ) Ω h | u | p 1 u v ( 1 q ) u 2 ( p q ) Ω h ( x ) | u | p + 1 d x
(2.12)

for all v H ( Ω ) .

Proof In view of Lemma 2.8, there exist ϵ > 0 and a differentiable functional ξ such that ξ ( 0 ) = 1 , ξ ( v ) ( u + v ) M λ ( Ω ) for all v B ( 0 ; ϵ ) H ( Ω ) and we have (2.12). By use of u M λ ( Ω ) , we have ψ λ ( u ) , u < 0 . In combination with the continuity of the functions ψ λ and ξ , we get ψ λ ( ξ ( v ) ( u + v ) ) , ξ ( v ) ( u + v ) < 0 as ϵ sufficiently small, this implies that ξ ( v ) ( u + v ) M λ ( Ω ) . □

3 Proof of Theorem 1.1

Firstly, we provide the existence of minimizing sequences for J λ on M λ ( Ω ) and M λ ( Ω ) as λ is sufficiently small.

Proposition 3.1 Let λ ( 0 , λ ) , then
  1. (i)
    there exists a minimizing sequence { u n } M λ ( Ω ) such that
    J λ ( u n ) = α λ ( Ω ) + o ( 1 ) and J λ ( u n ) = o ( 1 ) in  [ H ( Ω ) ] ;
     
  2. (ii)
    there exists a minimizing sequence { u n } M λ ( Ω ) such that
    J λ ( u n ) = α λ ( Ω ) + o ( 1 ) and J λ ( u n ) = o ( 1 ) in  [ H ( Ω ) ] .
     
Proof (i) By Lemma 2.7(ii) and the Ekeland variational principle [17], there exists a minimizing sequence { u n } M λ ( Ω ) such that
J λ ( u n ) < α λ ( Ω ) + 1 n
(3.1)
and
J λ ( u n ) < J λ ( w ) + 1 n w u n for each  w M λ ( Ω ) .
(3.2)
Taking n large, from Lemma 2.7(i) and (3.1), we have
J λ ( u n ) = ( 1 2 1 p + 1 ) u n 2 ( 1 q + 1 1 p + 1 ) λ Ω f ( x ) | u n | q + 1 d x < α λ ( Ω ) + 1 n < 1 q q + 1 t λ 2 β ( Θ ) .
(3.3)
This implies
| f | p S q + 1 u n q + 1 Ω f ( x ) | u n | q + 1 d x > ( p + 1 ) ( 1 q ) λ ( p q ) t λ 2 β ( Θ ) > 0 ,
(3.4)
that is,
u n > [ ( p + 1 ) ( 1 q ) λ ( p q ) t λ 2 β ( Θ ) S ( q + 1 ) | f | p 1 ] 1 q + 1 .
(3.5)
Now, we will show that
J λ ( u n ) , φ 0 as  n , φ H ( Ω ) .
Exactly as in Lemma 2.8 we may apply suitable functionals ξ n ( v ) > 0 to u n and obtain
ξ n ( v ) ( u n + v ) M λ ( Ω ) , v H ( Ω ) , v < ϵ n .
(3.6)
Hence, if φ H ( Ω ) and s > 0 small, substituting in (3.6) v = s φ and applying (3.2), we have
1 n [ | ξ n ( s φ ) 1 | u n + ξ n ( s φ ) s φ ] J λ ( u n ) J λ ( ξ n ( s φ ) ( u n + s φ ) ) = 1 2 u n 2 λ q + 1 Ω f ( x ) | u n | q + 1 d x 1 p + 1 Ω h ( x ) | u n | p + 1 d x 1 2 ξ n 2 ( s φ ) u n + s φ 2 + λ q + 1 ξ n q + 1 ( s φ ) Ω f ( x ) | u n + s φ | q + 1 d x + 1 p + 1 ξ n p + 1 ( s φ ) Ω h ( x ) | u n + s φ | p + 1 d x = ξ n 2 ( s φ ) 1 2 u n + s φ 2 1 2 ( u n + s φ 2 u n 2 ) + λ ξ n q + 1 ( s φ ) 1 q + 1 Ω f ( x ) | u n + s φ | q + 1 d x + λ q + 1 Ω f ( x ) ( | u n + s φ | q + 1 | u n | q + 1 ) d x + ξ n p + 1 ( s φ ) 1 p + 1 Ω h ( x ) | u n + s φ | p + 1 d x + 1 p + 1 Ω h ( x ) ( | u n + s φ | p + 1 | u n | p + 1 ) d x .
Dividing by s > 0 and passing to the limit as s 0 we derive
1 n [ | ξ n ( 0 ) φ | u n + φ ] [ ξ n ( 0 ) φ ] [ u n 2 λ Ω f ( x ) | u n | q + 1 d x Ω h ( x ) | u n | p + 1 d x ] Ω Δ u n Δ φ d x + λ Ω f ( x ) | u n | q 1 u n φ d x + Ω h ( x ) | u n | p 1 u n φ d x = Ω Δ u n Δ φ d x + λ Ω f ( x ) | u n | q 1 u n φ d x + Ω h ( x ) | u n | p 1 u n φ d x .
(3.7)
Since
ξ n ( 0 ) φ = 2 Ω Δ u n Δ φ λ ( q + 1 ) Ω f | u n | q 1 u n φ ( p + 1 ) Ω h | u n | p 1 u n φ ( 1 q ) u n 2 ( p q ) Ω h ( x ) | u n | p + 1 d x ,
by the boundedness of u n we get
ξ n ( 0 ) C 1 | ( 1 q ) u n 2 ( p q ) Ω h ( x ) | u n | p + 1 d x |
(3.8)

for a suitable positive constant C 1 .

Next, we show that | ( 1 q ) u n 2 ( p q ) Ω h ( x ) | u n | p + 1 d x | is bounded away from zero. Arguing by contradiction, assume that
( 1 q ) u n 2 ( p q ) Ω h ( x ) | u n | p + 1 d x = o ( 1 ) , n .
(3.9)
Since u n M λ ( Ω ) , we have
u n 2 = λ Ω f ( x ) | u n | q + 1 d x + Ω h ( x ) | u n | p + 1 d x ,
and consequently by (3.9),
p 1 1 q Ω h ( x ) | u n | p + 1 d x = λ Ω f ( x ) | u n | q + 1 d x + o ( 1 ) , n .
(3.10)
Then by (3.4), the Hölder inequality, Sobolev inequality and (3.9)-(3.10), we obtain
0 < ( λ λ ) Ω f ( x ) | u n | q + 1 d x p 1 1 q Ω h ( x ) | u n | p + 1 d x [ ( p q ) Ω h ( x ) | u n | p + 1 d x ( 1 q ) u n 2 ] q p p 1 λ Ω f ( x ) | u n | q + 1 d x = o ( 1 ) ,

moreover, u n = o ( 1 ) , which contradicts (3.5).

Thus, we get from (3.8) that
ξ n ( 0 ) C 2 , independent of  n .
Hence, by (3.7) it follows that
Ω Δ u n Δ φ d x λ Ω f ( x ) | u n | q 1 u n φ d x Ω h ( x ) | u n | p 1 u n φ d x C 3 n ,
which implies that J λ ( u n ) , φ 0 , as n .
  1. (ii)

    Similar to the arguments in (i), by Lemma 2.9 and Lemma 2.2, we can prove (ii). □

     

Now, we establish the existence of a local minimum for J λ on M λ + ( Ω ) .

Theorem 3.1 Let λ ( 0 , λ ) , then the functional J λ has a minimizer u 0 + in M λ + ( Ω ) and it satisfies
  1. (i)

    J λ ( u 0 + ) = α λ ( Ω ) = α λ + ( Ω ) ;

     
  2. (ii)

    u 0 + is a positive solution of problem (1.3);

     
  3. (iii)

    J λ ( u 0 + ) 0 as λ 0 .

     
Proof By Proposition 3.1(i), there is a minimizing sequence { u n } for J λ on M λ ( Ω ) such that
J λ ( u n ) = α λ ( Ω ) + o ( 1 ) and J λ ( u n ) = o ( 1 ) in  [ H ( Ω ) ] .
(3.11)
Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence { u n } and u 0 + H ( Ω ) such that
u n u 0 + weakly in  H ( Ω ) ,
(3.12)
u n u 0 + strongly in  L p + 1 ( Ω )
(3.13)
and
u n u 0 + strongly in  L q + 1 ( Ω ) .
(3.14)
First, we claim that
Ω f ( x ) | u 0 + | q + 1 d x > 0 .
If not, by (3.14) we conclude that
Ω f ( x ) | u n | q + 1 d x Ω f ( x ) | u 0 + | q + 1 d x 0 as  n .
Therefore, as n ,
J λ ( u n ) = 1 2 u n 2 λ q + 1 Ω f ( x ) | u n | q + 1 d x 1 p + 1 Ω h ( x ) | u n | p + 1 d x = ( 1 2 1 q + 1 ) λ Ω f ( x ) | u n | q + 1 d x + ( 1 2 1 p + 1 ) Ω h ( x ) | u n | p + 1 d x = ( 1 2 1 q + 1 ) λ Ω f ( x ) | u 0 + | q + 1 d x + ( 1 2 1 p + 1 ) Ω h ( x ) | u 0 + | p + 1 d x + o ( 1 ) ,

this contradicts J λ ( u n ) α λ ( Ω ) < 0 as n .

In combination with (3.11)-(3.14), it is easy to verify that u 0 + M λ ( Ω ) is a nontrivial weak solution of problem (1.3).

Now we prove that u n u 0 + strongly in H ( Ω ) . Supposing the contrary, then u 0 + < lim inf n u n and so
u 0 + 2 λ Ω f ( x ) | u 0 + | q + 1 d x Ω h ( x ) | u 0 + | p + 1 d x < lim inf n ( u n 2 λ Ω f ( x ) | u n | q + 1 d x Ω h ( x ) | u n | p + 1 d x ) = 0 ,
this contradicts u 0 + M λ ( Ω ) . Hence, u n u 0 + strongly in H ( Ω ) . This implies
J λ ( u n ) J λ ( u 0 + ) = α λ ( Ω ) as  n .
Moreover, we have u 0 + M λ + ( Ω ) . In fact, if u 0 + M λ ( Ω ) , by Lemma 2.4, there exist unique t 0 + and t 0 such that t 0 + u 0 + M λ + ( Ω ) and t 0 u 0 + M λ ( Ω ) , we get t 0 + < t 0 = 1 . Since
d J λ ( t u ) d t = 0 if and only if t = t 0 +  and  t 0
and
d 2 J λ ( t u ) d t 2 | t = t 0 + > 0 , d 2 J λ ( t u ) d t 2 | t = t 0 < 0 ,
there exists t ˜ ( t 0 + , t 0 ] such that J λ ( t 0 + u 0 + ) < J λ ( t ˜ u 0 + ) . By Lemma 2.4,
J λ ( t 0 + u 0 + ) < J λ ( t ˜ u 0 + ) J λ ( t 0 u 0 + ) = J λ ( u 0 + ) ,
which is a contradiction. Since J λ ( u 0 + ) = J λ ( | u 0 + | ) and | u 0 + | M λ + ( Ω ) , by Lemma 2.3 we may assume that u 0 + is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we find that u 0 + is one positive solution of problem (1.3). In addition, by Lemma 2.7,
0 > J λ ( u 0 + ) λ 2 1 q C ( p , q ) ( | f | p S q + 1 ) 2 1 q ,

which implies that J λ ( u 0 + ) 0 as λ 0 . □

Next, we establish the existence of a local minimum for J λ on M λ ( Ω ) .

Theorem 3.2 Let λ ( 0 , λ ) , then the functional J λ has a minimizer u 0 in M λ ( Ω ) and it satisfies
  1. (i)

    J λ ( u 0 ) = α λ ( Ω ) ;

     
  2. (ii)

    u 0 is a positive solution of problem (1.3).

     
Proof By Proposition 3.1(ii), there is a minimizing sequence { u n } for J λ on M λ ( Ω ) such that
J λ ( u n ) = α λ ( Ω ) + o ( 1 ) and J λ ( u n ) = o ( 1 ) in  [ H ( Ω ) ] .
Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence { u n } and u 0 H ( Ω ) such that
u n u 0 weakly in  H ( Ω ) , u n u 0 strongly in  L p + 1 ( Ω )
and
u n u 0 strongly in  L q + 1 ( Ω ) .

Connecting with Lemma 2.2, it is easy to see that u 0 M λ ( Ω ) is a nontrivial weak solution of problem (1.3).

Next we prove that u n u 0 strongly in H ( Ω ) . Supposing the contrary, then u 0 < lim inf n u n and so
u 0 2 λ Ω f ( x ) | u 0 | q + 1 d x Ω h ( x ) | u 0 | p + 1 d x < lim inf n ( u n 2 λ Ω f ( x ) | u n | q + 1 d x Ω h ( x ) | u n | p + 1 d x ) = 0 ,
this contradicts u 0 M λ ( Ω ) . Hence, u n u 0 strongly in H ( Ω ) . This implies
J λ ( u n ) J λ ( u 0 ) = α λ ( Ω ) as  n .

In addition, from Lemma 2.4(ii)-(iii), we have u 0 M λ ( Ω ) . Since J λ ( u 0 ) = J λ ( | u 0 | ) and | u 0 | M λ ( Ω ) , by Lemma 2.3 we may assume that u 0 is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we see that u 0 is one positive solution of problem (1.3). □

Proof of Theorem 1.1 It is an immediate consequence of Theorems 3.1 and 3.2. □

Declarations

Acknowledgements

This work is partly supported by NNSF (11101404, 11201204, 11361053) of China and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNU-LKQN-11-5).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Lanzhou University
(2)
Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University
(3)
College of Mathematics and Statistics, Northwest Normal University

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© Yang and Wang; licensee Springer. 2014

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