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

Abstract

In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is λf(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.

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 (N3), g λ :Ω×RR 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 =λu+ | 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 λf(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 (N4), 0<q<1<p< 2 ( 2 = N + 4 N 4 for N>4 and 2 = for N=4), λ>0 is a parameter, fC( Ω ¯ ) is a positive or sign-changing weight function and hC( 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 Su for all uH(Ω).

Now we define

J λ (u)= 1 2 u 2 λ q + 1 Ω f(x) | u | q + 1 dx 1 p + 1 Ω h(x) | u | p + 1 dx,uH(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)={uH(Ω){0} J λ (u),u=0}. Define

ψ λ (u)= J λ ( u ) , u = u 2 λ Ω f(x) | u | q + 1 dx Ω h(x) | u | p + 1 dx.

Then for u M λ (Ω),

ψ λ ( u ) , u =2 u 2 λ(q+1) Ω f(x) | u | q + 1 dx(p+1) Ω h(x) | u | p + 1 dx.

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 dx+ Ω h(x) | u | p + 1 dx
(2.1)

and

2 u 2 =λ(q+1) Ω f(x) | u | q + 1 dx+(p+1) Ω h(x) | u | p + 1 dx.
(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 dx 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 dxλ 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 dx> | 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 dx.

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 dx> | 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 θR 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 uH(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 uH(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 (tu);

  2. (ii)

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

  3. (iii)

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

  4. (iv)

    if Ω f(x) | u | q + 1 dx>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 (tu).

Proof (i) Fix uH(Ω){0}. Let

s(t)= t 1 q u 2 t p q Ω h(x) | u | p + 1 dx,t0.

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 dx0.

There is a unique t > t max such that s( t )=λ Ω f(x) | u | q + 1 dx 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 (tu).

Case II. Ω f(x) | u | q + 1 dx>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 dx=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 (tu), J λ ( t + u ) = min 0 t t J λ (tu).
  1. (ii)

    By the uniqueness of t (u) and the external property of t (u), we find that t (u) is continuous function of u0.

  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 uH(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 fC( Ω ¯ ) and changes sign in Ω, we have Θ={xΩf(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 dx,uH(Θ)

and the minimization problem

β(Θ)=inf { K ( u ) u N ( Θ ) } ,

where N(Θ)={uH(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 uH(Θ){0}, there exists a unique t(u)>0 such that t(u)uN(Θ). The maximum of K(tu) for t0 is reached at t=t(u), the map

t:H(Θ){0}(0,+);ut(u)

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

Proof For any given uH(Θ){0}, consider the function g(t)=K(tu), t0. Clearly,

g (t)=0tuN(Θ) u 2 = t p 1 Ω h(x) | u | p + 1 dx.
(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)uN(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), ut(u)u, is inverse to the retraction u u u . □

Define

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

where Γ={γC([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(tu)<0 for uH(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 (tu),u0, t[0,t(u)]. Then each γΓ 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 dx= Θ f(x) w 0 q + 1 dx>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 dx+ Ω h(x) | u | p + 1 dx. 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 vH(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 dx ξ p + 1 Ω h(x) | u + w | p + 1 dx.

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 vH(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 vB(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 dx> ( 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 0as n,φH(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),vH(Ω),v< ϵ n .
(3.6)

Hence, if φH(Ω) and s>0 small, substituting in (3.6) v=sφ 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 s0 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 |(1q) u n 2 (pq) Ω h(x) | u n | p + 1 dx| is bounded away from zero. Arguing by contradiction, assume that

(1q) u n 2 (pq) Ω h(x) | u n | p + 1 dx=o(1),n.
(3.9)

Since u n M λ (Ω), we have

u n 2 =λ Ω f(x) | u n | q + 1 dx+ Ω h(x) | u n | p + 1 dx,

and consequently by (3.9),

p 1 1 q Ω h(x) | u n | p + 1 dx=λ Ω f(x) | u n | q + 1 dx+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 Δφdxλ Ω f(x) | u n | q 1 u n φdx Ω h(x) | u n | p 1 u n φdx C 3 n ,

which implies that J λ ( u n ),φ0, 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 λ0.

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 dx>0.

If not, by (3.14) we conclude that

Ω f(x) | u n | q + 1 dx Ω f(x)| u 0 + | q + 1 dx0as 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 =0if and only ift= 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 λ0. □

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. □

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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).

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Yang, L., Wang, X. On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions. Bound Value Probl 2014, 117 (2014). https://doi.org/10.1186/1687-2770-2014-117

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