Open Access

Positive solutions for P-Laplace problems with nonlinear time-fractional differential equation

Journal of Inequalities and Applications20142014:262

https://doi.org/10.1186/1029-242X-2014-262

Received: 20 March 2014

Accepted: 13 June 2014

Published: 22 July 2014

Abstract

In this paper, we study the existence and multiplicity of positive solutions for semi-linear elliptic equations with a sign-changing weight function in weighted Sobolev spaces. By investigating the compact embedding theorem and based on the extraction of the Palais-Smale sequence in the Nehari manifold which is a subset of the weighted Sobolev spaces, we derive the existence of the multiple positive solutions of the equations by using the variational method. In the last part of this paper, by applying the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations for semi-linear elliptic equations are obtained.

MSC:35Q30, 76D05, 76D10.

Keywords

Nehari manifoldconcave-convex nonlinearitiespositive weight functionweighted Sobolev spacetime-fractional equationfixed point theorem

1 Introduction

In this paper, we consider the multiplicity results of positive solutions for the following semi-linear problem:
( E λ f , h ) { div ( a ( x ) | u ( x ) | p 2 u ( x ) ) + ( u ( x ) ) p 1 = λ f ( x ) | u ( x ) | q 1 + h ( x ) | u ( x ) | r 1 , in  Ω , u ( x ) = 0 , on  Ω ,

u W 0 1 , p ( a , Ω ) , where λ is a real positive parameter, 1 < q < p < r < p s ( p > 2 , p s = p s s + 1 , p s = N p s N p s , s ( N p , ) [ 1 p 1 , ) , p s < N ( s + 1 ) ). Ω is a bounded region with smooth boundary in R N ; a ( x ) , f ( x ) , h ( x ) are measurable functions and satisfy the following conditions:

(H1) 0 f ( x ) L H ( Ω ) , where L H ( Ω ) = L r r q ( Ω ) , q < r < p s , and f ( x ) has a compact support in Ω,

(H2) 0 h ( x ) L ( Ω ) , and it satisfies h ( x ) 1 as | x | ,

(H3) a ( x ) is a positive weight function, locally Hölder continuous, and almost everywhere with positive measure in the Sobolev space W 0 1 , p ( a ( x ) , Ω ) which comes with the standard norm u = { Ω ( a | u ( x ) | p + ( u ( x ) ) p ) d x } 1 p and there exists υ ( x ) if and only if υ ( x ) c 1 a ( x ) c 1 υ ( x ) , where c 1 1 and υ ( x ) is another weight function, which satisfies υ ( x ) L loc 1 ( Ω ) , υ ( x ) 1 p 1 L loc 1 ( Ω ) , υ ( x ) s L 1 ( Ω ) .

The problem ( E λ f , h ) is an important and basic mathematical model, widely used in many fields. For specific theoretical implications of the above model, one can refer to Drábek, Kufner and Nicolosi [1] and Adams and John [2] and references wherein.

Analogous equations with nonlinearities concave-convex in bounded domains are widely studied. For example, Ambrosetti [3] studied the problem below:
( E λ ) { Δ u = λ | u | q + | u | p , in  Ω , u > 0 , in  Ω , u = 0 , on  Ω ,
where 0 < q < 1 < p 2 1 . They proved the existence of λ 0 > 0 such that the problem ( E λ ) admits at least two positive solutions for λ ( 0 , λ 0 ) ; there is one positive solution for λ = λ 0 , and no positive solution exists for λ > λ 0 . Recently, for Ω = B N ( 0 , 1 ) , that is, Ω is a unit ball, Adimurthi and Yadava [4], Damascelli et al. [5] and Tang [6] proved that there are exactly two solutions for λ ( 0 , λ 0 ) ; one positive solution for λ = λ 0 and no positive solution exists for λ > λ 0 . When p 2 , h ( x ) = 1 and a ( x ) 1 , Wu [7] has investigated equation ( E λ f , 1 ), and he found that there exists λ 0 > 0 such that equation ( E λ f , 1 ) admits at least two positive solutions for λ ( 0 , λ 0 ) . Among other interesting results, Miotto and Miyagaki [8] have studied the following equation:
{ Δ u + u = λ f ( x ) | u | q 1 + h ( x ) | u | p 1 , in  Ω , u 0 , on  Ω ,
(1.1)
where λ > 0 , 1 < q < 2 < p < 2 ( 2 = 2 N N 2 if N 3 , 2 = if N = 2 ), Ω = Ω × R ( Ω R N 1 ) is an infinite strip domains, assuming that f ( x ) L r r q ( Ω ) = L 1 , where q < r 2 , with f + 0 and f is bounded and has a compact support in Ω. 0 h ( x ) L ( Ω ) satisfies lim | x N | h ( x , x N ) = 1 and there exists c 0 > 0 , θ 1 being the first eigenvalue of the Dirichlet problem −Δ in Ω , such that
h ( x , x N ) 1 c 0 e 2 1 + θ 1 | x N | ,
for all x = ( x , x N ) Ω . They proved that the existence of Λ = Λ ( q , p , h L , r ) such that the problem (1.1) has at least two positive solutions for all λ ( 0 , Λ f L 1 1 ) , Wu in [9] has studied (1.1) under the assumption that 0 f L 2 2 q ( Ω ) , 0 < h C ( Ω ) satisfying lim | x N | h ( x ) = 1 in Ω = Ω × R and there exist δ > 0 and 0 < c 0 < 1 such that
h ( x , x N ) 1 c 0 e 2 1 + θ 1 + δ | x N | ,

for all ( x , x N ) Ω . The existence of Λ 0 > 0 was obtained such that for λ ( 0 , Λ 0 ) the problem (1.1) possesses at least two positive solutions.

We consider the P-Laplace Dirichlet problem above. In the following we will switch our view point to investigate the existence of positive solutions for the corresponding nonlinear time-fractional differential equation of the problem ( E λ f , h ). We know that the subject of fractional differential equations has emerged as an important area of investigation by the fact that it has numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, porous media, and so on, thus the subject of fractional differential equations is gaining much importance and attention. For some recent developments on the subject, please see the literature [1017], and the references therein for more details. But not many people pay attention to the study of the P-Laplace problems with nonlinear partial differential equations of time-fractional order, except the literature such as [18, 19], but the aim of this paper is to do so, too. To the best of our knowledge, the results in this paper are new and original as we have not found any discussion in the existing literature.

The paper is organized as follows. In Sections 2 and 3, we show that equation ( E λ f , h ) in weighted Sobolev space has at least two positive solutions for λ sufficiently small. First, we recall the Nehari manifold which is a subset of the weighted Sobolev space, and analyze the behavior of the energy functional associated with our problems on the Nehari manifold. Moreover, by extracting the Palais-Smale sequences in the Nehari manifold and combining the properties of the compact embedding theorem in weighted Sobolev space, we obtain the result that there exist at least two positive solutions of the problem ( E λ f , h ). In Section 4, we shall consider the following time-fractional differential equations derived from ( E λ f , h ):
( E λ f , h , t ) { D α u ( x , t ) = div ( a ( x ) | u ( x , t ) | p 2 u ( x , t ) ) + ( u ( x , t ) ) p 1 D α u ( x , t ) = + λ f ( x ) | u ( x , t ) | q 1 + h ( x ) | u ( x , t ) | r 1 , in  Ω T , u ( x , t ) = 0 , on  Ω T , u ( x , 0 ) = ϕ ( x ) , in  Ω , u t ( x , 0 ) = ψ ( x ) , in  Ω ,

where Ω T = Ω × [ 0 , T ] , D α denotes the Caputo fractional derivative (e.g., see [13]), 1 < α < 2 is a parameter describing the order of the fractional time, and ϕ ( x ) , ψ ( x ) H 0 1 ( a ( x ) , Ω ) are given real-valued functions. Then the problem ( E λ f , h , t ) is deduced to an equivalent integral equation under the fractional order integral operator I α . Finally, we prove the existence of solution for the time-fractional differential equations by using the Arzela-Ascoli fixed point theorem. The conclusion is given by Section 5.

2 Notations and preliminaries

In the following, we first consider the positive solutions of the following problem.

Theorem 2.1 There exists λ 0 = λ 0 ( q , p , h , r , f L H 1 ) > 0 such that for λ ( 0 , λ 0 ) , the equation ( E λ f , h ) has at least two positive solutions.

In order to prove it, we need the following lemma.

Lemma 2.2 If 2 < p < r < p s , X L r ( Ω ) is the compact embedding, then X L p ( Ω ) is also the compact embedding, where X = W 0 1 , p ( a ( x ) , Ω ) (e.g., see Drábek, Kufner and Nicolosi [1]).

Throughout this section, we denote by S r the best Sobolev constant for the embedding of W 0 1 , p ( a ( x ) , Ω ) in L r ( Ω ) . We define
S r = sup u W 0 1 , p ( a ( x ) , Ω ) { 0 } { u L r u X } .

For the convenience we will denote W 0 1 , p ( a ( x ) , Ω ) by X, and X by if there is no confusion, unless otherwise stated, and the integrals are over Ω. Now we give the proof of Theorem 2.1.

Proof Associated with equation ( E λ f , h ), we define the energy functional I λ in X for given λ 0 , f ( x ) and h ( x ) by
I λ ( u ) = 1 p Ω ( a | u | p + u p ) d x 1 q λ Ω ( f ( x ) | u | q ) d x 1 r Ω ( h ( x ) | u | r ) d x .
It is clear that I λ is of class C 1 with Gâteaux derivative I λ ( u ) at each u X given by
I λ ( u ) , φ = Ω ( a | u | p 1 φ + u p 1 φ ) d x λ Ω ( f | u | q 2 u φ ) d x Ω ( h | u | r 2 u φ ) d x ,

for all φ X . Therefore the (weak) solutions of equation ( E λ f , h ) are the critical points of the energy functional I λ (see Rabinowitz [20]).

As the energy functional I λ is not bounded from below on X, it is useful to consider the functional on the Nehari manifold which has the best behavior subset of X (see Brown and Zhang [21]). For any λ > 0 , we define
N λ = { u X { 0 } : I λ ( u ) , u = 0 } .
Then u N λ if and only if
I λ ( u ) , u = u p λ Ω ( f | u | q ) d x Ω ( h | u | r ) d x = 0 .
(2.1)

 □

Note that any nonzero solution of the problem ( E λ f , h ) belongs to N λ . Furthermore, we have the following result.

Lemma 2.3 The functional I λ is coercive and bounded from below on N λ .

Proof Let u N λ be arbitrary. Then by (2.1) and by the Hölder and Sobolev inequalities we get
I λ ( u ) r p p r u p ( r q q r ) λ ( Ω | f | r r q d x ) r q r ( Ω | u | r d x ) q r r p p r u p ( r q q r ) λ f L H S r q u q .

Since q < p < r , it follows that I λ is bounded from below and coercive on N λ provided λ is small enough. □

Next, we consider the Nehari minimization problem; for λ 0 , define α λ = inf u N λ I λ ( u ) and
ψ λ ( u ) = I λ ( u ) , u = u p λ Ω ( f | u | q ) d x Ω ( h | u | r ) d x .
Then for u N λ by (2.1) we have
ψ λ ( u ) , u = p u p q λ Ω ( f | u | q ) d x r Ω ( h | u | r ) d x = ( p q ) u p ( r q ) Ω ( h | u | r ) d x
(2.2)
= ( p r ) u p ( q r ) λ Ω ( f | u | q ) d x .
(2.3)
Now, we split N λ into three parts (see Drábek, Kufner and Nicolosi [1] and Ambrosetti et al. [3]).
N λ + = { u N λ : ψ λ ( u ) , u > 0 } , N λ 0 = { u N λ : ψ λ ( u ) , u = 0 } , N λ = { u N λ : ψ λ ( u ) , u < 0 } .

Then we have the following result.

Lemma 2.4 There exists λ 1 > 0 such that for each λ ( 0 , λ 1 ) , we have N λ 0 = .

Proof We consider the following two cases.

Case (I) u N λ ( Ω ) and Ω ( f ( x ) | u | q ) d x = 0 . We then have
u p Ω ( h | u | r ) d x = 0 .
Thus
ψ λ ( u ) , u = ( p q ) u p ( r q ) Ω ( h | u | r ) d x = [ ( p q ) ( r q ) ] u p = ( p r ) u p < 0 .

Hence u N λ 0 ( Ω ) .

Case (II) u N λ ( Ω ) and Ω ( f ( x ) | u | q ) d x 0 . Suppose that N λ 0 for all λ > 0 . If u N λ 0 , then we have
0 = ψ λ ( u ) , u = p u p q λ Ω ( f | u | q ) d x r Ω ( h | u | r ) d x = ( p q ) u p ( r q ) Ω ( h | u | r ) d x .
Thus
0 < u p = r q p q Ω ( h | u | r ) d x ,
(2.4)
and
0 < λ Ω ( f | u | q ) d x = u p Ω ( h | u | r ) d x = r p p q Ω ( h | u | r ) d x .
(2.5)
Moreover, by the Hölder and Sobolev inequalities, for all u X , we obtain
λ Ω ( f | u | q ) d x λ f L H S r q u q .
(2.6)
Thus for any u N λ 0 , by (2.4)-(2.6) we obtain
u ( r q r p ) 1 p q S r q p q ( λ f L H ) 1 p q .
(2.7)
Let J λ : N λ ( Ω ) R be given by
J λ ( u ) = K ( q , r ) ( u r Ω ( h | u | r ) d x ) p r p λ Ω ( f | u | q ) d x ,
where K ( q , r ) = ( r p p q ) ( p q r q ) r r p . Then J λ ( u ) = 0 for all u N λ 0 . Indeed, from (2.4)-(2.5), it follows that, for u N λ 0 , we have
J λ ( u ) = K ( q , r ) ( u r Ω ( h | u | r ) d x ) p r p λ Ω ( f | u | q ) d x = ( r p p q ) ( p q r q ) r r p ( u p ) r r p ( Ω ( h | u | r ) d x ) p r p r p p q Ω ( h | u | r ) d x = ( r p p q ) ( p q r q ) r r p ( r q p q ) r r p ( Ω ( h | u | r ) d x ) r r p ( Ω ( h | u | r ) d x ) p r p r p p q Ω ( h | u | r ) d x = r p p q Ω ( h | u | r ) d x r p p q Ω ( h | u | r ) d x = 0 .
(2.8)
However, by (2.7) and the Hölder and Sobolev inequalities, for u N λ 0 ,
J λ ( u ) K ( q , r ) ( u r Ω ( h | u | r ) d x ) p r p λ f L H S r r u r K ( q , r ) ( c ˜ ) p r p λ f L H S r r u r ,

where c ˜ = ( h L S r r ) 1 . This implies that for λ sufficiently small we have J λ ( u ) > 0 for all u N λ 0 , this contradicts (2.8). Thus, we can conclude that there exists λ 1 > 0 such that for λ ( 0 , λ 1 ) , we have N λ 0 = . The proof is complete. □

Lemma 2.5 If u N λ + , then Ω ( f | u | q ) d x > 0 .

Proof We have
u p λ Ω ( f | u | q ) d x Ω ( h | u | r ) d x = 0 ,
and
u p > r q p q Ω ( h | u | r ) d x .
Thus
λ Ω ( f | u | q ) d x = u p Ω ( h | u | r ) d x > r p p q Ω ( h | u | r ) d x > 0 ,

and this completes the proof. □

By Lemma 2.4, for λ ( 0 , λ 1 ) we write
N λ = N λ + N λ ,
and we define
α λ + ( Ω ) = inf u N λ + I λ ( u ) , α λ ( Ω ) = inf u N λ I λ ( u ) .

The following results show that minimizers on N λ are the ‘usual’ critical points for I λ .

Lemma 2.6 For λ ( 0 , λ 1 ) , if u 0 N λ is a local minimizer point for I λ on N λ and u 0 N λ 0 , then I λ ( u 0 ) = 0 in X 1 ( Ω ) .

Proof If u 0 is a local minimizer point for I λ on N λ , then u 0 is a solution of the following optimization problem:
inf ψ λ ( u ) = 0 I λ ( u ) .
Hence, by the theory of Lagrange multipliers, there exists θ R such that
I λ ( u 0 ) = θ ψ λ ( u 0 ) ,
in X 1 ( Ω ) . Thus
I λ ( u 0 ) , u 0 = θ ψ λ ( u 0 ) , u 0 .
(2.9)
Since u 0 N λ , I λ ( u 0 ) , u 0 = 0 and so
u 0 p λ Ω ( f | u 0 | q ) d x Ω ( h | u 0 | r ) d x = 0 .
Hence
ψ λ ( u 0 ) , u 0 = ( p q ) u 0 p ( r q ) Ω ( h | u 0 | r ) d x .

Thus, if u 0 N λ 0 , ψ λ ( u 0 ) , u 0 0 and so by (2.9), θ = 0 . This completes the proof. □

For each u N λ X { 0 } , we have
u p λ Ω ( f | u | q ) d x Ω ( h | u | r ) d x = 0 ,
so we have λ Ω ( f | u | q ) d x = u p Ω ( h | u | r ) d x . By (2.2), we define the fiber map ϕ u ( t ) = I λ ( t u ) , and we let ϕ u ( t ) = 0 , i.e.,
t p 1 u p t q 1 λ Ω ( f | u | q ) d x t r 1 Ω ( h | u | r ) d x = 0 .
Hence
( t p q 1 ) u p ( t r q 1 ) Ω ( h | u | r ) d x = 0 .
By the Lagrange mean theorem, there exists a t ( ξ ) such that
u p Ω ( h | u | r ) d x = t r q 1 t p q 1 = ( r q ) t ( ξ ) r q 1 ( p q ) t ( ξ ) p q 1 = r q p q t ( ξ ) r p .
In particular, we have
0 < t max ( u ) = t ( ξ ) = ( ( p q ) u p ( r q ) Ω ( h | u | r ) d x ) 1 r p .
Lemma 2.7 Let H = r r q and λ 2 = ( r r q ) ( p q r q ) p q r p S 2 ( q r ) r p ( f L H ) 1 . Then for each u X { 0 } and λ ( 0 , λ 2 ) , we have
  1. (i)
    there is a unique t = t ( u ) > t max > 0 such that t ( u ) u N λ and
    I λ ( t u ) = sup t t max I λ ( t u ) > 0 ;
     
  2. (ii)
    if Ω ( f ( x ) | u | q ) d x > 0 , then there exists unique 0 < t + = t + ( u ) < t max such that t + ( u ) u N λ + and
    I λ ( t + u ) = inf 0 t t max I λ ( t u ) ;
     
  3. (iii)

    there exists a continuous bijection between U = { u X { 0 } : u = 1 } and N λ , in particular, t ( u ) is a continuous function for nonzero u.

     
Proof (i) Fix u X { 0 } , let
s ( t ) = t p q u p t r q Ω ( h ( x ) | u | r ) d x ,
for t 0 , we have s ( 0 ) = 0 , s ( t ) ( ) as t , and by (2.2), we have
s ( t ) = t p q u p t r q ( p q r q ) u p = ( t p q t r q p q r q ) u p ,
since 1 < q < p < r , so 0 < p q r q < 1 , p q < r q , hence, if 0 < t < 1 , then t p q > t r q , and so s ( t ) < 0 , therefore, s ( t ) can achieve its maximum at t max . Moreover,
s ( t max ) t p q u p t r q h L S r r u r ( p q r q ) p q r p u p ( r q ) r p ( c ˜ 1 u r ) p q r p ( c ˜ 1 ) ( p q r q ) r q r p u r 2 p q r p ( c ˜ 1 ) r q r p u r 2 q r r p ( p q r q ) p q r p ( c ˜ ) p q r p u q ( c ˜ 1 ) q p r p ( p q r q ) r q r p u q K ( q , r ) ( c ˜ ) p q r p u q S r q λ f L H u q λ Ω ( f | u | q ) d x .
(2.10)

Note that t u N λ , if and only if s ( t ) = λ Ω ( f | u | q ) d x .

Case (I) Suppose that λ Ω ( f | u | q ) d x 0 , there is a unique t > t max such that s ( t ) = λ Ω ( f | u | q ) d x and s ( t ) < 0 . Now
( p q ) t u p ( r q ) Ω ( h | t u | r ) d x = ( t ) q + 1 [ ( p q ) ( t ) p q 1 u p ( r q ) t r q 1 Ω ( h | t u | r ) d x ] = ( t ) q + 1 s ( t ) < 0 ,
and
I λ ( t u ) , t u = ( t ) p u p ( t ) q λ Ω ( f | u | q ) d x ( t ) r Ω ( h | u | r ) d x = ( t ) q [ s ( t ) λ Ω ( f | u | q ) d x ] = 0 .
Thus t u = t ( u ) u N λ , since for t > t max , we have
ψ ( t u ) , t u = ( p q ) t u p ( r q ) Ω ( h | t u | r ) d x < 0 , d 2 d t 2 I λ ( t u ) < 0 ,
and
d d t I λ ( t u ) = t p 1 Ω ( a | u ( x ) | p ) d x t q 1 λ Ω ( f | u | q ) d x t r 1 Ω ( h | u | r ) d x = t p 1 u p t q 1 λ Ω ( f | u | q ) d x t r 1 Ω ( h | u | r ) d x = 0 ,

for t = t . Therefore I λ ( t u ) = sup t t max I λ ( t u ) .

Case (II) If λ Ω ( f | u | q ) d x > 0 , by (2.10) and
s ( 0 ) = 0 < λ Ω ( f | u | q ) d x λ f L H S r q u q < s ( t max ) ,
for λ ( 0 , λ 2 ) , there are unique t + and t such that 0 < t + < t max < t ,
s ( t + ) = λ Ω ( f | u | q ) d x = s ( t ) ,
and s ( t + ) > 0 > s ( t ) we have t + ( u ) ( u ) N λ + , t ( u ) ( u ) N λ and
I λ ( t ( u ) u ) I λ ( t ( u ) u ) I λ ( t + ( u ) u )
for each t ( u ) [ t + ( u ) , t ( u ) ] and I λ ( t + ( u ) u ) I λ ( t ( u ) u ) for each t ( u ) [ 0 , t + ( u ) ] , thus I λ ( t u ) = sup t t max I λ ( t u ) , I λ ( t + u ) = inf 0 t t max I λ ( t u ) .
  1. (ii)

    by Case (II) of part (i).

     
  2. (iii)
    Fix u U , define G u : ( 0 , ) × U R by
    G u ( t , w ) = I λ ( t w ) , t w .
     
Since G u ( t ( u ) , u ) = I λ ( t ( u ) u ) , t ( u ) u = 0 , and
G u t ( t ( u ) , u ) = [ t ( u ) ] 1 ψ λ ( t ( u ) u ) , t ( u ) u < 0 ,

then by the implicit function theorem, there is a neighborhood W u of u in U and an unique continuous function T u : W u ( 0 , ) such that G u ( T u ( w ) , w ) = 0 for all w W u , in particular, T u ( u ) = t ( u ) . Since u U is arbitrary, we find that the function T : U ( 0 , ) , given by T u ( u ) = t ( u ) is continuous and one-to-one. Having T : U N λ , where T ( u ) = t ( u ) u , we find that T is continuous and one-to-one. Now if u N λ then we have T ( w ) = u , where w = u u , since t is continuous on U, it follows that t is continuous for nonzero u. Then the proof is complete. □

Lemma 2.8 There exists a positive number Λ 0 λ 1 ( λ 1 defined in Lemma  2.4) such that if λ ( 0 , Λ 0 ) , then
  1. (i)

    α λ + < 0 ,

     
  2. (ii)

    α λ > 0 and α λ + < α λ , in particular, α λ = α λ + .

     
Proof (i) Let u N λ + , by (2.3)
( r p r q ) u p < λ Ω ( f | u | q ) d x ,
(2.11)
and so
I λ ( u ) = r p p r u p ( r q q r ) λ Ω ( f | u | q ) d x < ( r p ) ( p q ) p q r u p < 0 .

Thus, α λ + < 0 .

(ii) Let u N λ , by (2.2) and the Sobolev embedding theorem,
( p q r q ) u p < Ω ( h | u | r ) d x S r r h L u r ,
and so
u > ( p q ( r q ) S r r h L ) 1 r p ,
for all u N λ , by the proof of Lemma 2.3
I λ ( u ) u q [ ( r p p r ) u p q ( r q q r ) λ f L H ] > ( p q ( r q ) S r r h L ) q r p [ ( r p p r ) ( p q ( r q ) S r r h L ) p q r p ( r q q r ) λ f L H ] .

Thus, there exists a positive number Λ 0 λ 1 such that if λ ( 0 , Λ 0 ) , then I λ ( u ) > 0 , i.e., α λ > 0 for all u N λ . Obviously, α λ + < α λ . This completes the proof. □

3 Proof of Theorem 2.1

First, by following the idea of Tarantello [22], we have the following result.

Lemma 3.1 For each u N λ , there exist ϵ > 0 and a differentiable function ξ : B ( 0 , ϵ ) X R + such that ξ ( 0 ) = 1 , the function ξ ( v ) ( u v ) N λ and
ξ ( u ) , v = p Ω ( a | u | p 1 v + u p 1 v ) d x + q λ Ω ( f | u | q 2 u v ) d x + r Ω ( h | u | r 2 u v ) d x ( p q ) u p ( r q ) Ω ( h | u | r ) d x for all  v ( x ) X .
(3.1)
Lemma 3.2 For each u N λ , there exist ϵ > 0 and a differentiable function ξ : B ( 0 , ϵ ) X R + such that ξ ( 0 ) = 1 , the function ξ ( v ) ( u v ) N λ and
ξ ( u ) , v = p Ω ( a | u | p 1 v + u p 1 v ) d x + q λ Ω ( f | u | q 2 u v ) d x + r Ω ( h | u | r 2 u v ) d x ( p q ) u p ( r q ) Ω ( h | u | r ) d x for all  v ( x ) X .
(3.2)

The proof of the two lemmas above is almost the same as given by Hsu [23] and thus we omit it.

Proposition 3.3 Let λ 0 = min { λ 1 , λ 2 , Λ 0 } . Then for λ ( 0 , λ 0 ) ,
  1. (i)
    there exists a minimizing sequence { u n } N λ such that
    I λ ( u n ) = α λ + o ( 1 ) = α λ + + o ( 1 ) , I λ ( u n ) = o ( 1 ) in  X 1 ;
     
  2. (ii)
    there exists a minimizing sequence { u n } N λ such that
    I λ ( u n ) = α λ + o ( 1 ) , I λ ( u n ) = o ( 1 ) in  X 1 .
     
Proof (i) by Lemma 2.3, and the Ekeland variational principle [24], there exists a minimizing sequence { u n } N λ + such that
I λ ( u n ) < α λ + + 1 n ,
(3.3)
I λ ( u n ) < I λ ( w ) + 1 n w u n w N λ + .
(3.4)
By taking n large, from Lemma 2.8(i), we have
I λ ( u n ) = ( 1 p 1 r ) u n p ( 1 q 1 r ) λ Ω ( f | u n | q ) d x < α λ + + 1 n < α λ + 2 ,
(3.5)
consequently u n 0 , and putting together (3.5), (2.6), and (2.7), we obtain for all n
[ α λ + 2 λ f L h S r q ( q r r q ) ] 1 q u n [ p q S r q ( r q r p ) ( λ f L H ) ] 1 p q .
(3.6)

Now, we will show that I λ ( u n ) 0 , as n .

Applying Lemma 3.1 with u n to obtain the functions ξ + : B ( 0 , ϵ ) R + for some ϵ n > 0 such that ξ + ( w ) ( u n w ) N λ + , choose 0 < ρ < ϵ n , let u X with u 0 and let w ρ = ρ u u . We set η ρ + = ξ n + ( w ρ ) ( u n w ρ ) , since η ρ + N λ + , we deduce from (3.4) that
I λ ( η ρ + ) I λ ( u n ) 1 n η ρ + u n ,
and by the mean value theorem, we have
I λ ( u n ) , η ρ + u n + o ( η ρ + u n ) 1 n η ρ + u n .
Thus
I λ ( u n ) , w ρ + ( ξ n + ( w ρ ) 1 ) I λ ( u n ) , ( u n w ρ ) 1 n η ρ + u n + o ( η ρ + u n ) .
(3.7)
From ξ n + ( w ρ ) ( u n w ρ ) N λ + and (3.7) it follows that
ρ I λ ( u n ) , u u + ( ξ n + ( w ρ ) 1 ) I λ ( u n ) I λ ( η ρ + ) , ( u n w ρ ) 1 n η ρ + u n + o ( η ρ + u n ) .
Thus
I λ ( u n ) , u u η ρ + u n n ρ + o ( η ρ + u n ) ρ + ( ξ n + ( w ρ ) 1 ) ρ I λ ( u n ) I λ ( η ρ + ) , ( u n w ρ ) .
(3.8)
Since η ρ + u n ρ | ξ n + ( w ρ ) | + | ξ n + ( w ρ 1 ) | u n and lim ρ 0 | ξ n + ( w ρ ) 1 | ρ ξ n ( w ) , if we let ρ 0 in (3.8) for a fixed n, then by (3.6) we can find a constant c > 0 , independent of ρ, such that
I λ ( u n ) , u u c n ( 1 + ξ n ( u ) ) ,
and we are done once we show that ( ξ n + ) ( u ) is uniformly bounded in n. By (3.1), (3.6), and (2.6), we have
( ξ n + ) ( u n ) , v b v | ( p q ) u n p ( r q ) Ω ( h | u n | r ) d x | ,
for some b > 0 . We only need to show that
| ( p q ) u n p ( r q ) Ω ( h | u n | r ) d x | > c ,
(3.9)
for some c > 0 , and n large enough. We argue by contradiction; assume that there exists a subsequence { u n } such that
( p q ) u n p ( r q ) Ω ( h | u n | r ) d x = o ( 1 ) ,
(3.10)
combining (3.10) with (3.6), we can find a suitable constant d > 0 such that
Ω ( h | u n | r ) d x d ,
(3.11)
for n sufficiently large. In addition (3.10), and the fact that u n N λ + also give
λ Ω ( f | u n | q ) d x = u n p Ω ( h | u n | r ) d x = r p p q Ω ( h | u n | r ) d x + o ( 1 ) ,
and the right side of (3.6) holds. This implies
I λ ( u ) = K ( q , r ) ( u r Ω ( h | u | r ) d x ) p r p λ Ω ( f | u | q ) d x = r p p q Ω ( h | u | r ) d x r p p q Ω ( h | u | r ) d x = o ( 1 ) .
(3.12)
However, by the right of (3.6), (3.11), and λ ( 0 , λ 0 ) ,
I λ ( u ) K ( q , r ) ( u r Ω ( h | u | r ) d x ) p r p λ f L H S r r u r K ( q , r ) ( c ˜ ) p r p λ f L H S r r u r > 0 ,
for λ sufficiently small, here c ˜ = ( h L S r r ) 1 . This contradicts (3.12), we get
I λ ( u n ) , u u c n ,

this completes the proof of (i).

(ii) Similarly, by using Lemma 3.2, we can prove (ii) and thus its proof and its details are omitted here. □

Theorem 3.4 Let λ 0 ( q , p , h , r , f L H 1 ) = λ 0 = min { λ 1 , λ 2 , Λ 0 } . Then for λ ( 0 , λ 0 ) , the functional I λ has a minimizer point u 0 + in N λ + and it satisfies
  1. (i)

    I λ ( u 0 + ) = α λ = α λ + ,

     
  2. (ii)

    u 0 + is a positive solution of equation ( E λ f , h ),

     
  3. (iii)

    u 0 + 0 as λ 0 .

     
Proof By Proposition 3.3(i), it follows that there exists { u n } N λ satisfying
I λ ( u n ) = α λ + o ( 1 ) = α λ + + o ( 1 ) , I λ ( u n ) = o ( 1 ) in  X 1 ( Ω ) .
We can infer that { u n } is bounded from below on X. Thus, passing a subsequence if necessary, there exists u 0 X , such that u n u 0 weakly in X. We get I λ ( u 0 ) = 0 . Suppose, by absurdity, that u 0 = 0 , then by (2.2), we have ( p q ) u n p ( r q ) ( h | u n | r ) d x = o ( 1 ) and by (2.1), we get
λ Ω ( f | u n | q ) d x = u n p Ω ( h | u n | r ) d x + o ( 1 ) = u n p p q r q u n p = r p r q u n p + o ( 1 ) ,
thus by the Egorov theorem we obtain Ω ( f | u n | q ) d x = o ( 1 ) . Since I λ ( u n ) = o ( 1 ) in X 1 ( Ω ) , we have
o ( 1 ) = I λ ( u n ) , u n = u n p Ω ( h | u n | r ) d x + o ( 1 ) .
By Lemma 2.8(i), it follows that α λ + < 0 , then considering n 0 N such that
1 r | u n p Ω ( h | u n | r ) d x | + 1 q | λ Ω ( f | u n | q ) d x | < α λ + 4 , I λ ( u n ) < α λ + 2 ,
for all n n 0 . Thus for all n n 0 , we get
( 1 p 1 r ) u n p < I λ ( u n ) α λ + 2 < 0 ,
which is an absurdity, because p < r . Hence, u 0 0 and since I λ ( u 0 ) = 0 , it follows that u 0 N λ and in particular, I λ ( u 0 ) α λ . We will show that, up to a subsequence, u n u 0 strongly in X. Suppose, for a contradiction, that u 0 < lim inf n u n . Since { u n } N λ and u 0 N λ , we have
α λ + I λ ( u 0 ) < lim inf n I λ ( u n ) = α λ + ,
which is a contradiction. Hence, we can suppose, up to a subsequence, that u n u 0 strongly in X. Note that u 0 N λ + , since u 0 N λ and I λ ( u 0 ) = α λ + < α λ . Considering u 0 + = | u 0 | , we get u 0 + 0 since u 0 X { 0 } . If u N λ + , then | u | N λ + and by I λ ( u 0 + ) = I λ ( u 0 ) = α λ we see that u 0 + N λ is a local minimum point of I λ on N λ . Then by Lemma 2.6. We find that u 0 + is a solution of the problem ( E λ f , h ). By the Harnack inequality according to Trudinger [25] we obtain u 0 + > 0 in Ω. Now by (2.3) we have
0 < ψ λ ( u 0 + ) , u 0 + = ( p r ) u 0 + p ( q r ) λ Ω ( f | u 0 + | q ) d x ( r p ) u 0 + p + ( r q ) λ f L H u 0 + q .
Then by (2.6), we infer that
( r p ) u 0 + p ( r q ) λ f L H u 0 + q ,
i.e.
u 0 + p q r q r p λ f L H ,
that is
u 0 + ( r q r p ) 1 p q ( λ f L H ) 1 p q = c ( λ f L H ) 1 p q ,
where c is a positive constant, independent of λ. So
u 0 + c ( λ f L H ) 1 p q ,

and thus we conclude the proof. □

Next, we establish the existence of a local minimum for I λ on N λ .

Theorem 3.5 Let λ 0 ( q , p , h , r , f L H 1 ) = λ 0 = min { λ 1 , λ 2 , Λ 0 } . Then for λ ( 0 , λ 0 ) , the functional I λ has a minimizer point u 0 in N λ and it satisfies:
  1. (i)

    I λ ( u 0 ) = α λ ,

     
  2. (ii)

    u 0 is a positive solution of equation ( E λ f , h ).

     
Proof By Proposition 3.3(ii), it follows that there exists a minimizing sequence { u n } for I λ on N λ such that
I λ ( u n ) = α λ + o ( 1 ) , I λ ( u n ) = o ( 1 ) in  X 1 ( Ω ) .
By Lemma 2.8 and Lemma 2.3 and the compact embedding theorem, there exists a subsequence { u n } and u 0 N λ is a nonzero solution of ( E λ f , h ) such that
u n u 0  weakly in  X , u n u 0  strongly in  L q ( Ω )  and  L r ( Ω ) .
We now prove that u n u 0 strongly in X. Suppose otherwise, then
u 0 < lim inf n u n ,
and so
u 0 p λ Ω ( f | u 0 | q ) d x Ω ( h | u 0 | r ) d x < lim inf n ( u n p λ Ω ( f | u n | q ) d x Ω ( h | u n | r ) d x ) = 0 .

This contradicts u 0 N λ . Hence, u n u 0 strongly in X. This implies I λ ( u n ) I λ ( u 0 ) = α λ , as n . Since I λ ( u 0 ) = I λ ( | u 0 | ) and | u 0 | N λ , by Lemma 2.6, we may assume that u 0 is a nonnegative solution. By Drábek, Kufner and Nicolosi [[1], Lemma 2.1], we have u 0 L ( Ω ) . Then we can apply the Harnack inequality due to Trudinger [25] in order to find that u 0 is positive in Ω.

Now we can complete the proof of Theorem 2.1: By Theorem 3.4 and Theorem 3.5, for ( E λ f , h ) there exist two positive solutions u 0 + and u 0 such that u 0 + N λ + , u 0 N λ . Since N λ + N λ = , this implies that u 0 + and u 0 are different. Thus the proof of Theorem 2.1 is complete. □

4 Time-fractional equations

In this section, we switch our view point to the fractional order equation ( E λ f , h , t ) in weighted Sobolev space H 0 1 ( a ( x ) , Ω ) with the standard norm
u H 0 1 ( a ( x ) , Ω ) = { Ω ( | a ( x ) u ( x ) | 2 + ( u ( x ) ) 2 ) d x } 1 2 .

In order to discuss the existence of the positive solution for the ( E λ f , h , t ), we need to present some basic notations, definitions, and preliminary results, which will be used throughout this section. We first have the following two definitions by [18].

Definition 4.1 The Caputo fractional derivative of order α of a function f ( t ) , t > 0 , is defined as
D α f ( t ) = 1 Γ ( 1 { α } ) 0 t 1 ( t s ) { α } f ( [ α ] + 1 ) d s ,

where { α } , [ α ] denotes the fractional and the integer part of the real number α, respectively, and Γ ( ) is the Gamma function.

Definition 4.2 The Riemann-Liouville fractional integral of order α of a function f ( t ) , t > 0 , is defined as
I 0 + α f ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 f ( s ) d s ,

provided that the right side is pointwise defined on ( 0 , ) .

Lemma 4.1 [18]

Assume y C [ 0 , T ] , T > 0 , 1 < α < 2 , then the problem
D α u ( t ) = y ( t ) , t [ 0 , T ] ,
(4.1)
has the unique solution
u ( t ) = u ( 0 ) + u ( 0 ) t + 1 Γ ( α ) 0 t ( t s ) α 1 y ( s ) d s .

Now we establish some results as regards the existence of positive solutions for ( E λ f , h , t ).

By Lemma 4.1, we may reduce ( E λ f , h , t ) to an equivalent integral equation as in the following problem:
( E λ f , h , integral ) { ϕ ( x ) ψ t + u ( x , t ) = 1 Γ ( α ) 0 t ( t s ) α 1 ( div ( a ( x ) | u ( x , s ) | p 2 u ( x , s ) ) + ( u ( x , s ) ) p 1 + λ f ( x ) | u ( x , s ) | q 1 + h ( x ) | u ( x , s ) | r 1 ) d s , in  Ω T , u ( x , t ) = 0 , on  Ω T .
The functional integral equations describe many physical phenomena in various areas of natural science, mathematical physics, mechanics, and population dynamics [2629]. The theory of integral equations is developing rapidly with the help of tools in functional analysis, topology, and fixed point theory (see, for instance, [3033]) and it serves as a useful tool in turn for other branches of mathematics, for example for differential equations (see [3436]). Now we define
( E λ f , h , fixed ) { Φ ( u ) = ϕ ( x ) + ψ ( x ) t Φ ( u ) = + 1 Γ ( α ) 0 t ( t s ) α 1 ( div ( a ( x ) | u ( x , s ) | p 2 u ( x , s ) ) Φ ( u ) = + | u ( x , s ) | p 2 u ( x , s ) + λ f ( x ) | u ( x , s ) | q 1 Φ ( u ) = + h ( x ) | u ( x , s ) | r 1 ) d s , in  Ω T , u ( x , t ) = 0 , on  Ω T .
Definition 4.3 We call u C ( [ 0 , T ] ; H 0 1 ( a ( x ) , Ω ) ) a weak solution of the fractional order equation ( E λ f , h , t ), if Ω ( u Φ ( u ) ) v d x = 0 , t [ 0 , T ] for every v H 0 1 ( a ( x ) , Ω ) , i.e.
Ω u v d x = Ω [ ϕ ( x ) + ψ ( x ) t + 1 Γ ( α ) 0 t ( t s ) α 1 ( | u ( x , s ) | p 2 u ( x , s ) + λ f ( x ) | u ( x , s ) | q 1 + h ( x ) | u ( x , s ) | r 1 ) d s ] v d x Ω 0 t ( t s ) α 1 ( a ( x ) | u ( x , s ) | p 2 u ( x , s ) ) d s v d x .

Lemma 4.2 The operator Φ ( u ) : H 0 1 ( a ( x ) , Ω ) H 1 ( a ( x ) , Ω ) is completely continuous.

Proof Put
F ( u ) = div ( a ( x ) | u ( x , s ) | p 2 u ( x , s ) ) + ( u ( x , s ) ) p 1 + λ f ( x ) | u ( x , s ) | q 1 + h ( x ) | u ( x , s ) | r 1 .
We can rewrite
Φ ( u ) = ϕ ( x ) + ψ ( x ) t + 1 Γ ( α ) 0 t ( t s ) α 1 F ( u ) d s .
For each v H 0 1 ( a ( x ) , Ω ) , and v H 0 1 ( a ( x ) , Ω ) = 1 , by integration by parts, we can get
| F ( u ) , v | = | ( a ( x ) | u | p 1 v ) + u p 1 v + λ f ( x ) | u | q 1 v + h ( x ) | u | r 1 v ) d x | .
Since 0 f ( x ) L H ( Ω ) , where L H ( Ω ) = L r r q ( Ω ) , q < r < p s , and f ( x ) has a compact support in Ω, 0 h ( x ) L ( Ω ) , and satisfies h ( x ) 1 as | x | , so F ( u ) C ( [ 0 , T ] ; H 0 1 ( a ( x ) , Ω ) ) . Since 2 < p , by Sobolev embedding theorem, we have W 0 1 , p ( a ( x ) , Ω ) H 0 1 ( a ( x ) , Ω ) , and thus, u H 0 1 ( a ( x ) , Ω ) C u W 0 1 , p ( a ( x ) , Ω ) , in the following, we denote u H 0 1 ( a ( x ) , Ω ) and u H 1 ( a ( x ) , Ω ) by u H 0 1 , u H 1 , respectively. Hence, by the Cauchy-Schwarz inequalities, the Poincaré inequalities, the Hölder inequalities, the Sobolev embedding theorem, and (2.7) and 1 < q < p < r , we have
| F ( u ) , v | = | ( a ( x ) | u | p 1 v ) + u p 1 v + λ f ( x ) | u | q 1 v + h ( x ) | u | r 1 v ) d x | ( | a ( x ) | u | p 1 | 2 d x ) 1 2 ( | v | 2 d x ) 1 2 + ( | u p 1 | 2 d x ) 1 2 ( | v | 2 d x ) 1 2 + ( | λ f ( x ) | u | q 1 | 2 d x ) 1 2 ( | v | 2 d x ) 1 2 + ( | h ( x ) | u | r 1 | 2 d x ) 1 2 ( | v | 2 d x ) 1 2 ( | a ( x ) | u | p 1 | 2 d x ) 1 2 v L 2 ( a ( x ) , Ω ) + ( | u p 1 | 2 d x ) 1 2 v L 2 ( a ( x ) , Ω ) + | λ | f ( x ) L H ( | | u | q 1 | 2 d x ) 1 2 v L 2 ( a ( x ) , Ω ) + ( | | u | r 1 | 2 d x ) 1 2 v L 2 ( a ( x ) , Ω ) ( ( | a ( x ) | u | p 1 | 2 d x ) 1 2 + ( | u p 1 | 2 d x ) 1 2 + | λ | f ( x ) L H ( | | u | q 1 | 2 d x ) 1 2 + ( | | u | r 1 | 2 d x ) 1 2 ) v H 0 1 ( | a ( x ) | u | p 1 | 2 d x ) 1 2 + ( | u p 1 | 2 d x ) 1 2 + | λ | f ( x ) L H ( | | u | q 1 | 2 d x ) 1 2 + ( | | u | r 1 | 2 d x ) 1 2 ( | a ( x ) u | 2 ( p 1 ) d x ) 1 2 ( p 1 ) ( p 1 ) + ( | u | 2 ( p 1 ) d x ) 1 2 ( p 1 ) ( p 1 ) + | λ | f ( x ) L H ( | u | 2 ( q 1 ) d x ) 1 2 ( q 1 ) ( q 1 ) + ( | u | 2 ( r 1 ) d x ) 1 2 ( r 1 ) ( r 1 ) = a ( x ) u L 2 ( p 1 ) ( a ( x ) , Ω ) ( p 1 ) + u L 2 ( p 1 ) ( a ( x ) , Ω ) ( p 1 ) + | λ | f ( x ) L H u L 2 ( q 1 ) ( a ( x ) , Ω ) ( q 1 ) + u L 2 ( r 1 ) ( a ( x ) , Ω ) ( r 1 ) C 0 u H 0 1 ( p 1 ) + C 1 | λ | f ( x ) L H u L 2 ( a ( x ) , Ω ) ( q 1 ) + C 2 u L 2 ( a ( x ) , Ω ) ( r 1 ) C u X ( p 1 ) + C 3 ( | λ | f ( x ) L H u H 0 1 ( q 1 ) + u H 0 1 ( r 1 ) ) C u X ( p 1 ) + C 4 ( | λ | f ( x ) L H u X ( q 1 ) + u X ( r 1 ) ) C ˜ { ( r q r p ) 1 p q S r q p q ( λ f L H ) 1 p q } ( p 1 ) + | λ | f ( x ) L H { ( r q r p ) 1 p q S r q p q ( λ f L H ) 1 p q } ( q 1 ) + { ( r q r p ) 1 p q S r q p q ( λ f L H ) 1 p q } ( r 1 ) = M .

Here, C 0 , C 1 , C 2 , C 3 , C 4 , C denote the best Sobolev constants, and C ˜ = max { C , C 4 } .

Thus, by Cauchy-Schwarz inequalities, we obtain
Φ ( u ) H 1 = sup v H 0 1 1 | Φ ( u ) , v | = sup v H 0 1 1 | ϕ ( x ) , v + ψ ( x ) , v t + 1 Γ ( α ) 0 t ( t s ) α 1 F ( u ) , v d s | | ϕ ( x ) , v | + | ψ ( x ) , v t | + | 1 Γ ( α ) 0 t ( t s ) α 1 F ( u ) , v d s | ϕ ( x ) L ( a ( x ) , Ω ) v H 0 1 + ψ ( x ) L ( a ( x ) , Ω ) v H 0 1 T + | F ( u ) , v | | 1 Γ ( α ) 0 t ( t s ) α 1 d s | ϕ ( x ) L ( a ( x ) , Ω ) + ψ ( x ) L ( a ( x ) , Ω ) T + M Γ ( α ) | 0 t ( t s ) α 1 d s | ϕ ( x ) L ( a ( x ) , Ω ) + ψ ( x ) L ( a ( x ) , Ω ) T + M α Γ ( α ) t α ϕ ( x ) L ( a ( x ) , Ω ) + ψ ( x ) L ( a ( x ) , Ω ) T + M α Γ ( α ) T α .

Hence, Φ ( u ) is bounded.

On the other hand, given ϵ > 0 , setting
δ = ( ψ ( x ) L ( a ( x ) , Ω ) + M Γ ( α ) T α 1 ) 1 ϵ ,
then, for every v H 0 1 ( a ( x ) , Ω ) , t 1 < t 2 , t 1 , t 2 [ 0 , T ] , and t 2 t 1 < δ , one has Φ u ( t 2 ) Φ u ( t 1 ) H 1 = sup v H 0 1 1 | Φ u ( t 2 ) Φ u ( t 1 ) , v | ϵ . That is to say, Φ ( u ) has equicontinuity. In fact,
Φ u ( t 2 ) Φ u ( t 1 ) H 1 = sup v H 0 1 1 | Φ u ( t 2 ) Φ u ( t 1 ) , v | = sup v H 0 1 1 | ψ ( x ) , v ( t 2 t 1 ) + 1 Γ ( α ) 0 t 2 ( t 2 s ) α 1 F ( u ) , v d s 1 Γ ( α ) 0 t 1 ( t 1 s ) α 1 F ( u ) , v d s | ψ ( x ) L ( a ( x ) , Ω ) v H 0 1 | t 2 t 1 | + | F ( u ) , v | | 1 Γ ( α ) t 1 t 2 ( t 2 s ) α 1 d s | + 1 Γ ( α ) 0 t 1 | F ( u ) , v | | ( t 2 s ) α 1 ( t 1 s ) α 1 | d s ψ ( x ) L ( a ( x ) , Ω ) | t 2 t 1 | + M α Γ ( α ) t 2 α M α Γ ( α ) t 1 α ψ ( x ) L ( a ( x ) , Ω ) | t 2 t 1 | + M α Γ ( α ) ( t 2 α t 1 α ) .

In the following, we divide the proof into two cases.

Case 1: δ t 1 < t 2 < T , since 1 < α < 2 , we get
Φ u ( t 2 ) Φ u ( t 1 ) H 1 = sup v H 0 1 1 | Φ u ( t 2 ) Φ u ( t 1 ) , v | ψ ( x ) L ( a ( x ) , Ω ) | t 2 t 1 | + M α Γ ( α ) ( t 2 α t 1 α ) = ψ ( x ) L ( a ( x ) , Ω ) | t 2 t 1 | + M α Γ ( α ) α t α 1 ( t 2 t 1 ) = ψ ( x ) L ( a ( x ) , Ω ) δ + M Γ ( α ) T α 1 δ ψ ( x ) L ( a ( x ) , Ω ) δ + M Γ ( α ) T α 1 δ = ( ψ ( x ) L ( a ( x ) , Ω ) + M Γ ( α ) T α 1 ) δ ϵ .

Here, t 1 < t < t 2 , and we apply the mean theorem t 2 β t 1 β = β t β 1 ( t 2 t 1 ) .

Case 2: 0 t 1 , t 2 < α 1 α δ ,
Φ u ( t 2 ) Φ u ( t 1 ) H 1 = sup v H 0 1 1 | Φ u ( t 2 ) Φ u ( t 1 ) , v | ψ ( x ) L ( a ( x ) , Ω ) | t 2 t 1 | + M α Γ ( α ) ( t 2 α t 1 α ) ψ ( x ) L ( a ( x ) , Ω ) δ + M α Γ ( α ) ( α 1 α δ ) α ψ ( x ) L ( a ( x ) , Ω ) δ + M Γ ( α ) T α 1 δ = ( ψ ( x ) L ( a ( x ) , Ω ) + M Γ ( α ) T α 1 ) δ ϵ .

By applying the Arzela-Ascoli theorem, we know that Φ ( u ) : H 0 1 ( a ( x ) , Ω ) H 1 ( a ( x ) , Ω ) is completely continuous. This completes the proof. □

By Lemma 4.2, we know that Ω ( u Φ ( u ) ) v d x = 0 , t [ 0 , T ] for every v H 0 1 ( a ( x ) , Ω ) . That is to say, the fractional order equation ( E λ f , h , t ) has a unique weak solution u C ( [ 0 , T ] ; H 0 1 ( a ( x ) , Ω ) ) .

5 Conclusion

In this paper, we study the existence of positive solutions for P-Laplace semi-linear elliptic equations and the corresponding time-fractional equations. That is, we first establish the multiplicity of positive solutions for nonlinear elliptic equations with a positive smooth weight function involving concave and convex nonlinearities in weighted Sobolev spaces, and the proof of the two positive solutions for the problem ( E λ f , h ) is given. Second, by applying the Arzela-Ascoli fixed point theorem, one existence result for the time-fractional equations is also obtained.

Finally we like to mention that for ( E λ f , h ) its corresponding time-fractional equations are the foundation models of the nonlinear problems in the field of PDES and it is worthwhile to pay more attention to their study.

Declarations

Acknowledgements

The third author is grateful to Professor Boling Guo and Professor Zhouping Xin for their support. The project is supported by the National Natural Science Foundation of China (Nos. 11161057, 10971164).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Xi’an Jiaotong University
(2)
Department of Mathematics, Yunnan Nationalities University
(3)
Institute of Mathematics, Yunnan Normal University
(4)
Department of Mathematics, Tongji University

References

  1. Drábek P, Kufner A, Nicolosi F: Quasilinear Elliptic Equations with Degenerations and Singularities. de Gruyter, Berlin; 1997.View ArticleMATHGoogle Scholar
  2. Adams RA, John JFF: Sobolev Space. Academic Press, New York; 2009.Google Scholar
  3. Ambrosetti A, Brezis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 1994, 122: 519-543. 10.1006/jfan.1994.1078MathSciNetView ArticleMATHGoogle Scholar
  4. Adimurthi FP, Yadava L: On the number of positive solutions of some semilinear Dirichlet problems in a ball. Differ. Integral Equ. 1997, 10: 1157-1170.MathSciNetMATHGoogle Scholar
  5. Damascelli L, Grossi M, Pacella F: Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1999, 16: 631-652. 10.1016/S0294-1449(99)80030-4MathSciNetView ArticleMATHGoogle Scholar
  6. Tang M: Exact multiplicity for semilinear Dirichlet problem involving concave and convex nonlinearities. Proc. R. Soc. Edinb., Sect. A 2003, 133: 705-717. 10.1017/S0308210500002614View ArticleMATHGoogle Scholar
  7. Wu TF: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 2006, 318: 253-270. 10.1016/j.jmaa.2005.05.057MathSciNetView ArticleMATHGoogle Scholar
  8. Miotto ML, Miyagaki OH: Multiple positive solutions for semilinear Dirichlet problems with sign-changing weight function in infinite strip domains. Nonlinear Anal. 2009, 71: 3434-3447. 10.1016/j.na.2009.02.010MathSciNetView ArticleMATHGoogle Scholar
  9. Wu TF: Multiple positive solutions for Dirichlet problems involving concave and convex nonlinearities. Nonlinear Anal. 2008, 69: 4301-4323. 10.1016/j.na.2007.10.056MathSciNetView ArticleMATHGoogle Scholar
  10. Lakshmikantham V, Leela S, Vasundhara DJ: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.MATHGoogle Scholar
  11. Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 2010,59(3):1363-1375. 10.1016/j.camwa.2009.06.029MathSciNetView ArticleMATHGoogle Scholar
  12. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.MATHGoogle Scholar
  13. Podlubny I Mathematics in Science and Engineering. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
  14. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon; 1993.MATHGoogle Scholar
  15. Tian Y, Chen A: The existence of positive solution to three-point singular boundary value problem of fractional differential equation. Abstr. Appl. Anal. 2009. 10.1155/2009/314656Google Scholar
  16. Zhang SQ: Positive solutions for boundary value problem of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006. Article ID 36, 2006: Article ID 36Google Scholar
  17. Bai Z, Lu H: Positive solutions for boundary value problem of nonlinear fractional differential equations. J. Math. Anal. Appl. 2005,311(2):495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
  18. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
  19. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems, II. Appl. Anal. 2002, 81: 435-493. 10.1080/0003681021000022032MathSciNetView ArticleMATHGoogle Scholar
  20. Rabinowitz PH Regional Conf. Ser. in Math. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.View ArticleGoogle Scholar
  21. Brown KJ, Zhang YP: The Nehari manifold for a semilinear elliptic problem with a sign-changing weight function. J. Differ. Equ. 2003, 193: 481-499. 10.1016/S0022-0396(03)00121-9MathSciNetView ArticleMATHGoogle Scholar
  22. Tarantello G: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1992, 9: 281-304.MathSciNetMATHGoogle Scholar
  23. Hsu TS: Multiplicity results for p -Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions. Abstr. Appl. Anal. 2009. Article ID 652109, 2009: Article ID 652109Google Scholar
  24. Ekland I: On the variational principle. J. Math. Anal. Appl. 1974, 17: 324-353.View ArticleGoogle Scholar
  25. Trudinger NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun. Pure Appl. Math. 1967, 20: 721-747. 10.1002/cpa.3160200406MathSciNetView ArticleMATHGoogle Scholar
  26. Corduneanu C: Integral Equations and Applications. Cambridge University Press, New York; 1973.MATHGoogle Scholar
  27. O’Regan D: Existence theory for nonlinear Volterra integrodifferential and integral equations. Nonlinear Anal. 1998, 31: 317-341. 10.1016/S0362-546X(96)00313-6MathSciNetView ArticleMATHGoogle Scholar
  28. Jeribi A: A nonlinear problem arising in the theory of growing cell populations. Nonlinear Anal., Real World Appl. 2002, 3: 85-105. 10.1016/S1468-1218(01)00015-3MathSciNetView ArticleMATHGoogle Scholar
  29. Ben Amar A, Jeribi A, Mnif M: Some fixed point theorems and application to biological model. Numer. Funct. Anal. Optim. 2008, 29: 1-23. 10.1080/01630560701749482MathSciNetView ArticleMATHGoogle Scholar
  30. Barroso CS: Krasnoselskii’s fixed point theorem for weakly continuous maps. Nonlinear Anal. 2003, 55: 25-31. 10.1016/S0362-546X(03)00208-6MathSciNetView ArticleMATHGoogle Scholar
  31. Ben Amar A, Jeribi A, Mnif M: On a generalization of the Schauder and Krasnosel’skii fixed point theorems on Dunford-Pettis spaces and applications. Math. Methods Appl. Sci. 2005, 28: 1737-1756. 10.1002/mma.639MathSciNetView ArticleMATHGoogle Scholar
  32. Latrach K, Taoudi MA, Zeghal A: Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations. J. Differ. Equ. 2006, 221: 256-271. 10.1016/j.jde.2005.04.010MathSciNetView ArticleMATHGoogle Scholar
  33. Zhang, SS: Integral Equation. Chongqing (1988)Google Scholar
  34. Zhang SS, Yang GS: Some further generalizations of Ky Fan’s minimax inequality and its applications to variational inequalities. Appl. Math. Mech. 1990,11(11):1027-1034. 10.1007/BF02015686MathSciNetView ArticleMATHGoogle Scholar
  35. Zhang SS, Yang GS: On the solution by determinantal series to Volterra integral equation of the second kind with convolution-type kernel. Math. Numer. Sin. 1988,10(2):146-157.MATHGoogle Scholar
  36. Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleMATHGoogle Scholar

Copyright

© Qiu et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.