Positive solutions for P-Laplace problems with nonlinear time-fractional differential equation
© Qiu et al.; licensee Springer. 2014
Received: 20 March 2014
Accepted: 13 June 2014
Published: 22 July 2014
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.
, where λ is a real positive parameter, (, , , , ). Ω is a bounded region with smooth boundary in ; , , are measurable functions and satisfy the following conditions:
(H1) , where , , and has a compact support in Ω,
(H2) , and it satisfies as ,
(H3) is a positive weight function, locally Hölder continuous, and almost everywhere with positive measure in the Sobolev space which comes with the standard norm and there exists if and only if , where and is another weight function, which satisfies , , .
The problem () 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  and Adams and John  and references wherein.
for all . The existence of was obtained such that for 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 (). 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 [10–17], 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.
where , denotes the Caputo fractional derivative (e.g., see ), is a parameter describing the order of the fractional time, and are given real-valued functions. Then the problem () is deduced to an equivalent integral equation under the fractional order integral operator . 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 such that for , the equation () has at least two positive solutions.
In order to prove it, we need the following lemma.
Lemma 2.2 If , is the compact embedding, then is also the compact embedding, where (e.g., see Drábek, Kufner and Nicolosi ).
For the convenience we will denote by X, and by if there is no confusion, unless otherwise stated, and the integrals are over Ω. Now we give the proof of Theorem 2.1.
for all . Therefore the (weak) solutions of equation () are the critical points of the energy functional (see Rabinowitz ).
Note that any nonzero solution of the problem () belongs to . Furthermore, we have the following result.
Lemma 2.3 The functional is coercive and bounded from below on .
Since , it follows that is bounded from below and coercive on provided λ is small enough. □
Then we have the following result.
Lemma 2.4 There exists such that for each , we have .
Proof We consider the following two cases.
where . This implies that for λ sufficiently small we have for all , this contradicts (2.8). Thus, we can conclude that there exists such that for , we have . The proof is complete. □
Lemma 2.5 If , then .
and this completes the proof. □
The following results show that minimizers on are the ‘usual’ critical points for .
Lemma 2.6 For , if is a local minimizer point for on and , then in .
Thus, if , and so by (2.9), . This completes the proof. □
- (i)there is a unique such that and
- (ii)if , then there exists unique such that and
there exists a continuous bijection between and , in particular, is a continuous function for nonzero u.
Note that , if and only if .
for . Therefore .
by Case (II) of part (i).
- (iii)Fix , define by
then by the implicit function theorem, there is a neighborhood of u in U and an unique continuous function such that for all , in particular, . Since is arbitrary, we find that the function , given by is continuous and one-to-one. Having , where , we find that is continuous and one-to-one. Now if then we have , where , since is continuous on U, it follows that is continuous for nonzero u. Then the proof is complete. □
and , in particular, .
Thus, there exists a positive number such that if , then , i.e., for all . Obviously, . This completes the proof. □
3 Proof of Theorem 2.1
First, by following the idea of Tarantello , we have the following result.
The proof of the two lemmas above is almost the same as given by Hsu  and thus we omit it.
- (i)there exists a minimizing sequence such that
- (ii)there exists a minimizing sequence such that
Now, we will show that , as .
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. □
is a positive solution of equation (),
and thus we conclude the proof. □
Next, we establish the existence of a local minimum for on .
is a positive solution of equation ().
This contradicts . Hence, strongly in X. This implies , as . Since and , by Lemma 2.6, we may assume that is a nonnegative solution. By Drábek, Kufner and Nicolosi [, Lemma 2.1], we have . Then we can apply the Harnack inequality due to Trudinger  in order to find that is positive in Ω.
Now we can complete the proof of Theorem 2.1: By Theorem 3.4 and Theorem 3.5, for () there exist two positive solutions and such that , . Since , this implies that and are different. Thus the proof of Theorem 2.1 is complete. □
4 Time-fractional equations
In order to discuss the existence of the positive solution for the (), 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 .
where , denotes the fractional and the integer part of the real number α, respectively, and is the Gamma function.
provided that the right side is pointwise defined on .
Lemma 4.1 
Now we establish some results as regards the existence of positive solutions for ().
Lemma 4.2 The operator is completely continuous.
Here, , , , , , C denote the best Sobolev constants, and .
Hence, is bounded.
In the following, we divide the proof into two cases.
Here, , and we apply the mean theorem .
By applying the Arzela-Ascoli theorem, we know that is completely continuous. This completes the proof. □
By Lemma 4.2, we know that , for every . That is to say, the fractional order equation () has a unique weak solution .
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 () 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 () 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.
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).
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