Research Article | Open Access
Existence of Weak Solutions for Nonlinear Time-Fractional -Laplace Problems
The existence of weak solution for -Laplace problem is studied in the paper. By exploiting the relationship between the Nehari manifold and fibering maps and combining the compact imbedding theorem and the behavior of Palais-Smale sequences in the Nehari manifold, the existence of weak solutions is established. By means of the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations of the -Laplace problem are obtained.
Fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be noninteger. The increasing interest of fractional equations is motivated by their applications in various fields of science such as physics, fluid mechanics, heat conduction with memory, chemistry, and engineering . In consequence, the subject of fractional differential equations is gaining diverse and continuous attention. For example, for fractional initial value problems, the existence and multiplicity of solutions (or positive solutions) were discussed in [2–4].
In this paper, we consider the following semilinear boundary value problem: where is a bounded region with smooth boundary in and is a positive weight function with positive measure of the Sobolev space , , is a smooth function which may change sign, and is a real positive parameter and assume throughout that is a fixed number such that . Thus, we will study a sublinear perturbation of a linear problem.
The problem (1) is an important and basic mathematical model, widely used in many fields. As for the specific theoretical implicity of the above model, one can see Drábek et al. , Adams and Fournier , and so on.
Similar problems have been studied by Brown and Zhang [7, 8] (when , , with ) and by Brown  (when , , but with ) by using variational viewpoint of the Nehari manifold. When , , Amann and Lopez-Gomez  have studied the existence of the equation by using global bifurcation theory and Binding et al. [11, 12] used variational methods. Huang and Pu in  have studied the following problem: where is a bounded region with smooth boundary in , is a real positive parameter, is a nonnegative function and satisfies or , and is a smooth function which may change sign in , and the existence of multiple positive solutions and the properties on Nehari manifold for (2) have been established under the assumption that , where , , , .
In the above, we investigated the -Laplace Dirichlet problem. Next, we will switch our viewpoint to consider the existence of weak solutions for the corresponding nonlinear time-fractional differential equation of the problem (1). Consider where , denotes the Caputo fractional derivatives , is a parameter describing the order of the fractional time, and , are given real-valued functions.
Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, porous media, engineering, and so forth. For some recent developments on the subject, one can see [14–21]. As far as we know, no contributions exist concerning the existence of weak solutions for the problems as we stated above.
2. Notations and Preliminary Results
Let be a weighted Sobolev space with a positive measurable weight function ; its norm is defined as , and the function satisfies , , where , is also a weight function (see [5, 6, 13]) and satisfies , , ; here, . Throughout this paper, we denote by the best Sobolev constant for the imbedding of in . In particular, , for all . For simplicity, we will denote by and denote by , and unless otherwise stated, integrals are over .
Let denote the positive principal eigenvalue of the problem: with corresponding positive principal eigenfunction . The Euler functional associated with (1) is
The next lemma shows the behavior of functional on .
Lemma 1. (i) Suppose that ; then is bounded below on .
(ii) If , then is no longer bounded below on .
Proof. (i) By the spectral theorem, we have
where . Hence, is bounded below on when .
(ii) If , then , so is unbounded below on .
In order to obtain existence results in the case of , we introduce the Nehari manifold: where denotes the usual duality. Thus, if and only if
Obviously, is a much smaller set than and so it is easier to study on .
On , we have The Nehari manifold is closely linked to the behavior of functions of the form . Such maps are known as fibering maps and were introduced by Drabek and Pohozaev  and are also discussed in Brown and Zhang . If , we have We can see that if and only if and more generally that if and only if ; that is, elements in correspond to stationary points of fibering maps. It follows from (11) and (12) that if , then . Thus, it is natural to subdivide into three parts corresponding to local minima, local maxima, and points of inflection. Consider So , , and correspond to minima, maxima, and points of inflection, respectively.
Let , then(i)if and have the same sign, then has a unique turning point at This turning point is a local minimum (maximum) so that if and only if .(ii)If and have different signs, then has no turning points and so no multiples of lie in .
Thus, if we define Analogously, we can define , , , and by replacing “” by “” or “.” As appropriate, we have the following:(i)if , then has a local minimum at and ,(ii) if , then has a local maximum at and ,(iii)if , then is strictly increasing and no multiple of lies in ,(iv) if , then is strictly decreasing and no multiple of lies in .
Next, we will prove the existence of solutions of (1) by investigating the existence of minimizers on . The following lemma proved that minimizers on are “usually” critical points for .
Lemma 2. Suppose that is a local maximum or minimum point for on , ; then in .
Proof. If is a local minimizer point for on , then is a solution of the optimization problem:
Hence, by the theory of Lagrange multipliers, there exists such that Thus, Since , and so Hence, Thus, if , and so by (18) . This completes the proof.
3. Properties of the Nehari Manifold
In this section, we will discuss the vital role played by the condition in determining the nature of the Nehari manifold. When , , for all , and so and . When , we have and and when is greater than , becomes nonempty and gets bigger as increases.
Theorem 3. Suppose there exists such that, for all , . Then, for ,(i) and so ,(ii) is bounded,(iii) and is closed,(iv).
Proof. (i) Suppose that the result is false. Then there exists such that ; if , then , and so which is a contradiction.
(ii) Suppose that is unbounded. Then there exists such that as . Let ; without loss of generality, we may assume that in and so in and in . Since , and so . Since , so, by (8), we have and divide by , so Suppose in . Then , and so . Thus, which is impossible as . Hence, in . Thus, and Thus, which is impossible as . Therefore, is bounded.
(iii) Suppose . Then there exists such that ; let and then we may assume that in and so in and in . Since , we have since the is bounded, it follows that and so . Suppose in ; then and so . Moreover and so or . Hence, and this is impossible as . Thus, we must have that in ; then Hence which is impossible and so . We now prove that is closed. Suppose and in . Then and so . Moreover, If both integrals equal , then which contradicts (i). Hence both integrals must be negative and so . Thus is closed.
(iv) Let , as , . Moreover, it is clear that and so which is impossible. Thus, .
The following theorem presents on and the behavior of on .
Theorem 4. Suppose the same hypotheses are satisfied as in Theorem 3; then(i) is bounded below on ,(ii) provided is nonempty.
Proof. The proof of (i) is an immediate consequence of the boundedness of .
(ii) Suppose . Then there exists such that . And it is clear from (9) that and as . Let ; since , is bounded away from . Hence , and ; we may assume that in and in and in . Then . If in , we have and ; that is, , whereas if in , ; that is, . In both cases, however, we must also have and this is a contradiction. Thus, .
4. The Existence of Weak Solution
We now show that there exists a minimizer on which is a critical point of and so a nontrivial positive solution of (1).
Theorem 5. Suppose for all ; then, for ,
(i) there exists a minimizer point for on ,
(ii) there exists a minimizer point for on provided that is nonempty.
Proof. (i) By Theorem 4, is bounded below on . Let be a minimizing sequence; that is,
Since is bounded, we may assume that in and in and in . Since , it follows that
and so . Hence, by Theorem 3, and so the fibering map has a unique minimum at , such that . Suppose in ; then
and so . Hence
which is impossible.
Hence in and so . It now follows easily that is a minimizer point for on , since and we may assume that is a nonnegative in , since , is a local minimum point for on . It follows from Lemma 2 that is a critical point of and so is a weak solution of (1).
(ii) Let be a minimizing sequence. Then by Theorem 4, we must have .
Suppose that is unbounded; we may suppose that as . Let ; since is bounded, it follows that and are bounded and so since is bounded, we may assume that in and in and in , so that . If in , it is easy to see that which is impossible because of Theorem 3(i). Hence, in and so hence, and , which is again impossible. Thus, is bounded and so we may assume that in and in and in . Suppose in . Then Hence , and so , where Moreover, but and so since the map attains its maximum at , Hence, which is impossible. Thus, in and it follows that is a minimizer point for on . Since , we may assume that is a nonnegative in ; since is closed, is a local minimum point for on . It follows from Lemma 2 that is a critical point of and so is a weak solution of (1).
5. The Corresponding Time-Fractional Equation
Definition 6 (see ). The Caputo fractional derivative of order of a function , , is defined as where and denote the fractional and the integer part of the real number , respectively, and is the Gamma function.
Definition 7 (see ). The Riemann-Liouville fractional integral of order of a function , , is defined as provided that the right side is pointwise defined on .
Lemma 8 (see ). Assume , , and ; then the problem has the unique solution
By Lemma 8, we may reduce to an equivalent integral equation as follows:
Now we define
Lemma 10. Let and be bounded; then the operator is completely continuous.
We can rewrite
For each and integration by parts, we can get
By Lemma 2, we know ; that is,
Since , so
And using the same proof as above, we can get . Thus, we deduce
And since , and is a positive sufficiently smooth function, there exists a positive constant , such that . Hence
We used Poincare’s inequality in the last inequality above. Thus, by Sobolev imbedding theorem , we have
In the following, we denote and by and , respectively. Hence, by Cauchy-Schwarz inequalities, Poincare inequalities, Hölder inequalities, Sobolev imbedding theorem, and (49), for , we can get
Here, and , from the Sobolev imbedding theorem.
Thus, by Cauchy-Schwarz inequalities, we obtain Hence, is bounded.
On the other hand, given , setting then, for every , , , and , one has . That is to say, is equicontinuity. In fact,
In the following, we divide the proof into two cases.
Case 1. ; since , we get here, , and we apply the mean theorem .
Case 2. , ,