A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space
We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem in , on , in a suitable weighted Sobolev space, where is a bounded domain and is a continuous bounded nonlinearity.
Let be a bounded domain with boundary . Let be an operator in divergence form: with coefficients which are symmetric and satisfy the degenerate ellipticity condition: for all , and is an -weight . Let and and let be a real valued continuous function defined on . In this paper, we study the existence of weak solution of the BVP: under suitable hypotheses on the functions , , and . The present work is inspired by a semilinear problem in bounded domain given in the book by Zeidler . In general, the Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic PDEs. For degenerate problems with various types of singularities in the coefficients it is natural to look for solutions in weighted Sobolev spaces; for example, see [2–8]. Section 2 deals with preliminaries and some basic results. Section 3 contains the main result and is about the existence of a weak solution to (3) in a suitable weighted Sobolev space.
We need the following preliminaries for the ensuing study. Let be a bounded domain (open connected set). Let be a locally integrable function with a.e. We say that belongs to the Muckenhoupt class , , or that is an -weight, if there is a constant such that for all balls in , where denotes the -dimensional Lebesgue measure in . We assume that , . We will denote by the usual Banach space of measurable real valued functions, , defined in for which For and a positive integer , the weighted Sobolev space is defined by with the associated norm In order to avoid too many suffices, at each step, a generic constant is denoted by or . We need the following result.
Proposition 1 (the weighted Sobolev inequality). Let be a bounded domain and let . Then, there exist positive constants and such that, for all and all satisfying ,
A proof of the above statement can be found in [5, Theorem 1.3].
For and in the above inequality, we have where Further, we use function space which is defined as the closure of with respect to the norm (correctness of definition of this norm follows from inequality (9)). We also note that and are Hilbert spaces.
For more details on -weight and weighted Sobolev spaces, we refer to [5, 7, 9–11].
Proposition 2. Let , , and let be a bounded open set in . If in , then there exist a subsequence and a function such that (i) as , -a.e. on ;(ii), -a.e. on .
Proof. The proof of this theorem follows in the lines of Theorem in .
Let denote the value of linear functional at .
Definition 3. Let be the operators on the real separable reflexive Banach space . Then, (i) satisfies condition () if (ii) is demicontinuous if and only if as implies as ,(iii) is asymptotically linear if is linear and
In Section 3, we use the following result.
Proposition 4. Let be operators on the real separable reflexive Banach space . Then, (1)the operator is linear and continuous;(2)the operator is demicontinuous and bounded;(3) is asymptotically linear;(4)for each and for each , the operator defined by satisfies condition in . If implies , then, for each , the equation has a solution in .
For a detailed proof of the above theorem, we refer to  or to [1, Theorem ].
Definition 5. One says that is a weak solution of (3) if for every .
We need the following hypotheses for further study.
() Let be continuous in , where is a bounded function (i.e., for a constant , let ).
() Let . Assume , , and .
() Assume , where and .
We define the functionals by Also define by A function is a weak solution of (3) if By noting and by Hölder’s inequality, we get where is a constant arising out of (9). Now, is linear and bounded. Then, there exists an operator defined by , for all . Also, by and , it follows from Hölder’s inequality that and hence by the weighted Sobolev inequality (9) Now is linear and so there exists an operator such that Further, we have Then, problem (3) is equivalent to solving the operator equation
3. Main Results
The main result of this section is to establish the existence of a solution for the degenerate nonlinear elliptic BVP (3), when is not an eigenvalue of with certain restrictions. Also, two results are established related to the cases when does not change sign.
Theorem 6. Assume the hypotheses and the inequality where is a constant arising out of (9). Let not be an eigenvalue of (24). Then, the BVP (3) has a solution .
Proof. Idea of proof is such. First we write a weak solution of the BVP (3) as solution of operator equation
where is linear and continuous, and is demicontinuous and bounded and satisfies few more conditions. Further, we put Proposition 4 to this operator equation. The realization of this idea is split into 5 steps for convenience.
Step 1. We note that the operator is linear. It follows from (17) and (20) that the operators are bounded.
Step 2. Let in . We now claim or If in , then in and in . Using Proposition 2, there exist a subsequence and functions and in such that Now Since is bounded, we get Also, By and (29), we infer that Letting , by dominated convergence theorem, we obtain Hence, we have claim (28) or equivalently is demicontinuous.
Step 3. Next, we claim that is asymptotically linear. Since is bounded, we observe that, for all , which implies , where . Consequently, which shows that is asymptotically linear.
Step 4. We denote . Let in and We claim that strongly in or satisfies condition . By (2) and inequality (9) of Proposition 1, it follows that where , a positive constant depending on . Since is linear, from (38) we have By hypothesis , we have now From (37), (39), and (40), we note Since, by condition (25), , we have Consequently, , as , which implies that, for each , satisfies condition (S).
Step 5. Since, by given hypothesis, is not an eigenvalue of (24), implies . By Proposition 4, has a solution which equivalently shows that the BVP (3) has a solution .
Remark 7. In the following results, we dispense with condition (25), when does not change sign. The two results are related to the cases when with and with .
The proof of the following results is similar to Theorem 6 and, hence, we give a sketch of the proof.
Theorem 8. Suppose that hold. Let and ; then the BVP (3) has a solution .
Proof. As in Theorem 6, the basic idea is to reduce the problem (3) to an operator equation . We note that the operator is linear and continuous and is bounded. Let in and as in (37) Since and , then, by (2) and the weighted Sobolev inequality (9), it follows that Since is linear, from (44) we have By hypothesis , we have now From (43), (45), and (46), we note that Since , we have and hence , as , which implies that, for each , satisfies condition (S). Also, we note that is asymptotically linear. Now implies or which shows that (since and ). By Proposition 4, has a solution .
With suitable changes in the proof of Theorem 8, we arrive at the following result.
Theorem 9. Let the hypotheses of Theorem 8 hold, except that and . Then, the BVP (3) has a solution .
Proof. The proof of this result is in the same lines to that of Theorem 8 and hence omitted.
Remark 10. In Theorems 8 and 9, we have studied the BVP (3) both positive and negative values of . In Theorem 9, with positive value of we do not need the extra condition (25) at the cost of negative.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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