Existence of Solutions to a Class of Semilinear Elliptic Problem with Nonlinear Singular Terms and Variable Exponent

Abstract

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation with and . The results show the dependence of the summability of in some Lebesgue spaces and on the values of .

1. Introduction

In this paper, we study the existence of solutions for the following semilinear elliptic problem with nonlinear singular terms and variable exponent:where is a bounded domain in with smooth boundary , is a continuous function on , , , , is a nonnegative function belonging to the Lebesgue space , for some suitable , and is a bounded positive definite matrix; that is, there exist such thatfor every in , for almost every in .

Problem (1) has been widely applied in many areas, such as the contexts of chemical heterogeneous catalysts, non-Newtonian fluids, and also the theory of heat conduction in electrically conducting materials; see [1–4] for detailed discussion.

Problem (1) has been extensively studied in the past. In [5], Lazer and Mckenna dealt with model (1) with , a continuous function; they proved that the solution was in if and only if , while it was not in if .

Later, Lair and Shaker in [6] studied the existence of solutions to the elliptic equation

They proved that problem (3) with has a unique weak positive solution in if is a nonnegative nontrivial function in .

Moreover, the results of Lair and Shaker were generalized by Shi and Yao (see [7]); they studied the following problem:where is a bounded domain in , , may take the value 0 on , and is possibly singular near They proved the existence and the uniqueness of positive solutions without assuming monotonicity or strict positivity on .

Recently, Boccardo and Orsina in [8] studied the existence, regularity, and nonexistence of solutions for the following problem:They discussed the dependence of the results on the summability of and the values of . For the other results of singular elliptic equations, see [9, 10]. In this paper, we generalize the results in [8] to the case when is a variable exponent by applying the method of regularization, Schauder fixed point theorem, the integrability of solution to the approximate problem with , and a necessary compactness argument to overcome some difficulties arising from the singular terms with variable exponent.

2. Preliminaries

Firstly, we give the definition of weak solutions to problem (1).

Definition 1. A function is called a weak solution of problem (1), if the following identity holds:

In order to prove our results, we will consider the following approximation problem:where , .

Lemma 2. Problem (7) has a nonnegative solution in .

Proof. Let be fixed and a function in . It is not difficult to prove that the following problem has a unique solution (see [11, 12]):So, for any , we define the mapping as . Taking as a test function, we have, using (2),By Poincaré inequality (on the left hand side) and Hölder’s inequality (on the right hand side), we get that for some constant independent of . This implies that Therefore, the ball of of radius is invariant under the mapping Since the embedding is compact, we obtain that is a compact operator and . It is also easy to prove that is continuous on , so by Schauder’s fixed point theorem, we get that there exists a function , for every fixed , such that ; that is, problem (7) has a solution. Since , the maximum principle implies that . Since the right hand side of (7) belongs to , the result of Theorem  4.2 in [13] implies that .

Lemma 3. The sequence is increasing with respect to , in , and for every there exists (independent of ) such that

Proof. Due to and , we have that so that Choosing as a test function, observing that and applying (2), we get that This implies that a.e. in ; that is, for every . Since the sequence is increasing with respect to , we only need to prove that (12) holds for . Applying Lemma 2, we know that ; that is, there exists a constant (depending only on and ) such that and then Due to , , the strong maximum principle implies that in and (12) holds for . The monotonicity of implies that (12) holds for .

Remark 4. If and are two solutions of (7), following the lines of the proof of the first part in Lemma 3 we may show that . By symmetry, this implies that the solution of (7) is unique.

Lemma 5. The solution to problem (7) with satisfies

Proof. By and Lemma  2.2 in [14], we know that there exists such that and , which implies that the gradient of exists everywhere; then Hopf Lemma in [15] shows that in , where is the outward unit normal vector of at . Moreover, following the lines of the proof of Lemma in [5], we get that

We know clearly that the estimates on depend on and , and we will discuss it in different cases.

3. The Case

In this case, we obtain a priori estimates on in only if is more regular than , and we have the following results.

Lemma 6. Let be the solution of (7) with , and suppose that with . Then the sequence is bounded in .

Proof. Choosing as a test function in (7), by Hölder’s inequality, (2), and the fact that , we getBy the assumption of , we have , and using Sobolev Embedding Theorem (on the left hand side), we have thatthat is, Since , (22) yields the boundedness of in . By this estimate and (22), the conclusion follows.

Once we have the boundedness of , we can prove an existence result for (1).

Theorem 7. Suppose that is a nonnegative function in , with , and let . Then problem (1) has a solution satisfying (6).

Proof. Since is bounded in by Lemma 6 and converges to pointwise in (by Lemma 3), then we know that there exists such that So we have thatSince satisfies (12), we get that where . Then by Lebesgue Dominated Convergence Theorem, we have thatSince is a solution of (7), this implies that Letting , combining (25) with (27), we get that which proves that (1) has a solution in .

The summability of depends on the summability of , which is proved in the next Lemma.

Lemma 8. Suppose that , , and let . Then the solution of (1) given by Theorem 7 is such that(i)if , then ;(ii)if , then , .

Proof. To prove (i), let and define . Taking as a test function in (7), using (2), we get Since , it follows thatStarting from inequality (31), Theorem  4.2 in [13] shows that there exists a constant (independent of ), such that which implies that belongs to .
To prove (ii), noting that if , , since , the result when is true by Sobolev Embedding Theorem. If , letting and choosing as a test function in (7), using Hölder’s inequality, we get thatBy Sobolev inequality (on the left hand side), we have thatwhere is the constant of the Sobolev embedding; combining with (33) and (34), we have thatWe choose in such a way that ; that is, which yields that if and only if , and that . Therefore, (35) becomes which implies thatUsing Young’s inequality on the right hand side in (38), we have that where Thus, we get thatTherefore, we know that is bounded in , and so does .

Theorem 9. Suppose that , ⩽ , and . Then problem (1) has a solution in , .

Proof. The lines of our proof are that if we can prove that is bounded in (with as in the statement), the existence of a solution in of (1) will be proved by passing to the limit in (7) as in the proof of Theorem 7. To prove that is bounded in , we begin by proving that it is bounded in , with . To attain this goal, we choose as a test function in (7) as in the statement of Lemma 8, where ; however will be singular at , and therefore, we choose as a test function in (7), where for fixed; by (2) and , we have thatBy Sobolev Embedding Theorem () on the left hand side, it follows thatwhere is the best constant of the Sobolev Embedding Theorem. Combining (41) with (42), we have thatUsing Hölder’s inequality on the right hand side, we get Letting , we get (35); that is, where is chosen in such a way that ; that is, If , choosing in (43), and letting , we have that Using Hölder’s inequality and Young’s inequality, we get where . Thus we have that Therefore we obtain that is bounded in , where is the value of for .
If , it is clear that the inequality on holds true if and only if , starting from (35) and arguing as in the proof of Lemma 8, we also get that is bounded in with .
The right hand side of (41) is bounded with respect to (and , which we take smaller than 1) by using the estimate on in and the choice of .
Since , If , by Hölder’s inequality, we have thatThe choice of and the value of are such that , so that the right hand side of (51) is bounded with respect to and . Hence, is bounded in .

Theorem 10. Suppose that , < / with < , and . Then problem (1) has a solution in , .

Proof. The lines of our proof are similar to that in the proof of Theorem 9. We also begin by proving that is bounded in , with . To this aim, we also choose as a test function in (7), where , for fixed. Since and (2), we have that Using Sobolev Embedding Theorem () on the left hand side, it follows that where is the constant of the Sobolev Embedding Theorem.
Using Hölder’s inequality and Lemma 5 on the right hand side, we get Letting , we obtain thatwhere is chosen in such a way that ; that is If , it is clear that the inequality on holds true if and only if , and arguing as to the case in the proof of Theorem 9, we also obtain that is bounded in , with .
Since , If , similarly to the proof of Theorem 9, we have by Hölder’s inequality that Since the choice of and the value of , the right hand side of the above inequality is bounded with respect to and . Hence, is bounded in .

4. The Case

The case has many analogies with the case . In this case, we can also prove that is bounded in only if is more regular than and and are close to 1.

Lemma 11. Suppose that , and let be the solution of (7) with . Then is bounded in .

Proof. Taking as a test function in (7), by (2), we obtain that Using Lemmas 2 and 3, we know that and there exists a constant s.t. . Hence , and we have Using Hölder’s inequality on the right hand side, and Lemma 5, we obtain Therefore, is bounded in .

Once we have the boundedness of , we can prove the following existence theorem along the lines of Theorem 7.

Theorem 12. Suppose that and . Then problem (1) has a solution in .

The summability of can be proved along the lines of Lemma 8 with little changes.

Lemma 13. Suppose that and . Then the solution of (1) given by Theorem 12 is such that(i)if , then ;(ii)if , then , .

Proof. The proof of (i) is similar to the proof of Lemma 8(i); we omit the details here.
To prove (ii) we choose as a test function with in (7); similarly to the proof of Lemma 8, we obtain thatIf , choosing in (62), by Hölder’s inequality, we get We choose in such a way that ; that is, . Since being , it follows that is bounded in .
If , starting from inequality (62) and Hölder’s inequality, we have that We also choose in such a way that , which yields that , if and only if , and that . So, since being , we have the boundedness of in , and so does .

Moreover, we can only prove that a positive power of is bounded in only if is more regular than and is close to and we only have the boundedness of in .

Lemma 14. Suppose that , and let be the solution of (7) with and . Then is bounded in , and is bounded in and in , with .

Proof. Taking as a test function in (7), since and , by Lemma 5 and (2), we get that Since we have that Thus, we have that is bounded in
Applying Sobolev Embedding Theorem to , we have that where is the best constant of the Sobolev embedding. Since the boundedness of in , we thus have the boundedness of in .
To prove the boundedness of in , we choose as a test function in (7), where , . By (2) and (12), we have that By Young’s inequality, we get that Since is bounded in (where ), by Hölder inequality, we obtain that and hence is bounded in .

Once we have the boundedness of , we can prove the following existence theorem along the lines of Theorem 7.

Theorem 15. Suppose that is a nonnegative function in , , and . Then problem (1) has a solution in . Furthermore, belongs to .

The summability of can be proved along the lines of Lemma 8 with little changes.

Lemma 16. Suppose that , , and . Then the solution of (1) given by Theorem 15 is such that(i)if , then ;(ii)if , then , .

Proof. The proof of (i) is similar to the proof of Lemma 8(i); we omit the details here.
To prove (ii), we choose as a test function in (7); applying (2) and Sobolev Embedding Theorem, we have thatIf , we choose ; using Hölder inequality, we have that We also choose in such a way that , since being , and we have the boundedness of in which is the value of for .
If , starting from (72) using Hölder inequality, we get that We also choose in such a way that , which yields that , if and only if , and that . Since being , we have the boundedness of in , and so does .

5. The Case

If , the boundedness of in can also be obtained only if is more regular than , and the proof has many analogies with the case . We have the following results.

Lemma 17. Suppose that , with , and let be the solution of (7) with . Then the sequence is bounded in .

Proof. We choose as a test function in (7), by Hölder’s inequality, (2), and Lemma 5, since , we have that Applying Sobolev Embedding Theorem on the left hand side, we get Let , and it follows that By Young’s inequality, we get that Thus, we have that We choose to get So the boundedness of in is obtained; using the estimate and (75) again, we have the estimate of in .

Once the boundedness of in is obtained, we can prove the following existence theorem.

Theorem 18. Suppose that with , and . Then problem (1) has a solution in .

Lemma 19. Suppose that with , and . Then the solution of (1) given by Theorem 18 is such that(i)if , then ;(ii)if , then , .

Proof. The proof of (i) is similar to that for Lemma 8(i), and we omit the details here.
To prove (ii), if , the proof is identical to that for Lemma 13, and we also omit it here.
If , we can prove the results by Sobolev Embedding Theorem.
If , we choose , and we use once again as a test function in (7). Using , as well as Hölder’s inequality, Sobolev Embedding Theorem, Lemma 5, and (2), we get The choice of in such a way that yields that , if and only if , and that . The choice of implies that . Thus we have the boundedness of in , and so does the limit in .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The project is supported by NSFC (11271154), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education, and by the 985 Program of Jilin University.