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Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms
Boundary Value Problemsvolume 2013, Article number: 229 (2013)
The authors of this paper deal with the existence of weak solutions to the homogenous boundary value problem for the equation with and . The authors prove the existence of solutions in for suitable m and α.
MSC:35J62, 35B25, 76D03.
In this paper, we study the existence of solutions for the following quasi-linear elliptic problem:
where Ω is a bounded domain in () with smooth boundary ∂ Ω. , , , .
Model (1.1) may describe many physical phenomena such as chemical heterogeneous catalysts, nonlinear heat transfers, some biological experiments, etc. [1–3]. In the case when , Lazer and Mckenna in  studied the following problem:
where is singular at . They obtained similar results as that of . Moreover, Boccardo and Orsina in  discussed how the summability of f and the values of α affected the existence, regularity and nonexistence of solutions. For more results, the interested readers may refer to [9, 10]. When , , Giacomoni, Schindler and Takáč in  applied lower and upper-solution method and the mountain pass theorem to prove that the problem
where , , has multiple weak solutions. And then, the authors in  not only improved the results in  but also obtained that the solution was not in if . However, we need to point out that all the papers mentioned discussed the existence of solutions by means of upper-lower solution techniques. In this paper, we apply the method of regularization and Schauder’s fixed point theorem as well as a necessary compactness argument to overcome some difficulties arising from the nonlinearity of the differential operator, the singularity of nonlinear terms and the summability of the weighted function and then prove the existence of positive solutions in for suitable m and α when and , which implies that the summability of the weighted function determines whether or not problem (1.1) has a solution in .
2 Main results
In this section, we apply the method of regularization and Schauder’s fixed point theorem to prove the existence of solutions. In order to prove the main results of this section, we consider the following auxiliary problem:
Definition 2.1 A function is called a solution of problem (1.1) if the following identity holds:
Since the proof of the following lemmas are similar to that in , we only give a sketch of the proof.
Lemma 2.1 Problem (2.1) has a unique nonnegative solution for any fixed , .
Proof Let be fixed. For any , we get that the following problem has a unique solution by applying the variational method to
We may refer to [2, 10] for the existence and uniqueness of the solution for problem (2.2). So, for any , we may define the mapping as . In fact, multiplying the first identity in (2.2) by v, and integrating over Ω, we have
Applying the embedding theorem , we obtain
which implies that
Due to the embedding , we get that Γ is a compact operator. Moreover, if for some , then and hence for a constant C independent of λ. Then by Schauder’s fixed point theorem, we know that there exists such that , i.e., problem (2.1) has a solution. Noting that , the maximum principle in [13, 14] shows that , . □
Lemma 2.2 The sequence is increasing with respect to n. in for any , and there exists a positive constant (independent of n) such that for all ,
Proof Choosing as a test function, observing that
This inequality yields a.e. in Ω, that is, for every . Since the sequence is increasing with respect to n, we only need to prove that satisfies inequality (2.3). According to Lemma 2.1, we know that there exists a positive constant C (only depending on , N, p) such that , then
Noting that , , the strong maximum principle implies that in Ω, i.e., inequality (2.3) holds. □
Theorem 2.1 Suppose that f is a nonnegative function in and , then problem (1.1) has a solution in .
Proof We consider the existence of solutions in the case when . Multiplying the first identity in problem (2.1) by and integrating over Ω, we get
Then we know that there exist and such that
For every , we get from inequality (2.3) that
where . Then applying Lebesgue’s dominated convergence theorem, one has that
as satisfies the following identity:
Combining with (2.4)-(2.6), we have that
Next, we shall prove that a.e. in Ω. It is easy to see that both (2.6) and (2.7) hold for all with compact support. Thus in (2.6) we choose , where , , and , to obtain
Noting that , we obtain that
Letting in (2.9) and using identity (2.7), we get
which implies that
Let in (2.10), where ψ is an arbitrary function in and is a constant, we get that
Let , we have that
Since ψ is an arbitrary function, we obtain that
We choose , where is a constant, and we have that
which yields that a.e. in Ω. This proves that u is a weak solution of problem (1.1) when . □
The first question is what happens to the solution if the inhomogeneous function is not in but a nonnegative bounded Radon measure μ. Since a nonnegative Radon measure μ may always be approximated by a sequence of functions, we want to know whether the approximate solutions may converge to a nontrivial function in or whether the approximate solutions converge. The existence of solutions in this case is still unknown, but we have the following result.
Theorem 2.2 Suppose that μ is a nonnegative Radon measure concentrated on a Borel set E of zero p-capacity, and that is a bounded sequence of nonnegative functions which converges to μ in the narrow topology of measures. Let be the solution of problem (2.1) with the non-homogeneous function . Then
Proof By the conclusion of Theorem 2.1, we get that the solution of problem (2.1) with is bounded in . Since the set E has zero p-capacity, by [, Lemma 5.1], for any real number , there exists a function satisfying
Noting that converges to μ in the narrow topology of measure, one has from (2.12) that
Define . Choosing as a test function in (2.1) with a non-homogeneous function , we obtain that
Using , we assume that is any subsequence such that in and in . We show that the two limits in the theorem hold for any such subsequence. This completes the proof. Note that
By (2.12)-(2.15) and weak lower semi-continuity, we have
Letting , we have
which implies a.e. in Ω. This completes the proof of the theorem. □
The above theorem shows that problem (1.1) has a solution in when and . But if f is only a Radon measure, the solution may not exist. At least, the solution can not be approximated by the solution of problem (2.1). The second question we are interested in is whether this problem has a solution in when () and . We have the following.
Theorem 2.3 Let f be a nonnegative function in () (). If , then problem (1.1) has a solution satisfying
In order to prove this theorem, we need the following lemma.
Lemma 2.3 The solution to problem (2.1) with satisfies
Proof By , and Lemma 2.2 in , we know that there exists such that and , which implies that the gradient of exists everywhere, then the Hopf lemma in  shows that , in , where ν is the outward unit normal vector of ∂ Ω at x. Moreover, following the lines of proof of the lemma in , we get
Proof of Theorem 2.3 Multiplying the first identity in problem (2.1) by , integrating over Ω, and applying Hölder’s inequality and Lemma 2.3, we get
as by the assumption ; hence
From (2.18), we know that there exist and such that
For every , from Lemma 2.2, we get that
Then applying Lebesgue’s dominated convergence theorem, we have
since satisfies the following identity:
In (2.21), letting and using (2.19) and (2.20), we have
Following the lines of proof of Theorem 2.1, we get that problem (1.1) has a solution in . □
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The work was supported by the Natural Science Foundation of China (11271154) and by the 985 program of Jilin University. We are very grateful to the anonymous referees for their valuable suggestions that improved the article.
The authors declare that they have no competing interests.
Both authors collaborated in all the steps concerning the research and achievements presented in the final manuscript.