Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms
© Chu and Gao; licensee Springer. 2013
Received: 17 April 2013
Accepted: 30 September 2013
Published: 7 November 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.
where Ω is a bounded domain in () with smooth boundary ∂ Ω. , , , .
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
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 , .
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 , . □
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 .
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.
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.
In order to prove this theorem, we need the following lemma.
Following the lines of proof of Theorem 2.1, we get that problem (1.1) has a solution in . □
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.
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