Existence of solutions for quasilinear elliptic equations with superlinear nonlinearities
© Gao et al.; licensee Springer 2012
Received: 23 March 2012
Accepted: 6 August 2012
Published: 10 August 2012
Working in a weighted Sobolev space, a new result involving superlinear nonlinearities for a quasilinear elliptic boundary value problem in a domain in is established. The proofs rely on the Galerkin method, Brouwer’s theorem and a new weighted compact Sobolev-type embedding theorem due to V.L. Shapiro.
MSC:35J25, 35J62, 65L60.
The nonlinear part in Eq. (1.1) satisfies certain superlinear conditions.
There have been many results for quasilinear elliptic equations under the conditions of which the nonlinearities satisfy sublinear or linear growth in a weighted Sobolev space. One can refer to [1–6].
However, there seem to be relatively few papers that consider the quasilinear elliptic equations with superlinearity, because the compactly embedding theorem cannot be obtained easily.
The aim of this paper is to obtain an existence result for problem (1.1). Our methods combine the Galerkin-type techniques, Brouwer’s fixed-point theorem, and a new compactly embedding theorem established by V.L. Shapiro in .
This paper is organized as follows. In Section 2, we introduce some necessary assumptions and main results. In Section 3, four fundamental lemmas are established. In Section 4, the proofs of the main results are given.
2 Assumptions and main results
In this section, we introduce some assumptions and give the main results in this paper.
Let be a fixed closed set (it may be the empty set) and be weight functions. is nonnegative (maybe identically zero). Denote as the vector function .
Definition 2.2 is a simple- region if the following conditions ()-() hold:
() There exists a complete orthonormal system in . Also, , ∀n;
() There exists a sequence of eigenvalues , corresponding to the orthonormal sequence , and satisfying as , such that , ;
() , where is an open set for ;
() For each and in ()-(), associated with each there are positive functions satisfying , and , , for ;
It is assumed throughout the paper that () meets:
() is weakly sequentially continuous;
() , s.t. , .
meets the following conditions:
() satisfies the Caratheodory conditions;
where . K is a nonnegative constant and .
Remark 2.2 Observing that for , , where is a positive function, and meets both () and ().
Now we state our main results in this paper.
To derive out Theorem 2.1, we first discuss the problem in , which is the subspace of spanned by . Then by virtue of the Galerkin method, the results will be extended to .
3 Fundamental lemmas
In this section, we introduce and establish four fundamental lemmas. Lemmas 3.1 and 3.2 give two useful embedding theorems. Lemma 3.3 constructs some approximation solutions in . Lemma 3.4 studies the properties of the approximation solutions.
Lemma 3.1 ()
Assume that is given by (2.3) and is a simple- region. For , then is compactly imbedded in for ∀θ (); for , then is compactly imbedded in for ∀θ ().
Lemma 3.2 ()
Assume that is given by (2.3) and is a simple- region. Then is compactly imbedded in .
where is a positive constant depending on m.
The remaining proof is separated into two parts. The first part is to prove the claim (3.8) for . The second part is to get the conclusion by leaving based on (3.8).
Consequently, where (here is a large enough constant). By virtue of the generalized Brouwer’s theorem , there exists , such that , . Taking , then (3.8) holds.
Part 2. We claim that (n fixed) is uniformly bounded according to m.
Moreover, there exists a and a subsequence such that , a.e. for ∀j.
Now replacing m by in (3.8) and taking the limit as on both sides of the equation, we consequently obtain that (3.1) holds and Lemma 3.3 is completed. □
Lemma 3.4 Let all the assumptions in Theorem 2.1 hold. Then the sequence obtained in Lemma 3.3 is uniformly bounded in .
Proof For in Lemma 3.3, set where .
Lemma 3.4 is completed. □
4 Proof of Theorem 2.1
Hence, the proof of Theorem 2.1 is complete. □
The authors express their sincere thanks to the referees for their valuable suggestions. This work was supported financially by the National Natural Science Foundation of China (11171220).
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