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.
Keywords:weighted Sobolev space; superlinear; quasilinear elliptic equation
Consider the following quasilinear elliptic problem:
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.
Consider the following pre-Hilbert spaces
with the inner product
where , . Let be the Hilbert space obtained through the completion of by using the method of Cauchy sequences with respect to the norm , and be the completion of the space with the norm . Similarly, we may have () and . Consequently, (2.1) may lead to
For the linear differential operator,
the two-form is
Now we state our main results in this paper.
Theorem 2.1Assume thatis a simple-region, the operatorsatisfies ()-(), and is near-related to the operator, (2.5) holds for bothand, fmeets ()-(), and (the dual of). Then the problem (1.1) has at least one nontrivial weak solution, that is, there exists asuch that
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 ()
Lemma 3.2 ()
From (3) and (4) of (2.5), we have
For (3.10), by (3.2), we obtain
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.
Arguing by contradiction, without loss of generality, suppose that
holds, that is,
(3.14) and (3.16) imply that
Dividing both sides of (3.17) by and leaving , we obtain from the fact that is near-related to , , and (3.5) together with Lemma 3.1 that . However, n is a positive integer. So, we have arrived at a contradiction. (3.12) does not hold. Then
(3.5) and (3.18) imply that there is a subsequence of (for ease of notation take the full sequence) and a  such that
Therefore, from (3.19), we obtain
And recall Lemma 3.1 that
We suppose that Lemma 3.4 is false. Without loss of generality, suppose that
And we can get
also, (1) and (2) of (2.5) to show
So, we have arrived at a contradiction. Thus, there holds
Lemma 3.4 is completed. □
4 Proof of Theorem 2.1
Proof Since is a separable Hilbert space, from Lemma 3.1 and Lemma 3.2, we conclude that there exist a subsequence of (which for ease of notation we take the full sequence) and a function with the following properties :
Also, by (3.1), we have
Hence, the proof of Theorem 2.1 is complete. □
The authors declare that they have no competing interests.
The paper is the result of joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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).
Berestycki, H, Figueredo, DG: Double resonance in semilinear elliptic problems. Commun. Partial Differ. Equ.. 6(1), 91–120 (1981). Publisher Full Text
Rumbos, A, Shapiro, VL: Jumping nonlinearities and weighted Sobolev spaces. J. Differ. Equ.. 214, 326–357 (2005). Publisher Full Text
Rumbos, A: A semilinear elliptic boundary value problem at resonance where the nonlinearity may grow linearly. Nonlinear Anal. TMA. 16, 1159–1168 (1991). Publisher Full Text
Lefton, L, Shapiro, VL: Resonance and quasilinear parabolic differential equations. J. Differ. Equ.. 101, 148–177 (1993). Publisher Full Text
Shapiro, VL: Resonance, distributions and semilinear elliptic partial differential equations. Nonlinear Anal. TMA. 8, 857–871 (1984). Publisher Full Text
Jia, G, Zhao, Q: Existence results in weighted Sobolev spaces for some singular quasilinear elliptic equations. Acta Appl. Math.. 109, 599–607 (2010). Publisher Full Text
Shapiro, VL: Special functions and singular quasilinear partial differential equations. SIAM J. Math. Anal.. 22, 1411–1429 (1991). Publisher Full Text