Abstract
Working in a weighted Sobolev space, a new result involving superlinear nonlinearities
for a quasilinear elliptic boundary value problem in a domain in
MSC: 35J25, 35J62, 65L60.
Keywords:
weighted Sobolev space; superlinear; quasilinear elliptic equation1 Introduction
Consider the following quasilinear elliptic problem:
where Ω is an open (possibly unbounded) set in
The nonlinear part
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 [16].
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 Galerkintype techniques, Brouwer’s fixedpoint theorem, and a new compactly embedding theorem established by V.L. Shapiro in [7].
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
Consider the following preHilbert spaces
with inner product
with the inner product
Definition 2.1 For the quasilinear differential operator
For the linear differential operator,
the twoform is
Definition 2.2
(
(
(
(
(
where
There are many examples to illustrate the Simple
Remark 2.1 From (
Definition 2.3
We make the following assumptions concerning the operators
It is assumed throughout the paper that
(
(
(
(
where
(
Remark 2.2 Observing that for
Now we state our main results in this paper.
Theorem 2.1Assume that
To derive out Theorem 2.1, we first discuss the problem in
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 3.1 ([7])
Assume that
Lemma 3.2 ([7])
Assume that
Lemma 3.3Let all the assumptions in Theorem 2.1 hold. Then for
Proof For fixed n (
From (3) and (4) of (2.5), we have
where
For
Note from (
where
The remaining proof is separated into two parts. The first part is to prove the claim
(3.8) for
Part 1. Fix m (
we set
It is clear that
For (3.9), observing the fact that the operator
and
For (3.10), by (3.2), we obtain
Consequently,
Part 2. We claim that
Arguing by contradiction, without loss of generality, suppose that
Taking
holds, that is,
where
On the other hand, using (
Similarly, we can also obtain the same conclusion where
(3.14) and (3.16) imply that
Dividing both sides of (3.17) by
(3.5) and (3.18) imply that there is a subsequence of
Therefore, from (3.19), we obtain
And recall Lemma 3.1 that
Moreover, there exists a
Since by virtue of the Hölder inequality, (
Now replacing m by
Lemma 3.4Let all the assumptions in Theorem 2.1 hold. Then the sequence
Proof For
We suppose that Lemma 3.4 is false. Without loss of generality, suppose that
To lead to a contradiction, taking
And we can get
In view of
Dividing both sides of (3.25) by
Apply (3) and (4) of (2.5) in conjunction with (
also, (1) and (2) of (2.5) to show
where
Using (3.29) and (
From (3.4), we get
Dividing both sides of (3.31) by
So, we have arrived at a contradiction. Thus, there holds
Lemma 3.4 is completed. □
4 Proof of Theorem 2.1
Proof Since
Let
Observing (1) of (2.5), (
On the other hand, applying (
Also, by (3.1), we have
For (4.10), leaving
Next, given
where
Replacing
Hence, the proof of Theorem 2.1 is complete. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
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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