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:
where Ω is an open (possibly unbounded) set in ( ), is the first eigenvalue of ((2.3) below), and is a singular quasilinear elliptic operator defined by
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 .
Consider the following pre-Hilbert spaces
with inner product , , and
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
Definition 2.1 For the quasilinear differential operator , the two-form is
For the linear differential operator,
the two-form is
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 ;
( ) For each , , ( ), there exists for with the property
Remark 2.1 From ( ) and ( ), it is easy to see that , are positive and
Definition 2.3 is near-related to if the following condition holds:
We make the following assumptions concerning the operators and : ( ) and satisfy (so do and ):
It is assumed throughout the paper that ( ) meets:
( ) is weakly sequentially continuous;
( ) , s.t. , .
meets the following conditions:
( ) satisfies the Caratheodory conditions;
( ) (superlinear growth condition) There exists θ with , such that
where . K is a nonnegative constant and .
( ) There exists a nonnegative function and a constant , such that
Remark 2.2 Observing that for , , where is a positive function, and meets both ( ) and ( ).
Now we state our main results in this paper.
Theorem 2.1Assume that is a simple- region, the operator satisfies ( )-( ), and is near-related to the operator , (2.5) holds for both and , fmeets ( )-( ), and (the dual of ). Then the problem (1.1) has at least one nontrivial weak solution, that is, there exists a such that
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 .
Lemma 3.3Let all the assumptions in Theorem 2.1 hold. Then for , there exists a such that
Proof For fixed n ( ) and , set . From simple- conditions, we obtain
From (3) and (4) of (2.5), we have
where . Combining (3.3) with (3.4), we get
For , a positive integer, we put
Note from ( ) that for , a.e. . Also, from , the Hölder inequality, Minkowski inequality, and (2.4), for , we get
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).
Part 1. Fix m ( ). To show there exists such that
It is clear that , where
For (3.9), observing the fact that the operator is near-related to , (3.2), (3.5), (3.7), and Lemma 3.1, we conclude that
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.
Part 2. We claim that (n fixed) is uniformly bounded according to m.
Arguing by contradiction, without loss of generality, suppose that
Taking in (3.8),
holds, that is,
On the other hand, using ( ), for , we have
Similarly, we can also obtain the same conclusion where or . As a result,
(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
Moreover, there exists a and a subsequence such that , a.e. for ∀j.
Since by virtue of the Hölder inequality, ( ), ( ) and the Lebesgue dominated convergence theorem, we get
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 3.4Let 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 .
We suppose that Lemma 3.4 is false. Without loss of generality, suppose that
To lead to a contradiction, taking in (3.1), then
And we can get
In view of , ( ), and (3.24),
Dividing both sides of (3.25) by and leaving , from the fact that is near-related to and Lemma 3.1, we get
Apply (3) and (4) of (2.5) in conjunction with ( ) to show
also, (1) and (2) of (2.5) to show
where and are positive constants. Setting and , it is obvious . By (3.27) and (3.28), we conclude that
Using (3.29) and ( ), from (3.25), we obtain
From (3.4), we get . Set . It is easy to obtain
Dividing both sides of (3.31) by and leaving , by (3.26), we get
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 :
Let where is a fixed but arbitrary positive integer. In fact, for , we have
Observing (1) of (2.5), ( )-( ), (4.3)-(4.5) and , we have
On the other hand, applying ( ), ( ), (4.2), (4.3), the Hölder inequality, and the Lebesgue dominated convergence theorem, we obtain that
Also, by (3.1), we have
For (4.10), leaving , from (4.1), (4.8), and (4.9), we have
Next, given , we define a projection , that is,
where . It is easy to get . As a result, there hold
Replacing by in (4.11), passing to the limit as on both sides, and using (4.13), we can obtain
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).
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