A positive solution of a p-Laplace-like equation with critical growth
© Yang et al.; licensee Springer 2012
Received: 14 May 2012
Accepted: 13 September 2012
Published: 2 October 2012
The existence of a positive solution of a p-Laplace-like equation with critical growth is established by the generalization to the concentration-compactness principle and the Sobolev inequality under some proper assumptions. Moreover, we achieve some regularity results of the solution.
where Ω is a smooth bounded domain in , , and the functions a, f satisfy some proper conditions, the details of which are described later.
There were many papers about the existence of the solution of p-Laplacian problems involving critical growth such as [1–6]. In them, and f are some concrete functions with critical growth, which means that does not converge to zero as , where is the critical exponent, i.e., . The concentration-compactness principle, which was built by Lions in [7, 8], plays an important role in achieving the existence of a nontrivial solution of the problems in them.
The authors proved the existence of a nonnegative and nontrivial solution for a Dirichlet problem for p-mean curvature operate with critical growth in , where , , and f is some concrete function involving a critical exponent. Since the function a has an explicit form, the authors can use the concentration-compactness principle to achieve their results, too. But if a is an abstract function in problem (1.1), then the problem becomes more complicated and interesting, and Lions’ C-C principle cannot be directly applied to it. Thanks to the generalization of the C-C principle in , we can establish the existence of a nonnegative and nontrivial solution of equation (1.1) if we impose some proper conditions on the functions a and f and make more careful estimates. Moreover, we achieve some regularity result of the solution and prove the solution is positive under some proper assumptions. The results can be easily extended to a more general p-Laplace-like equation with critical growth and singular weights by the Caffarelli-Kohn-Nirenberg inequality and the method in .
Recently, there have been some articles on stochastic partial differential equations (SPDEs) involving p-Laplace operator; see [12, 13]. Some estimates and properties of the solution of the corresponding elliptic equations are important to the research on SPDEs. So, the results in this paper may be useful in the study of p-Laplace SPDEs with critical growth.
In this paper, we suppose that the potential satisfies the following assumptions.
Let be of continuous derivative with respect to ξ with and satisfy the following conditions:
(A3) is strictly convex in ξ, that is, for any ,
(A4) there exists a positive number such that .
We impose some assumptions on the critical nonlinear term , which is continuous, as follows:
Moreover, we suppose a and f satisfy the next correlation.
where X is , i.e., the completion of with the norm .
It is not difficult to see that both and with satisfy (A1)-(A4), and the problems in  and  are concrete examples of problem (1.1). Moreover, we can consider some more generalized problem with a singular nonlinear term with critical growth by the similar method in this paper and . Then we can achieve more generalized results than those in  and , which will be considered in another paper. Since is not an explicit function, we need to use the generalization of the C-C principle in  and more subtle estimates to study problem (1.1).
The first main result in this paper is
Theorem 1.1 Suppose problem (1.1) satisfies assumptions (A1)-(A4), (B1)-(B4) and (C1). Moreover, there exists a nontrivial such that and
(C2) , where , with ().
Then problem (1.1) has a nonnegative and nontrivial solution.
Since the condition (C2) is difficult to check, we give another easily checked theorem.
Theorem 1.2 Assume conditions (A1)-(A4), (B1)-(B4) and (C1) are satisfied and
(A5) for any ,
(B5) there exists a nonempty set such that for any ,
Then problem (1.1) has a nonnegative and nontrivial solution in X.
To establish the regularity of the solution u and prove in Ω, we need to impose stronger assumptions on the potential a and the nonlinear term f, which are as follows:
(D2) a admits the form . Moreover, for any .
Theorem 1.3 Assume the assumptions in Theorem 1.1 or those in Theorem 1.2 hold, then there exists a constant such that the solution if assumption (D1) holds. Moreover, if assumption (D2) is satisfied, then in Ω.
In Section 2, we will prove the main results. Some corollaries and examples are shown in Section 3.
2 The proof of the main results
First, we present the main tool in this paper - the generalized concentration-compactness principle, which is easily deduced from Theorem 2.1 in .
Lemma 2.1 Suppose assumptions (A1), (A2) and (A3) hold, weakly converges to u in X and , converge to μ, ν weakly in the sense of measures, respectively.
where denotes the Dirac measure at the point .
It is clear that if . Next, we deduce satisfies the geometrical result of the mountain pass theorem without the (PS) condition, i.e.,
where , denotes the boundary of .
Proof Let in (2.6), and we have .
Hence, if is small enough. So, we have showed the existence of and in (2.8).
Hence, if t is large enough, then we can set satisfying (2.8). □
Γ denotes the class of continuous paths joining 0 to in X, denotes the dual space of X.
Lemma 2.3 The sequence is bounded in X.
In the last inequality, we have used (2.2) and (2.5). If we fix a small enough σ such that in the above inequality, then the conclusion in this lemma is obvious. □
As a result of the above preparations, we can prove Theorem 1.1.
Applying Lemma 2.1, we have the corresponding conclusions of Lemma 2.1.
Hence, is still bounded in X and the boundary is independent of ε, j.
In the above inequality, first letting , then taking , and finally taking , we deduce . So, (2.1) implies (2.13). Since ν is a bounded measure, J is at most finite.
According to (2.17), (2.18), we deduce that converges to 0 a.e. in Ω as , maybe extracting a subsequence. Since is strictly convex, by the same method as , we claim a.e. in Ω, and there exists a subsequence, still denoted by itself, such that weakly converges to in . Hence, (2.7), (2.9) and (2.11) imply that weakly converges to in and . So, it is not difficult to see that and , which means that u is a weakly solution of equation (1.1).
According to the definition of and assumption (C2), we have , which means . □
As a result of the preparations, we can prove Theorem 1.2 as follows.
Proof of Theorem 1.2 Without loss of generalization, we suppose . For convenience, let A and B denote some positive constants which may be different in different places.
Hence, (2.22) implies is bounded, and the bound is independent of ε.
So, assumption (C3) implies that assumption (C2) holds if we choose ε small enough, and the conclusion in Theorem 1.2 follows from Theorem 1.1. □
Thus, follows from the standard Moser iteration method.
3 Some corollaries and examples
In this section, we firstly consider when (C3) is true through analyzing and , then we give some concrete examples and corollaries.
Firstly, we analyze the effect of to :
Lemma 3.1 Suppose satisfies assumptions (A1)-(A5) and
(A6) There exist positive numbers , , A, B such that for any and for any .
Secondly, we analyze how affects . The proof is similar to the above, and we omit it.
Lemma 3.2 Suppose defined in Theorem 1.2 satisfies
(B6) There are positive numbers A, B and such that when .
Lemma 3.3 Suppose defined in Theorem 1.2 satisfies
(B7) There exist positive numbers A, B and such that if .
In the following, we can utilize Theorem 1.2 and Lemmas 3.1-3.3 to prove the following results about some concrete problems. The proof is trivial and we omit it.
Then problem (1.1) possesses a nontrivial solution.
Proof Take and in Lemma 3.1, and we can deduce the conclusion. □
Then problem (3.1) possesses a positive solution in ().
Then problem (3.2) possesses a positive solution in ().
Proof Letting and in Lemma 3.1, and combining Lemma 3.2, Lemma 3.3 and Theorem 1.3, we can derive the conclusion. □
The project is supported by National Natural Science Foundation of China (No. 11271143, 10901060, 10971073), the Natural Science Foundation of Zhejiang Province (No. Y6110775, Y6110789).
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