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Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity

Published online by Cambridge University Press:  02 April 2015

Yinbin Deng  and
Wei Shuai
Yinbin Deng
Affiliation:
Department of Mathematics, Huazhong Normal University, Wuhan 430079, People’s Republic of China, (ybdeng@mail.ccnu.edu.cn)
Wei Shuai
Affiliation:
Department of Mathematics, Huazhong Normal University, Wuhan 430079, People’s Republic of China, (ybdeng@mail.ccnu.edu.cn)
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Abstract

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In this paper, we study the existence of non-trivial solutions for the following class of semilinear biharmonic problem with critical nonlinearity:

Here Δ 2u = Δ(Δu), N ≥ 5, μ > 0 is a parameter, 2** = 2N/(N − 4) is the critical Sobolev exponent, V (x) and K (x) are positive continuous functions that vanish at infinity, f is a function with a subcritical growth and P(x) is a bounded, non-negative continuous function. By working in weighted Sobolev spaces and using a variational method, we prove that the problem has at least one non-trivial solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 2 , April 2015 , pp. 281 - 299
Copyright
Copyright © Royal Society of Edinburgh 2015