Juhua He, Ke Wu, Fen Zhou
Abstract:
We consider the biharmonic equation with critical Sobolev exponent,
$$
\Delta^2u-\Delta u-\Delta(u^2)u+V(x)u=|u|^{2^{**}-2}u+\alpha |u|^{p-2}u,\quad
\text{in }\mathbb{R}^N,
$$
where \(N> 4\), \(\alpha>0\), \(V(x)\) is a given potential, \(2^{**}=\frac{2N}{N-4}\) is the
Sobolev critical exponent and \(2< p< 2^{**}\). Under the combined influence of the
biharmonic, quasilinear terms, and critical nonlinearities, looking for
solutions with \(N\in \{5,6\}\) is totally different
from the case when \(N\geq 7\). For the case \(N\in \{5,6\}\), we show that
this equation has a nontrivial solution, using a variational method.
Submitted January 6, 2025. Published July 7, 2025.
Math Subject Classifications: 35J35, 35J60, 35J62.
Key Words: Critical exponent; nontrivial solutions; biharmonic operator; quasilinear problem.
DOI: 10.58997/ejde.2025.69
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Juhua He Department of Mathematics Yunnan Normal University Kunming 650500, China email: 1745190963@qq.com |
|---|---|
| Ke Wu Department of Mathematics Yunnan Normal University Kunming 650500, China email: wuke2002@126.com | |
| Fen Zhou Department of Mathematics Yunnan Normal University Kunming 650500, China email: zhoufen_85@163.com |
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