Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 69, pp. 1-19.

Title: Existence of nontrivial solutions for biharmonic equations with critical growth

Authors: Juhua He (Yunnan Normal Univ., Kunming, China)
         Ke Wu  (Yunnan Normal Univ., Kunming, China)
         Fen Zhou (Yunnan Normal Univ., Kunming, China)

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 07, 2025.
Math Subject Classifications: 35J35, 35J60, 35J62.
Key Words: Critical exponent; nontrivial solutions; biharmonic operator; quasilinear problem.