Multiple solutions for a class of fourth-order elliptic equations with critical growth

Abstract

In this article, we study the semilinear biharmonic problem with a critical growth: Δ2u=μQ(x)∣u∣p−2u+V(x)∣u∣2**−2uinΩ,u=Δu=0on∂Ω,\left\{\begin{array}{ll}{\Delta }^{2}u=\mu Q\left(x){| u| }^{p-2}u+V\left(x){| u| }^{{2}^{* * }-2}u\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\Delta u=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω⊂RN(N>4)\Omega \subset {{\mathbb{R}}}^{N}\hspace{0.25em}\left(N\gt 4) is a smooth bounded domain, μ>0\mu \gt 0, 1<p<21\lt p\lt 2, and 2**=2N⁄(N−4){2}^{* * }=2N/\left(N-4) is the critical Sobolev exponent. By using the Nehari method and the critical point theory, at least kk nontrivial solutions of the above equations for μ\mu sufficiently small and under some appropriate assumptions on QQ and VV are obtained.

Keywords

• biharmonic equation
• critical growth
• nehari method
• nontrivial solutions
• 35j35
• 35j40
• 47j65