Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem
The paper is concerned with the Kirchhoff type problem with critical growth \[ \begin{cases} -M\left(\displaystyle\int_{\Omega}|\nabla u|^2\,dx\right)\Delta u=\lambda |u|^2 u+|u|^4u,\,\,\,\text{in}\,\,\,\Omega,\\ u=0,\,\,\,\text{on}\,\,\,\partial\Omega, \end{cases}\tag{P} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^3\), \(M\in C(\mathbb{R}^+,\mathbb{R})\) and \(\lambda >0\). By using the contraction-compactness principle and an abstract critical point theorem, the authors investigate the existence and multiplicity of nontrivial solutions for problem (P), when \(\lambda\) belongs to some neighborhood of the eigenvalue of the nonlinear operator \(-M(\int_{\Omega}|\nabla u|^2\,dx)\Delta\).
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