On Compactness Conditions for the $p$-Laplacian
Pavel Jirásek · Communications on Pure & Applied Analysis · 2016
We investigate the geometry and validity of various compactness conditions (e.g. Palais-Smale condition) for the energy functional\begin{eqnarray} J_{\lambda_1}(u)=\frac{1}{p}\int_\Omega | abla u|^p \ \mathrm{d}x- \frac{\lambda_1}{p}\int_\Omega|u|^p \ \mathrm{d}x - \int_\Omega fu \ \mathrm{d}x onumber\end{eqnarray} for $u \in W^{1,p}_0(\Omega)$, $1 < p < \infty$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $f \in L^\infty(\Omega)$ is a given function and $-\lambda_1<0$ is the first eigenvalue of the Dirichlet $p$-Laplacian $\Delta_p$ on $W_0^{1,p}(\Omega)$.