An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$

Abstract

An improved Poincaré inequality is shown for $p>2$: There exists a constant $c>0$ such that for all $u\in W_0^{1,p}(\Omega)$, \begin{equation*} \tag*{(P)} \int_\Omega |\nabla u|^p \,{\rm d}x - \lambda_1 \int_\Omega |u|^p \,{\rm d}x \geq c \Big( | u^\parallel |^{p-2} \int_\Omega |\nabla\varphi_1|^{p-2} |\nabla u^\top|^2 \,{\rm d}x + \int_\Omega |\nabla u^\top|^p \,{\rm d}x \Big) . \end{equation*} Here, a function $u\in L^2(\Omega)$ is decomposed as an orthogonal sum \[ u = u^\parallel\cdot \varphi_1 + u^\top \;\mbox{ where }\; u^\parallel {\stackrel{{\mathrm {def}}}{=}} \|\varphi_1\|_{ L^2(\Omega) }^{-2} \langle u, \varphi_1 \rangle \;\mbox{ and }\; \langle u^\top, \varphi_1 \rangle = 0 , \] $\lambda_1$ denotes the first eigenvalue of the positive Dirichlet $p$-Laplacian $-\Delta_p$, $\Delta_p u\equiv {\mathop{\mathrm {div}}} ( |\nabla u|^{p-2} \nabla u )$, $\lambda_1$ is simple, and $\varphi_1$ stands for the corresponding eigenfunction. Inequality (P) is applied to show the existence of a weak solution to the following degenerate quasi\-linear boundary value problem at resonance, where $f\in L^2(\Omega)$ with $\langle f,\varphi_1 \rangle = 0$: \[ - \Delta_p u = \lambda_1 |u|^{p-2} u + f(x) \;\mbox{ in } \Omega ;\qquad u = 0 \;\mbox{ on } \partial\Omega . \]