Ground state alternative for p-Laplacian with potential term

Abstract

Let $\Omega$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(\Omega)$. Consider the functional $Q$ and its G\^{a}teaux derivative $Q^\prime$ given by $$Q(u):=\int_\Omega (|\nabla u|^p+V|u|^p)\dx, \frac{1}{p}Q^\prime (u):=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u.$$ If $Q\ge 0$ on $C_0^{\infty}(\Omega)$, then either there is a positive continuous function $W$ such that $\int W|u|^p \mathrm{d}x\leq Q(u)$ for all $u\in C_0^{\infty}(\Omega)$, or there is a sequence $u_k\in C_0^{\infty}(\Omega)$ and a function $v>0$ satisfying $Q^\prime (v)=0$, such that $Q(u_k)\to 0$, and $u_k\to v$ in $L^p_\mathrm{loc}(\Omega$). In the latter case, $v$ is (up to a multiplicative constant) the unique positive supersolution of the equation $Q^\prime (u)=0$ in $\Omega$, and one has for $Q$ an inequality of Poincar\'e type: there exists a positive continuous function $W$ such that for every $\psi\in C_0^\infty(\Omega)$ satisfying $\int \psi v \mathrm{d}x \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^p \mathrm{d}x\le Q(u)+C|\int u \psi \mathrm{d}x|^p$ for all $u\in C_0^\infty(\Omega)$. As a consequence, we prove positivity properties for the quasilinear operator $Q^\prime$ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.