Ground state alternative for p-Laplacian with potential term

Abstract

Let $\Omega$ be a domain in ${\mathbb{R}}^d$, $d\ge 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^{\infty }(\Omega )$. Consider the functional $Q$ and its Gâteaux derivative $Q^{\prime }$ given by $$Q(u):=\int_Ω(|\nabla u|^p+V|u|^p)\dx, \frac{1}{p}Q^\prime (u):=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u.$$ If $Q\ge 0$ on $C_0^{\infty }(\Omega )$, then either there is a positive continuous function $W$ such that $\int W|u{|}^p\mathrm{d}x\le Q(u)$ 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 $L_{\mathrm{loc}}^p(\Omega$). 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 Poincaré type: there exists a positive continuous function $W$ such that for every $\psi \in C_0^{\infty }(\Omega )$ satisfying $\int \psi v\mathrm{d}x\ne 0$ 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.

arXiv commentCorrected error in Appendix

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• Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations