On positive solutions of minimal growth for singular 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):=\frac{1}{p}\int_Ω(|\nabla u|^p+V|u|^p)\dx, Q^\prime (u):=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u. It is assumed that $Q\ge 0$ on $C_0^{\infty }(\Omega )$. In a previous paper we discussed relations between the absence of weak coercivity of the functional $Q$ on $C_0^{\infty }(\Omega )$ and the existence of a generalized ground state. In the present paper we study further relationships between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^{\prime }(u)=0$.

28 pages