Ground state alternative for p-Laplacian with potential term

Abstract

Let Ω be a domain in $$\mathbb{R}^d$$ , d ≥ 2, and 1 < p < ∞. Fix $$V \in L_{\mathrm{loc}}^\infty(\Omega)$$ . Consider the functional Q and its Gâteaux derivative Q′ given by $$ Q(u) := \mathop \int_\Omega (|\nabla u|^p+V|u|^p){\rm d}x,\,\, \frac{1}{p}Q^\prime (u) := -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u.$$ If Q ≥ 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′ (v) = 0, such that Q(u k ) → 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′ (u) = 0 in Ω, 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 $$\int \psi v\,{\rm d}x \neq 0$$ there exists a constant C > 0 such that $$C^{-1}\int W|u|^p\,\mathrm{d}x\le Q(u)+C\left|\int u \psi\,\mathrm{d}x\right|^p$$ . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.