Nodal regions for solutions of nonlinear elliptic problems

Abstract

In questo lavoro, mediante la teoria di Morse, viene data una stima del numero delle regioni nodali delle soluzioni del problema $-\Delta u=\lambda c(x)u+\mid u\mid^{p-2}u\: in\:\Omega,\: u\epsilon H_{0}^{1}(\Omega),\: dove\:\Omega\subset\mathbf{R^{\textrm{n}}},N\geq3$, è un aperto connesso, limitato e regolare, $p\epsilon(2,2N/(N-2],$ c(x) $\epsilon L^{q}(\Omega),$ q > p/(p-2) e $\lambda\epsilon\mathbf{R}$.

In this paper we are concerned with the problem $-\Delta u=\lambda c(x)u+\mid u\mid^{p-2}u\: in\:\Omega,\: u\epsilon H_{0}^{1}(\Omega),\: where\:\Omega\subset\mathbf{R^{\textrm{n}}},N\geq3$, is a smooth bounded domain, $p\epsilon(2,2N/(N-2],$ c(x) $\epsilon L^{q}(\Omega),$ q > p/(p-2) and $\lambda\epsilon\mathbf{R}$. Using the Morse theory, we estimate the number of the nodal regions of the solutions of the above problem.