EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

Abstract

We study existence of positive solutions of the classical nonlinear Schr<TEX>$\ddot{o}$</TEX>dinger equation <TEX>$-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$</TEX>, u > 0 in <TEX>$\mathbb{R}^n$</TEX> <TEX>$u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$</TEX>. In fact, we consider the following more general quasi-linear Schr<TEX>$\ddot{o}$</TEX>odinger equation <TEX>$-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$</TEX> <TEX>$-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$</TEX>, u > 0 in <TEX>$\mathbb{R}^n$</TEX> <TEX>$u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$</TEX>, where m <TEX>$\in$</TEX> (1, n) is a positive number and <TEX>$m^*\;:=\;\frac{mn}{n-m}\;</TEX><TEX>></TEX><TEX>\;0$</TEX>, is the corresponding critical Sobolev embedding number in <TEX>$\mathbb{R}^n$</TEX>. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.