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EXISTENCE AND NON-EXISTENCE FOR SCHR¨ODINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS
Henghui Zou 대한수학회 2010 대한수학회지 Vol.47 No.3
We study existence of positive solutions of the classical nonlinear Schr¨odinger equation [수식]In fact, we consider the following more general quasi-linear Schr¨odinger equation [수식]where m ∈ (1, n) is a positive number and [수식]is the corresponding critical Sobolev embedding number in Rn. 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.
EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS
Zou, Henghui Korean Mathematical Society 2010 대한수학회지 Vol.47 No.3
We study existence of positive solutions of the classical nonlinear Schr$\ddot{o}$dinger equation $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$. In fact, we consider the following more general quasi-linear Schr$\ddot{o}$odinger equation $-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$ $-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$, where m $\in$ (1, n) is a positive number and $m^*\;:=\;\frac{mn}{n-m}\;>\;0$, is the corresponding critical Sobolev embedding number in $\mathbb{R}^n$. 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.