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PAPER: Positive solutions for elliptic problems with critical nonlinearity and combined singularity

REF. DATE:24 March 2010 CREATED/MODIFIED: 14 January 2016 242 views No Comment

Positive solutions for elliptic problems with critical nonlinearity and combined singularity (with J. Chen), Mathematica Bohemica, 135:4(2010), 413-422.


ABSTRACT: We study the existence of multiple positive weak solutions of the equation

    \[\left\{\begin{array}{rll} -\Delta u - {\lambda\over {|x|^2}}u &= u^{2^\ast-1}  + \mu u^{-q} & \hbox{in}\quad \Omega\backslash\{0\},\\ u(x) > 0 & \hbox{in}\quad \Omega\backslash\{0\},\quad\quad  u(x)= 0 & \hbox{on}\quad \partial\Omega,\end{array}\right.\eqno{(P_{\lambda,\mu})}\]

where 0\in \Omega and \Omega\subset\mathbb{R}^N(N\geq 3) is a bounded domain with smooth boundary, 2^{\ast} =2N/(N-2) is the critical Sobolev exponent, 0 < \lambda < \Lambda=(N-2)^2/4 and 0 < q < 1. We use variational methods to prove that for suitable \mu, the problem has at least two positive weak solutions.


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