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PAPER: Multiplicity result for a class of elliptic equations with singular term

REF. DATE:1 September 2012 CREATED/MODIFIED: 14 January 2016 189 views No Comment

Multiplicity result for a class of elliptic equations with singular term (with J. Chen, K. Murillo), Nonlinear Analysis TMA 75:15(2012), 5797–5814.


ABSTRACT: We consider the existence of nontrivial solutions of the equation

(1)   \begin{equation*} -\Delta u -{\frac{\lambda }{{|x|^{2}}}}u= |u|^{2^{\ast}-2}u + \mu|x|^{\alpha-2}u + f(x)|u|^{\gamma},\quad x\in \Omega\backslash\{0\},\:\:\: u\in H^1_0(\Omega), \end{equation*}

where 0\in\Omega is a smooth bounded domain in \mathbb{R}^N (N\geq 3). By variational methods and Nehari set techniques, we show that this problem, under some additional hypotheses on \lambda>0, \mu>0, \alpha>0, 0\leq \gamma<1 and f\in L^\infty(\Omega), has four nontrivial solutions in H^1_0(\Omega), and that least one of them is sign-changing. Classes of elliptic equations which include (1) have a lost of compactness phenomena, since the nonlinearity has a critical growth imposed by the critical exponent 2^{\ast } of the Sobolev embedding H^1_0(\Omega) into L^{2^{\ast }}(\Omega). This means that we could not use standard variational methods. On the other hand, due to the presence of the singular term \frac{\lambda }{{|x|^{2}}}, the problem has a strong singularity at 0\in \Omega.


Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.na.2012.05.023)
Extra info: (P102/2012-06-23)

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