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PAPER: Four solutions of an elliptic problem with double singularity

REF. DATE:9 September 2012 CREATED/MODIFIED: 14 January 2016 369 views No Comment

Four solutions of an elliptic problem with double singularity (with J. Chen, K. Murillo) Lib. Math. 32:2(2012), 169-192.


ABSTRACT: We consider the existence of nontrivial solutions u\in H_0^1(\Omega) of a Dirichlet problem with equation

    \[-\Delta u -\frac{\mu}{|x|^2}u = \lambda f(x)|u|^{q-2}u + \frac{|u|^{\frac{4-2s}{N-2}}u}{|x|^s},\quad x\in\Omega\backslash\{0\},\]

where 0\in\Omega\subset\mathbb{R}^N is a bounded domain with smooth boundary, f:\Omega \rightarrow \mathbb{R} is a real continuous and positive function, 0\leq s<2, 0\leq \mu < \bar{\mu} -4 (necessarily N> 6), and \frac{N+\sqrt{{\bar{\mu}-\mu}}}{\sqrt{\bar{\mu}} + \sqrt{\bar{\mu}-\mu}} < q < 2, with \bar{\mu} = (N-2)^2/4. Note that \frac{4-2s}{N-2} is the critical Sobolev-Hardy exponent minus 2. By variational methods we show that there are (at least) two positive solutions and (at least) one pair of sign-changing solutions in H_0^1(\Omega) for any \lambda\in (0,\Lambda^*), where \Lambda^* is a positive value suitably defined.


Download: Preprint version (RIA |  download) / Publisher version (Lib. Math.)
Extra info: (P066/2012-06-23)

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