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PAPER: A positive solution of a Schrödinger-Poisson system with critical exponent

REF. DATE:5 April 2013 CREATED/MODIFIED: 15 January 2016 196 views No Comment

A positive solution of a Schrödinger-Poisson system with critical exponent (with L. Huang), Commun. Math. Anal. 15:1(2013), 29-43.


ABSTRACT: We use variational methods to study the existence of at least one positive solution of the following Schrödinger-Poisson system

    \[\left\{\begin{array}{ll}        -\Delta u +u +l(x)\phi u = k(x)|u|^{{2^*}-2}u +\mu h(x)|u|^{q-2}u  \quad & \ \hbox{in}\ {\bkR}^3,\\ -\Delta \phi = l(x)u^2\quad & \ \hbox{in} \ {\bkR}^3,      \end{array}\right.\]

under some suitable conditions on the non-negative functions l, k, h and constant \mu > 0, where 2\leq q < 2^* (critical Sobolev exponent). Note that the nonlinearity involves a critical exponent, the Sobolev embedding H^1(\bkR^3)\hookrightarrow L^s(\bkR^3) (2\leq s\leq 6) is not compact. This will create additional difficulies in the proof of the Palais-Smale condition. We will transform the problem into a nonlocal elliptic equation in \mathbb{R}^3, where we also consider the limiting case q=2.


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Extra info:> (P081/2013)

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