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PAPER: Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity

REF. DATE:18 October 2013 CREATED/MODIFIED: 18 January 2016 12 views No Comment

Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity (with L. Huang, J. Chen), J. Differential Equations 255(2013), 2463–2483.


ABSTRACT: We study the existence and multiplicity of positive solutions of a class of Schrödinger-Poisson systems of the form

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

where k\in C(\bkR^3) changes sign in \bkR^3, \lim_{|x|\rightarrow \infty} k(x)=k_\infty < 0, and the nonlinearity g behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that \mu > \mu_1 and near \mu_1, where \mu_1 is the first eigenvalue of -\Delta+id in H^1(\bkR^3) with weight function h, whose corresponding positive eigenfunction is denoted by e_1. An interesting phenomenon here is that we do not need the condition \int_{\mathbb{R}^3} k(x) e_1^p dx <  0, which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. S. Alama, G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993) 439–475).


Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.jde.2013.06.022)
Extra info: (P083/2013-10)

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