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PAPER: On the Schrödinger-Poisson system with a general indefinite nonlinearity

REF. DATE:29 September 2015 CREATED/MODIFIED: 18 January 2016 14 views No Comment

On the Schrödinger-Poisson system with a general indefinite nonlinearity (with L. Huang, J. Chen), Nonlinear Analysis: Real World Applications 28(2016), 1–19.

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. D.G. Costa, H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in \bkR^N, Calc. Var. Partial Differential Equations 13(2001) 159–189.)

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.nonrwa.2015.09.001)
Extra info: (P084/2016-04)

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