Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

Abstract

<p style='text-indent:20px;'>In this paper, we consider the following Schrödinger-Poisson equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\triangle u + u + \phi u = u^{5}+\lambda g(u), &amp;\hbox{in}\ \ \Omega, \\\ &amp; -\triangle \phi = u^{2}, &amp; \hbox{in}\ \ \Omega, \\\ &amp; u, \phi = 0, &amp; \hbox{on}\ \ \partial\Omega.\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{3} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> and the nonlinear growth of <inline-formula><tex-math id="M4">\begin{document}$ u^{5} $\end{document}</tex-math></inline-formula> reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution.</p>