Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $

Abstract

<p style='text-indent:20px;'>This work concerns with the existence and multiplicity of positive solutions for the following quasilinear Schrödinger equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u+V(x)u-\Delta(u^2)u = a(\epsilon x)g(u), \; \; \; \; x\in\mathbb R^N, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ V(x)&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ u&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ g $\end{document}</tex-math></inline-formula> are continuous functions and <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula> has <inline-formula><tex-math id="M8">\begin{document}$ m $\end{document}</tex-math></inline-formula> maximum points. With the change of variables we show that this equation has at least <inline-formula><tex-math id="M9">\begin{document}$ m $\end{document}</tex-math></inline-formula> nontrivial solutions by using variational methods, the Ekeland's variational principle, and some properties of the Nehari manifold. Some recent results are improved.</p>