Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

Abstract

<p style='text-indent:20px;'>In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0, \mu&gt;0, h\in C(\mathbb{R}, \mathbb{R}) $\end{document}</tex-math></inline-formula> is superlinear at infinity, the potentials <inline-formula><tex-math id="M3">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful <inline-formula><tex-math id="M5">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula>-estimates. For the subcritical case (<inline-formula><tex-math id="M6">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula>) we can deal with large <inline-formula><tex-math id="M7">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula>. For the critical case we treat that <inline-formula><tex-math id="M8">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula> is small.</p>