Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities

Abstract

<p style='text-indent:20px;'>It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u +V(x) u = (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u \, {\rm{\;in\;}}\, \mathbb{R}^N, \\ \ u\in H^1( \mathbb{R}^N) \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \lambda > 0, N \geq 3, \alpha \in (0, N) $\end{document}</tex-math></inline-formula>. The potential <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a continuous function and <inline-formula><tex-math id="M5">\begin{document}$ I_\alpha $\end{document}</tex-math></inline-formula> denotes the standard Riesz potential. Assume also that <inline-formula><tex-math id="M6">\begin{document}$ 1 < q < 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ 2_\alpha < p < 2^*_\alpha $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M8">\begin{document}$ 2_\alpha = (N+\alpha)/N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ 2_\alpha = (N+\alpha)/(N-2) $\end{document}</tex-math></inline-formula>. Our main contribution is to consider a specific condition on the parameter <inline-formula><tex-math id="M10">\begin{document}$ \lambda > 0 $\end{document}</tex-math></inline-formula> taking into account the nonlinear Rayleigh quotient. More precisely, there exists <inline-formula><tex-math id="M11">\begin{document}$ \lambda^* > 0 $\end{document}</tex-math></inline-formula> such that our main problem admits at least two positive solutions for each <inline-formula><tex-math id="M12">\begin{document}$ \lambda \in (0, \lambda^*] $\end{document}</tex-math></inline-formula>. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter <inline-formula><tex-math id="M13">\begin{document}$ \lambda^*> 0 $\end{document}</tex-math></inline-formula> is optimal in some sense which allow us to apply the Nehari method.</p>