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>

A study of extremal parameter for fractional singular Choquard problem

“…In this paper, we are focused to extend the Nehari manifold idea to discuss the multiplicity of solutions for the problem ($$P_{\lambda }$$) beyond the extremal value of parameter $$\lambda ^*$$ (see definition in (2.1) below), inspired by the work of Il'yasov and Silva (see [20]). It is worth mentioning the works of [10, 33, 35] in the direction of singular and Choquard problems. To study the minimizers of the energy functional $$\mathcal {E}_\lambda$$, the functional $$\mathcal {E}_\lambda$$ must be bounded from below in X 0 , which unfortunately is missing in our problem.…”

On elliptic problems with Choquard term and singular nonlinearity

“…There are many articles pertaining to elliptic nonlocal problems driven by the Choquard term with various assumptions on the potential function V . For example, the readers may refer to Moroz-Schaftingen [8], Carvalho et al [1], Mukherjee-Sreenadh [9], and the references therein.…”