Semilinear elliptic problems via the nonlinear Rayleigh quotient with two nonlocal nonlinearities

Abstract

It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: $\{\begin{array}{l}-\mathrm{\Delta }u+V(x)u=\mu (I_{{\alpha }_1}\ast |u{|}^q)|u{|}^{q-2}u-\lambda (I_{{\alpha }_2}\ast |u{|}^p)|u{|}^{p-2}u\text{in}{\mathbb{R}}^N,\\ u\in H^1({\mathbb{R}}^N),\end{array}$where $p>q,\lambda ,\mu >0$, ${\alpha }_1\le {\alpha }_2$; ${\alpha }_1,{\alpha }_2\in (0,N),N\ge 3$; $p\in (2_{{\alpha }_2},2_{{\alpha }_2}^{\ast })$; $q\in (2_{{\alpha }_1},2_{{\alpha }_1}^{\ast })$, $2_{{\alpha }_j}=(N+{\alpha }_j)/N$ and $2_{{\alpha }_j}^{\ast }=(N+{\alpha }_j)/(N-2),j=1,2$. Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp ${\mu }_n>0$ and ${\lambda }^{\ast },{\lambda }_{\ast }>0$ such that our main problem admits at least two solutions for each $\mu >{\mu }_n$ where $\lambda \in (0,min({\lambda }^{\ast },{\lambda }_{\ast }))$. The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each $\mu <{\mu }_n$ and $\lambda >0$.

Keywords:

• Choquard problem
• double parameters
• nonlinear Rayleigh quotient
• nonlocal elliptic problems

Mathematics Subject Classifications:

• 35A15
• 35A25
• 35J15
• 35J20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was also partially supported by CNPq with grants 309026/2020-2.