Stein-Weiss problems via nonlinear Rayleigh quotient for concave-convex nonlinearities

Abstract

In the present work, we consider existence and multiplicity of positive solutions for nonlocal elliptic problems driven by the Stein-Weiss problem with concave-convex nonlinearities defined in the whole space ${\mathbb{R}}^N$. More precisely, we consider the following nonlocal elliptic problem: \begin{equation*} - Δu + V(x)u = λa(x) |u|^{q-2} u + \displaystyle \int \limits_{\mathbb{R}^N}\frac{b(y)\vert u(y) \vert^p dy}{\vert x\vert^α\vert x-y\vert^μ\vert y\vert^α} b(x)\vert u\vert^{p-2}u, \,\, \hbox{in}\ \mathbb{R}^N, \,\, u\in H^1(\mathbb{R}^N), \end{equation*} where $\hat{I}»>0,\hat{I}\pm \in (0,N),N\ge 3,0<\hat{I}¼<N,0<\hat{I}¼+2\hat{I}\pm <N$. Furthermore, we assume also that $V:{\mathbb{R}}^N\to \mathbb{R}$ is a bounded potential, $a\in L^r({\mathbb{R}}^N),a>0$ in ${\mathbb{R}}^N$ and $b\in L^t({\mathbb{R}}^N),b>0$ in ${\mathbb{R}}^N$ for some specific $r,t>1$. We assume also that $1\le q<2$ and $2_{\hat{I}\pm ,\hat{I}¼}<p<2_{\hat{I}\pm ,\hat{I}¼}^{\ast }$ where $2_{\hat{I}\pm ,\hat{I}¼}=(2N-2\hat{I}\pm -\hat{I}¼)/N$ and $2_{\hat{I}\pm ,\hat{I}¼}^{\ast }=(2N-2\hat{I}\pm -\hat{I}¼)/(N-2)$. Our main contribution is to find the largest $\hat{I}{»}^{\ast }>0$ in such way that our main problem admits at least two positive solutions for each $\hat{I}»\in (0,\hat{I}{»}^{\ast })$. In order to do that we apply the nonlinear Rayleigh quotient together with the Nehari method. Moreover, we prove a Brezis-Lieb type Lemma and a regularity result taking into account our setting due to the potentials $a,b:{\mathbb{R}}^N\to \mathbb{R}$.

In the present work, we consider existence and multiplicity of positive solutions for nonlocal elliptic problems driven by the Stein-Weiss problem with concave-convex nonlinearities defined in the whole space ${\mathbb{R}}^N$