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

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a class of nonlocal elliptic problems with concave-convex nonlinearities, using nonlinear Rayleigh quotient and Nehari methods, in the whole space setting.

Contribution

It introduces a novel approach combining nonlinear Rayleigh quotient and Nehari method to find multiple solutions for nonlocal elliptic problems with specific nonlinearities.

Findings

Established existence of at least two positive solutions for certain parameter ranges.

Identified the largest parameter value for which solutions exist.

Proved a Brezis-Lieb type lemma and regularity results tailored to the problem.

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*} - \Delta u + V(x)u = \lambda a(x) |u|^{q-2} u + \displaystyle \int \limits_{\mathbb{R}^N}\frac{b(y)\vert u(y) \vert^p dy}{\vert x\vert^\alpha\vert x-y\vert^\mu \vert y\vert^\alpha} b(x)\vert u\vert^{p-2}u, \,\, \hbox{in}\ \mathbb{R}^N, \,\, u\in H^1(\mathbb{R}^N), \end{equation*} where $\lambda >0, \alpha \in (0,N), N\geq3, 0<\mu<N, 0 < \mu + 2 \alpha < N$. 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 $1\leq q<2$ and $2_{\alpha,\mu} < p<2_{\alpha,\mu}^*$ where $2_{\alpha ,\mu}=(2N-2\alpha-\mu)/N$ and $2_{\alpha,\mu}^*= (2N-2\alpha-\mu)/(N-2)$. Our main contribution is to find the largest $\lambda^* > 0$ in such way that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda^*)$. 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 $a, b : \mathbb{R}^N \to \mathbb{R}$.