Stein-Weiss problems via nonlinear Rayleigh quotient for concave-convex nonlinearities
Edcarlos D. Silva, Marcos. L. M. Carvalho, Márcia S. B. A. Cardoso
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 $λ>0, α\in (0,N), N\geq3, 0<μ<N, 0 < μ+ 2 α< N$. 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 $2_{α,μ} < p<2_{α,μ}^*$ where $2_{α,μ}=(2N-2α-μ)/N$ and $2_{α,μ}^*= (2N-2α-μ)/(N-2)$. Our main contribution is to find the largest $λ^* > 0$ in such way that our main problem admits at least two positive solutions for each $λ\in (0, λ^*)$. 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 $a, b : \mathbb{R}^N \to \mathbb{R}$.