On doubly nonlocal $p$-fractional coupled elliptic system

Abstract

\noi We study the following nonlinear system with perturbations involving p-fractional Laplacian \begin{equation*} (P)\left\{ \begin{split} (-\De)^s_p u+ a_1(x)u|u|^{p-2} &= α(|x|^{-μ}*|u|^q)|u|^{q-2}u+ β(|x|^{-μ}*|v|^q)|u|^{q-2}u+ f_1(x)\; \text{in}\; \mb R^n,\\ (-\De)^s_p v+ a_2(x)v|v|^{p-2} &= γ(|x|^{-μ}*|v|^q)|v|^{q-2}v+ β(|x|^{-μ}*|u|^q)|v|^{q-2}v+ f_2(x)\; \text{in}\; \mb R^n, \end{split} \right. \end{equation*} where $n>sp$, $0<s<1$, $p\ge 2$, $\mu \in (0,n)$, $\frac{p}{2}\left(2-\frac{\mu }{n}\right)<q<\frac{p_s^{\ast }}{2}\left(2-\frac{\mu }{n}\right)$, $\alpha ,\beta ,\gamma >0$, $0< a_i \in C^1(\mb R^n, \mb R)$, $i=1,2$ and $f_1,f_2: \mb R^n \to \mb R$ are perturbations. We show existence of atleast two nontrivial solutions for $(P)$ using Nehari manifold and minimax methods.

26 pages

References in corpus (1)

• Nodal solutions for the Choquard equation