On doubly nonlocal $p$-fractional coupled elliptic system

Tuhina Mukherjee, Konijeti Sreenadh

DOI: http://dx.doi.org/10.12775/TMNA.2018.018

Abstract


We study the following nonlinear system with perturbations involving $p$-fractional Laplacian: \begin{equation} \begin{cases} (-\Delta)^s_p u+ a_1(x)u|u|^{p-2} = \alpha(|x|^{-\mu}*|u|^q)|u|^{q-2}u\\ \hskip 2.5 cm + \beta (|x|^{-\mu}*|v|^q)|u|^{q-2}u+ f_1(x)&\text{in } \mathbb R^n, \\ (-\Delta)^s_p v+ a_2(x)v|v|^{p-2} = \gamma(|x|^{-\mu}*|v|^q)|v|^{q-2}v \\ \hskip 2.5cm + \beta (|x|^{-\mu}*|u|^q)|v|^{q-2}v+ f_2(x)& \text{in } \mathbb R^n, \end{cases} \tag{P} \end{equation} where $n> sp$, $0< s< 1$, $p\geq2$, $\mu \in (0,n)$, ${p}( 2-{\mu}/{n})/2 < q < {p^*_s}( 2-{\mu}/{n})/2$, $\alpha,\beta,\gamma > 0$, $0< a_i \in C(\mathbb R^n, \mathbb R)$, $i=1,2$ and $f_1,f_2\colon \mathbb R^n \to \mathbb R$ are perturbations. We show existence of at least two nontrivial solutions for (P) using Nehari manifold and minimax methods.

Keywords


p-fractional Laplacian; Choquard equation; Nehari manifold

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