Non-homogeneous $(p_1,p_2)$-fractional Laplacian systems with lack of compactness

Abstract

The present paper studies the existence of weak solutions for following type of non-homogeneous system of equations (S)where Ω ⊂ R N is smooth bounded domain, s1, s2 ∈ (0, 1), 1 < p1, p2 < ∞, N > max{p1s1, p2s2}, α > −1 and β > −1. We employ the variational techniques where the associated energy functional is minimized over Nehari manifold set while imposing appropriate bound on dual norms of f1, f2.

Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity

“…We highlight here that only very few results are available for systems in the nonlinear and nonlocal case, that is, (s 1 , s 2 )-fractional (p 1 , p 2 )-Laplacian operators, that is, with s 1 < 1, s 2 < 1, p 1 ≠ 2, and p 2 ≠ 2, and it concerns the nonsingular case. We refer in particular Mukherjee and Mukherjee 35 and Xiang et al 36 and in the nonhomogeneous case Mukherjee and Mukherjee 37 where existence of solutions are investigated with variational methods in case of subcritical and critical growths.…”