Existence and non-existence results to a mixed anisotropic Schrödinger system in a plane

Abstract

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type − ∂ x x u + ( − Δ ) y s 1 u + u − u 2 s 1 − 1 = κ α h ( x , y ) u α − 1 v β in  R 2 , − ∂ x x v + ( − Δ ) y s 2 v + v − v 2 s 2 − 1 = κ β h ( x , y ) u α v β − 1 in  R 2 , u , v ⩾ 0 in  R 2 , where s 1 , s 2 ∈ ( 0 , 1 ) , α, β > 1, α + β ⩽ min { 2 s 1 , 2 s 2 }, and 2 s i = 2 ( 1 + s i ) 1 − s i , i = 1 , 2. The existence of a ground state solution entirely depends on the behaviour of the parameter κ > 0 and on the function h. In this article, we prove that a ground state solution exists in the subcritical case if κ is large enough and h satisfies (H). Further, if κ becomes very small, then there is no solution to our system. The study of the critical case, i.e., s 1 = s 2 = s, α + β = 2 s , is more complex, and the solution exists only for large κ and radial h satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on h.