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

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type{−∂xxu+(−Δ)ys1u+u−u2s1−1=καh(x,y)uα−1vβinR2,−∂xxv+(−Δ)ys2v+v−v2s2−1=κβh(x,y)uαvβ−1inR2,u,v⩾0inR2, where s1,s2∈(0,1),α[jls-end-space/], β>1[jls-end-space/], α+β⩽min{2s1,2s2}[jls-end-space/], and 2si=2(1+si)1−si[jls-end-space/], i=1,2[jls-end-space/]. 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., s1=s2=s[jls-end-space/], α+β=2s[jls-end-space/], 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. © 2025 Elsevier B.V., All rights reserved.