Affiliations: [a] Department of Mathematics, California State University at Los Angeles, Los Angeles, CA 90032, USA | [b] Department of Mathematics, Indian Institute of Technology Jodhpur, Rajasthan 342030, India | [c] Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Abstract: 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−1vβin R2,−∂xxv+(−Δ)ys2v+v−v2s2−1=κβh(x,y)uαvβ−1in R2,u,v⩾0in R2,
where s1,s2∈(0,1),α, β>1, α+β⩽min{2s1,2s2}, and 2si=2(1+si)1−si, 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., s1=s2=s, α+β=2s, 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.
Keywords: Mixed Schrödinger operator, system of PDEs in plane, variational methods, concentration-compactness, non-existence
