Existence of solutions to Kirchhoff type equations involving the nonlocal p₁& … &pm fractional Laplacian with critical Sobolev-Hardy exponent

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal p1&…&pm fractional Laplacian with critical Sobolev-Hardy exponent M1∫R2N|u(x)−u(y)|p1|x−y|N+p1s1dxdy(−Δ)p1s1u+⋯+Mm∫R2N|u(x)−u(y)|pm|x−y|N+pmsmdxdy(−Δ)pmsmu=f(x,u)+γ|u|ps∗(α)−2u|x|α+β|u|q−2u|x|αin Ω,u=0in RN∖Ω, where 0<sm<⋯<s1=s<1, 1<pm≤⋯≤p1=p<Ns, m≥1, β,γ are nonnegative constants and ps∗(α)=p(N−α)N−sp≤ps∗(0) is called the critical Sobolev-Hardy exponent, 1<q<p, 0≤α<ps. Here (−Δ)rs, with r∈{p1,…,pm} is the fractional r-Laplace operator. Ω is an open bounded subset of RN with smooth boundary and 0∈Ω. M1,…,Mm are continuous functions and f is a Carathéodory function which does not satisfy the Ambrosetti-Rabinowitz condition. By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem. Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when the γ=0. We also study the existence of two nontrivial solutions for Kirchhoff type equation involving the fractional p-Laplacian via Morse theory. Finally, equation we consider the case N=ps, and study a degenerate Kirchhoff involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.