Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (−Δ+m2−m)u=|u|p−1uinΩ,with the Dirichlet boundary condition u=0 on ∂Ω. Here, p∈(1,∞) and the operator (−Δ+m2−m) is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of p, m and Ω. Precisely, we show that (i) if p is not H1 subcritical (p≥n+2n−2) and Ω is star-shaped, the equation has no nontrivial solution for all m>0; (ii) if p is not H1∕2 supercritical (10 and any bounded domain Ω; (iii) finally, in the intermediate range (n+1n−1