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, (root-Delta + m(2) - m)u = |u|(p-1) u in Omega, with the Dirichlet boundary condition u = 0 on partial derivative Omega. Here, p ? (l,infinity) and the operator (root-Delta + m(2) - 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 Omega. Precisely, we show that (i) if p is not H-1 subcritical (P >= n+2/n-2) and Omega is star-shaped, the equation has no nontrivial solution for all m > 0; (ii) if p is not H-1/2 supercritical (1 < p <= n+1/n-1), then there exists a least energy solution for all m > 0 and any bounded domain Omega; (iii) finally, in the intermediate range (n+1/n-1 < p < n+2/n-2), the problem has a nontrivial solution, provided that m is sufficiently large and the problem - Delta u - |u|(p-1) u in Omega, u = 0 on partial derivative Omega (0.1) admits a non-degenerate nontrivial solution, for example, when Omega is a ball or an annulus. (C) 2018 Elsevier Ltd. All rights reserved.