Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain. (August 2018) - Choi, Woocheol; Hong, Younghun; Seok, Jinmyoung
Abstract: We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, ( − Δ + m 2 − m ) u = | u | p − 1 u in Ω, with the Dirichlet boundary condition u = 0 on ∂ Ω . Here, p ∈ ( 1, ∞ ) and the operator ( − Δ + 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 Ω . Precisely, we show that ( i ) if p is not H 1 subcritical ( p ≥ n + 2 n − 2 ) and Ω is star-shaped, the equation has no nontrivial solution for all m > 0 ; ( i i ) 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 Ω ; ( i i i ) 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 (0.1) − Δ u = | u | p − 1 u in Ω, u = 0 on ∂ Ω admits a non-degenerate nontrivial solution, for example, when Ω is a ball or an annulus.