Semilinear elliptic equations with a relativistic Laplacian on bounded domains

We study semilinear elliptic equations involving a relativistic Laplacian on bounded domains \begin{equation} (\sqrt{-\Delta + m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~\Omega,\end{equation} with the Dirichlet boundary condition $u=0$ on $\partial \Omega$. Here $p \in (1,\infty)$ and the operator $(\sqrt{-\Delta + m^2} - m)$ is defined in terms of spectral decomposition. In the first part, we obtain existence and nonexistence result. Then, the second part is devoted to study asymptotic behaviors of solutions when the mass parameter $m$ of equations goes to $0^{+}$ or diverges to infinity.