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

TL;DR

This paper investigates the existence and nonexistence of solutions to semilinear equations involving the pseudo-relativistic operator on bounded domains, depending on parameters like p, m, and domain shape, revealing critical thresholds for solutions.

Contribution

It provides new results on solution existence for pseudo-relativistic equations, identifying critical exponents and conditions on domain and parameters that influence solvability.

Findings

No nontrivial solutions for supercritical p on star-shaped domains.

Existence of least energy solutions in subcritical and certain intermediate ranges.

Solutions depend on domain shape, parameter m, and critical exponents.

Abstract

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\sqrt{-\Delta + m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~\Omega, 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 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 ($p \geq \frac{n+2}{n-2}$) and $\Omega$ is star-shaped, the equation has no nontrivial solution for all $m > 0$; $(ii)$ if $p$ is not $H^{1/2}$ supercritical ($1 <p \leq \frac{n+1}{n-1}$), then there exists a least energy solution for all $m>0$ and any bounded domain $\Omega$; $(iii)$ finally, in the intermediate range ($\frac{n+1}{n-1}<p<\frac{n+2}{n-2}$), the problem has a nontrivial solution, provided that $m$ is sufficiently large and the problem -\Delta u = |u|^{p-1}u \quad \textrm{in}~\Omega, \qquad u =0\quad \textrm{on}~\partial \Omega admits a non-degenerate nontrivial solution, for example, when $\Omega$ is a ball or an annulus.