Semilinear nonlocal elliptic equations with critical and supercritical exponents

TL;DR

This paper investigates semilinear nonlocal elliptic equations involving the fractional Laplacian with critical and supercritical exponents, establishing regularity, decay estimates, nonexistence, symmetry, and existence results.

Contribution

It provides new regularity results, decay estimates, and existence/nonexistence theorems for solutions of fractional elliptic equations with critical and supercritical nonlinearities.

Findings

Solutions are regular and classical when they exist.

No solutions exist at the critical exponent due to Pohozaev identity.

Solutions exist for supercritical exponents in the whole space and bounded domains.

Abstract

We study the problem \begin{eqnarray*} (-\Delta)^s u &=& u^p - u^q \quad\text{in }\quad \mathbb{R}^N, u &\in& \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), u&>0& \quad\text{in}\quad\mathbb{R}^N, \end{eqnarray*} where $s\in(0,1)$ is a fixed parameter, $(-\Delta)^s$ is the fractional laplacian in $\mathbb{R}^N$, $q>p\geq \frac{N+2s}{N-2s}$ and $N>2s$. For every $s\in(0,1)$, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all $s\in(0,1)$. Using those decay estimates, we prove Pohozaev type identity in $\mathbb{R}^N$ and show that the above problem does not have any solution when $p=\frac{N+2s}{N-2s}$. We also discuss radial symmetry and decreasing property of the solution and prove that when $p>\frac{N+2s}{N-2s}$, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every $p\geq \frac{N+2s}{N-2s}$ and every solution is a classical solution.