Semilinear nonlocal elliptic equations with critical and supercritical exponents

Abstract

We study the problem $\left\{ \begin{align} &{{(-\Delta lta )}^{s}}u={{u}^{p}}-{{u}^{q}}\ \text{in}\ \text{ }{{\mathbb{R}}^{N}}, \\ &u\in {{{\dot{H}}}^{s}}({{\mathbb{R}}^{N}})\cap {{L}^{q+1}}({{\mathbb{R}}^{N}}), \\ &u>0\ \ \text{in}\ \ {{\mathbb{R}}^{N}}, \\ \end{align} \right.$ where $s∈(0,1)$ is a fixed parameter, $(-Δ)^s$ is the fractional Laplacian in $\mathbb{R}^N$, $q>p≥q \frac{N+2s}{N-2s}$ and $N>2s$. For every $s∈(0,1)$, 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 $s∈(0,1)$. Using those decay estimates, we prove Pohozaev type identity in ${{\mathbb{R}}^{N}}$ and we 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 $p≥q \frac{N+2s}{N-2s}$ and every solution is a classical solution.