Nonlocal Hénon equation with nonlinearities involving Sobolev critical and supercritical growth

Abstract

In this paper, we study the following class of fractional Hénon problems involving exponents critical or supercritical\begin{equation*} \begin{cases}\displaystyle (-\Delta)^s u = \lambda |x|^{\mu} u +|x|^{\alpha}|u|^{ (p_{\alpha,s}^*+ \varepsilon) -1} u & \mbox{in } \Omega, \\ u = 0 & \mbox{in } \mathbb{R}^N\setminus\Omega, \end{cases}\end{equation*}where $p_{\alpha,s}^*= \frac{N + 2 \alpha + 2s}{N-2s}$ is the critical exponent for a nonlinearity with Hénon weight in nonlocal context, $\varepsilon \geq 0$, $\Omega$ is either a ball or an annulus in $\mathbb R^N$, $s \in (0,1)$ and $\mu, \alpha > -2s$.We used the Emden--Fowler transformation to make the one-dimensional reduction of problems and under appropriate hypotheses on the constant $\lambda$, we prove the existence of at least one non-trivial radial solution for these problems using the concentration compactness principle or Linking Theorem.