Nonlocal Hénon type problem with nonlinearities involving slightly subcritical growth

Abstract

In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem(0.1)$\{\begin{array}{cc}{(-\Delta )}^{s}u=\beta \left(x\right)u^{p-\epsilon }& \text{in }\Omega ,\\ u=0& \text{in }{R}^n﹨\Omega ,\end{array}$ where $s\in (0,1),n>2s,p:=\frac{n+2s}{n-2s}$ is the Sobolev critical exponent, $\Omega \subset R^n$ is a smooth bounded domain with Lipschitz boundary, ${(-\Delta )}^s$ is the fractional Laplace operator and $\beta \in C^2\left(\bar{\Omega }\right)$ is a bounded positive continuous function. We assume that there exists a nondegenerate critical point ${\xi }^{⁎}\in \partial \Omega$ of the restriction of β to the boundary ∂Ω such that$\nabla \left(\beta {\left({\xi }^{⁎}\right)}^{-\frac{n-2s}{2s}}\right)\cdot \eta \left({\xi }^{⁎}\right)>0.$ Given any integer $k\ge 1$, we show that for $\epsilon >0$ small enough, problem (0.1) has a positive solution, which is a sum of k bubbles which concentrate at ${\xi }^{⁎}$ as ε tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point ${\xi }_{⁎}$. This work can be seen as a nonlocal analogue of the result by Dávila, Faya and Mahmoudi, see [28].