A critical fractional equation with concave-convex power nonlinearities

Abstract

In this work we study the following fractional critical problem $$ (P_λ)=\left\{\begin{array}{ll} (-Δ)^s u=λu^{q} + u^{2^*_{s}-1}, \quad u{>}0 & \mbox{in} Ω\\ u=0 & \mbox{in} \RR^n\setminus Ω\,, \end{array}\right. $$ where $\Omega \subset {\mathbb{R}}^n$ is a regular bounded domain, $\lambda >0$, $0<s<1$ and $n>2s$. Here $(-\Delta {)}^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$ -(-Δ)^s u(x)={\rm P. V.} \int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \RR^n. $$ Our main results show the existence and multiplicity of solutions to problem $(P_{\lambda })$ for different values of $\lambda$. The dependency on this parameter changes according to whether we consider the concave power case ($0<q<1$) or the convex power case ($1<q<2_s^{\ast }-1$). These two cases will be treated separately.

29 pages

References in corpus (2)

• From the long jump random walk to the fractional Laplacian
• Normal and Anomalous Diffusion: A Tutorial

Cited by in corpus (1)

• Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity