A non-local scalar field problem: existence, multiplicity and asymptotic behavior

Abstract

The main purpose of this work is to study a non-local scalar field equation in bounded domains. More precisely we consider the semi-linear fractional elliptic problem: 0 \\hbox{ in } \\Omega, \\hbox{ and } u= 0 \\hbox{ in }\\mathbb{R}^N\\setminus \\Omega,\\] ]]> ( − Δ ) s u = λ u p − 1 − u m − 1 in Ω , u > 0 in Ω , and u = 0 in R N ∖ Ω , where Ω is a regular bounded domain of R N , m>p>2 and 0 $ ]]> λ > 0 . We discuss the existence, non-existence, and multiplicity of solutions, for the largest possible range of the parameters λ , p , m . Optimal results are obtained in the sub-critical and critical regimes, and partial results are obtained in the super-critical case. Furthermore, a careful asymptotic analysis when m tends to ∞ will be provided and we identify the limiting profiles as solutions to a free boundary type problem with a nonlinear right-hand side.