Multiplicity results for a non-local problem with concave and convex nonlinearities

Abstract

We shall prove a multiplicity result for a non-local problem with a super-critical nonlinearity of the form, (1)(−Δ)su=u|u|p−2+μu|u|q−2,inΩ,u=0,onRn∖Ω,where Ω⊂ℝn is a bounded domain, 0<s<1 and 1<q<2<p. As a consequence of our results, for each p>2, there exists μ∗>0 such that for each μ∈(0,μ∗) problem (1) has a sequence of solutions with a negative energy. This result is new for super-critical values of p. We shall also explore the existence of symmetric solutions for symmetric domains. Our methodology is based on a variational principle established by one of the authors that allows one to deal with super-critical problems when the standard Euler–Lagrange functional is restricted to certain convex sets.