We shall prove a multiplicity result for a non-local problem with a super-critical nonlinearity of the form, (−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.

Nonlinear elliptic problems, Variational methods
Nonlinear Analysis, Theory, Methods and Applications
School of Mathematics and Statistics

Kouhestani, N. (Najmeh), Mahyar, H. (Hakimeh), & Momeni, A. (2019). Multiplicity results for a non-local problem with concave and convex nonlinearities. Nonlinear Analysis, Theory, Methods and Applications, 182, 263–279. doi:10.1016/j.na.2018.12.006