A note on the combination between local and nonlocal p-Laplacian operators
Bingzhong Hu, Yang Yang · Complex Variables and Elliptic Equations · 2019
In this article, we give some results on a combination between local and nonlocal p-Laplacian operators. On the one hand, we investigate the Dancer-Fučík spectrum which is defined as the set of all points (a,b)∈R2 such that −Δpu+(−Δ)psu=b(u+)p−1−a(u−)p−1,in Ω;u=0,in RN∖Ω, has a nontrivial solution u. Here Δpu is the standard local p-Laplacian operator, (−Δ)psu is the fractional p-Laplacian, which is a nonlocal operator and Ω is a bounded domain in RN with Lipschitz boundary. Via an appropriate minimax scheme, we construct an unbounded sequence of decreasing curves in the spectrum. On the other hand, we use an abstract critical point theorem to prove a bifurcation and multiplicity result for the following critical problem −Δpu+(−Δ)psu=λ|u|p−2u+|u|p∗−2u,in Ω;u=0,in RN∖Ω, where p∗=Np/(N−p) is the critical Sobolev exponent. This extends the result for the nonlocal quasilinear case.