Existence results for fractional \(p\)-Laplacian problems via Morse theory. (English) Zbl 06567151

Iannizzotto, Antonio; Liu, Shibo; Perera, Kanishka; Squassina, Marco

Adv. Calc. Var. 9, No. 2, 101-125 (2016).

Summary: We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.

MSC:

35P15	Estimates of eigenvalues in context of PDEs
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35R11	Fractional partial differential equations

Keywords:

fractional \(p\)-Laplacian problems; Morse theory; existence and multiplicity of weak solutions; regularity of solutions