Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2018
Existence and Multiplicity of Solutions
for Fractional Elliptic Problems with
Abstract. We consider the following fractional elliptic problem:
u = f(u)H(u − μ)inΩ,
u =0 on IR
,s ∈ (0, 1) is the fractional Laplacian, Ω is a bounded
domain of IR
, (n ≥ 2s) with smooth boundary ∂Ω,His the Heaviside
step function, f is a given function and μ is a positive real parameter.
The problem (P ) can be considered as simpliﬁed version of some models
arising in diﬀerent contexts. We employ variational techniques to study
the existence and multiplicity of positive solutions of problem (P ).
Mathematics Subject Classiﬁcations. 34R35, 35J25, 35B38.
Keywords. Fractional Laplacian, discontinuous nonlinearity, free bound-
ary, mountain pass theorem.
The problems involving the nonlocal operators of elliptic type arise in several
contexts for their interesting theoretical structure and in contrast of concrete
applications in many things such as anomalous diﬀusion in plasmas, ﬂames
propagation, geophysical ﬂuid dynamics and American options in ﬁnances.
See [12, 18], as well as the references therein.
This work is motivated by the large interest in the current literature
using variational techniques combining the nonlocal analysis and also by the
fact that in the nature, the nonlinearities can be discontinuous. Also, one
of the reasons to study the partial diﬀerential equations with discontinuous
nonlinearities is due to many free boundary problems which can be reduced
to boundary value problems with a discontinuities in the second term. Let us
recall that in his 1989 seminal paper, Ambrosetti and Badiale provedfor
the ﬁrst time the existence of solutions of problem with discontinuous non-
linearity using the dual variational principle inspired by the work of Clarke
. In concrete, they considered the following problem: