Affiliations: [a] Faculdade de Matemática-ICEN, Universidade Federal do Pará, Belém, Brazil | [b] Dipartimento PAU, Università ‘Mediterranea’ di Reggio Calabria, Reggio Calabria, Italy | [c] Dipartimento di Scienze di Base e Fondamenti, Università degli Studi di Urbino ‘Carlo Bo’, Urbino (Pesaro e Urbino), Italy
Correspondence:
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Corresponding author: Raffaella Servadei, Dipartimento di Scienze di Base e Fondamenti, Università degli Studi di Urbino ‘Carlo Bo’, Piazza della Repubblica 13, 61029 Urbino (Pesaro e Urbino), Italy. E-mail: raffaella.servadei@uniurb.it.
Abstract: In this paper we study a highly nonlocal problem involving a fractional operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Precisely, we consider the following nonlocal problem
−M(∥u∥X02)LKu=f(x,u)(∫Ω(∫0u(x)f(x,τ)dτ)dx)rin Ω,u=0in Rn∖Ω,
where r⩾0, s∈(0,1), Ω is an open bounded subset of Rn, n>2s, with continuous boundary, f:Ω‾×R→R and M:[0,+∞)→[0,+∞) are functions verifying suitable conditions and LK is the integrodifferential operator defined as follows
LKu(x):=∫Rn(u(x+y)+u(x−y)−2u(x))K(y)dy,x∈Rn,
where the kernel K satisfies some natural conditions. By working in the fractional Sobolev space X0, which encodes Dirichlet homogeneous boundary conditions, and exploiting the genus theory introduced by Krasnoselskii, we derive the existence of infinitely many weak solutions for this problem.
Keywords: Kirchhoff-type equations, fractional Laplacian, nonlocal problems, variational methods, critical point theory, Krasnoselskii’s genus
