Mediterr. J. Math. (2017) 14:185
published online August 14, 2017
Springer International Publishing AG 2017
Existence of Sign-Changing Solutions
for a Nonlocal Problem of p-Kirchhoﬀ Type
S. H. Rasouli, H. Fani and S. Khademloo
Abstract. This work is devoted to study the existence of sign-changing
solutions to nonlocal problems involving the p-Laplacian. Our approach
is based on the variational method and quantitative deformation lemma.
Mathematics Subject Classiﬁcation. 35J50, 35J60, 35J65.
Keywords. p-Kirchhoﬀ-type equations, sign-changing solutions,
deformation lemma, degree theory.
The purpose of this paper is to investigate existence of sign-changing solutions
for the following class of nonlocal Dirichlet problems:
a + b
u = f(u),x∈ Ω,
u =0,x∈ ∂Ω,
where Ω is a bounded domain in R
, N =1, 2, 3, with smooth boundary,
a, b are positive constants, Δ
denotes the p-Laplacian operator deﬁned by
∇u), p>1, and f ∈ C
Problem (1) is a general version of a model presented by Kirchhoﬀ .
More precisely, Kirchhoﬀ introduced a model
where ρ, ρ
,h,E,L are constants, which extends the classical D’Alembert’s
wave equation by considering the eﬀects of the changes in the length of the
strings during the vibrations. The problem
a + b
Δu = f(x, u)inΩ
u =0 on∂Ω (3)