Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2018
Some Remarks on a Class of p(x)-Laplacian
Robin Eigenvalue Problems
Nguyen Thanh Chung
Abstract. We consider the p(x)-Laplacian Robin eigenvalue problem
u = λV (x)|u|
u, x ∈ Ω,
u =0,x∈ ∂Ω,
where Ω is a bounded domain in R
with smooth boundary ∂Ω, N ≥ 2,
is the outer normal derivative of u with respect to ∂Ω, p, q ∈ C
p(x) ≤ p
p(x) <N, β ∈ L
β(x) > 0, and λ>0 is a parameter. Under some suit-
able conditions on the functions q and V , we establish the existence
of a continuous family of eigenvalues in a neighborhood of the origin
using variational methods. The main results of this paper improve and
generalize the previous ones introduced in Deng (J Math Anal Appl
360:548–560, 2009), Keﬁ (Zeitschrift f¨ur Analysis und ihre Anwendun-
gen (ZAA) 37:25–38, 2018).
Mathematics Subject Classiﬁcation. 35J70, 35J60, 35D05.
Keywords. p(x)-Laplacian, eigenvalue problems, Robin boundary con-
ditions, variational methods.
In recent years, attention has been increasingly paid to the study of diﬀeren-
tial and partial diﬀerential equations involving variable exponent conditions.
The interest in studying such problems was stimulated by their applications in
electrorheological ﬂuids, image processing or theory of elasticity, see . The
p(x)-Laplacian operator Δ
(.), in which p(.) is a continuous function pos-
sesses more complicated properties than the p-Laplacian operator, mainly due
to the fact that it is not homogeneous. This leads to the fact that the study
of various mathematical problems with variable exponents is very interest-
ing and raises many diﬃcult mathematical problems, see . p(x)-Laplacian
problems with Dirichlet, Neumann, and Steklov boundary conditions and