Abstract
Abstract
We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the $p(x)$
p
(
x
)
-Laplacian operator with Dirichlet boundary condition:
$$ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , $$
−
Δ
p
(
x
)
u
+
V
(
x
)
|
u
|
q
(
x
)
−
2
u
=
f
(
x
,
u
)
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a smooth bounded domain in $\mathbb{R}^{N}$
R
N
, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, $f(x,t)$
f
(
x
,
t
)
is a Carathéodory function satisfying some growth conditions. Depending on the assumptions, the solutions set may consist of a bounded infinite sequence of solutions or a unique one. Our technique is based on a symmetric version of the mountain pass theorem.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
Cited by
4 articles.
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