Affiliation:
1. Department of Mathematics University of Craiova 200585 Craiova Romania
Abstract
Abstract
Let Ω ⊂ R
N
(N ≥ 2) be a simply connected bounded domain, containing the origin, with C
2 boundary denoted by ∂Ω. Denote by
Ω
e
x
t
:=
R
N
∖
Ω
¯
$\Omega^{\mathrm{ext}}:=\mathbb{R}^{N} \backslash \bar{\Omega}$
the exterior of Ω. We consider the perturbed eigenvalue problem
−
Δ
p
u
−
Δ
q
u
=
μ
K
(
x
)
|
u
|
p
−
2
u
for
x
∈
Ω
ext
u
(
x
)
=
0
for
x
∈
∂
Ω
u
(
x
)
→
0
,
as
|
x
|
→
∞
,
$$\left\{\begin{array}{lcl}-\Delta_{p} u-\Delta_{q} u=\mu K(x)|u|^{p-2} u & \text { for } & x \in \Omega^{\text {ext }} \\u(x)=0 & \text { for } & x \in \partial \Omega \\u(x) \rightarrow 0, & \text { as } & |x| \rightarrow \infty,\end{array}\right.$$
where p, q ∈ (1,N),
p
≠
q
$p \neq q$
and K is a positive weight function defined on Ωext having the property that K ∈ L∞
(Ωext) ∩ LN/p
(Ωext) . We show that the set of parameters μ for which the above eigenvalue problem possesses nontrivial weak solutions is exactly an unbounded open interval.