DOI: 10.68381/jca32059 ISSN: 0944-6532
Schrödinger-Poisson System Involving Potential Vanishing at Infinity and Unbounded Below
Genivaldo P. Corrêa, Gelson C. G. dos Santos
This article concerns the following class of system
\left\{ \begin{array}{lr} -\Delta u +V(x)u+\ell(x)\phi u = f(u) + \lambda|u|^{q-2}u & \text{in } \mathbb{R}^3,\\[2mm] -\Delta \phi = \ell(x)u^{2} & \text{in } \mathbb{R}^3,\\[2mm] u,\phi\in D^{1,2}(\mathbb{R}^3), \ u,\phi\geq0 & \text{in } \mathbb{R}^3, \end{array} \right.
{
−
Δ
u
+
V
(
x
)
u
+
ℓ
(
x
)
ϕ
u
=
f
(
u
)
+
λ
∣
u
∣
q
−
2
u
in
R
3
,
−
Δ
ϕ
=
ℓ
(
x
)
u
2
in
R
3
,
u
,
ϕ
∈
D
1
,
2
(
R
3
)
,
u
,
ϕ
≥
0
in
R
3
,
where
\lambda\geq0
λ
≥
0
and
q\geq2^*=6
q
≥
2
∗
=
6
is the critical Sobolev exponent in dimension 3, the nonlinearity
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
is superlinear and has subcritical growth,
V,\ell: \mathbb{R}^3\rightarrow \mathbb{R}
V
,
ℓ
:
R
3
→
R
are measurable functions with
\ell\in L^2(\mathbb{R}^3)
ℓ
∈
L
2
(
R
3
)
, the potential
V
V
can change sign in
\mathbb{R}^3
R
3
and vanish at infinity, that is,
V (x) \rightarrow 0
V
(
x
)
→
0
as
|x|\rightarrow\infty
∣
x
∣
→
∞
. Our approach is based on variational method combined with Benci-Fortunato's reduction argument [ Topol. Methods Nonlinear Anal. 11 (1998) 283–293], Del Pino-Felmer's penalization technique [ Calc. Var. Partial Diff. Equations 4 (1996) 121–137] and
L^\infty
L
∞
-estimate.