Author:
Zhang Weiqiang,Zuo Jiabin,Rădulescu Vicenţiu D.
Abstract
AbstractThe present paper is devoted to the study of the following double-phase equation $$\begin{aligned} -\text {div}(|\nabla u|^{p-2}\nabla u+\mu _{\varepsilon }(x)|\nabla u|^{q-2}\nabla u)+V_{\varepsilon }(x)(|u|^{p-2}u+\mu _{\varepsilon }(x)|u|^{q-2}u)=f(u)\quad \text{ in }\quad \mathbb {R}^{N}, \end{aligned}$$
-
div
(
|
∇
u
|
p
-
2
∇
u
+
μ
ε
(
x
)
|
∇
u
|
q
-
2
∇
u
)
+
V
ε
(
x
)
(
|
u
|
p
-
2
u
+
μ
ε
(
x
)
|
u
|
q
-
2
u
)
=
f
(
u
)
in
R
N
,
where $$N\ge 2$$
N
≥
2
, $$1<p<q<N$$
1
<
p
<
q
<
N
, $$q<p^{*}$$
q
<
p
∗
with $$p^{*}=\frac{Np}{N-p}$$
p
∗
=
Np
N
-
p
, $$\mu :\mathbb {R}^{N}\rightarrow \mathbb {R}$$
μ
:
R
N
→
R
is a continuous non-negative function, $$\mu _{\varepsilon }(x)=\mu (\varepsilon x)$$
μ
ε
(
x
)
=
μ
(
ε
x
)
, $$V:\mathbb {R}^{N}\rightarrow \mathbb {R}$$
V
:
R
N
→
R
is a positive potential satisfying a local minimum condition, $$V_{{{\,\mathrm{\varepsilon }\,}}}(x)=V({{\,\mathrm{\varepsilon }\,}}x)$$
V
ε
(
x
)
=
V
(
ε
x
)
, and the nonlinearity $$f:\mathbb {R}\rightarrow \mathbb {R}$$
f
:
R
→
R
is a continuous function with subcritical growth. Under natural assumptions on $$\mu $$
μ
, V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.
Funder
Basic and Applied Basic Research Foundation of Guangdong Province
Ministry of Education and Research, Romania
Publisher
Springer Science and Business Media LLC