Affiliation:
1. School of Mathematics and Computation Science, Huaihua University , Huaihua , Hunan 418008 , P. R. China
2. Key Laboratory of Intelligent Control Technology for Wuling-Mountain Ecological Agriculture in Hunan Province , Huaihua , Hunan 418008 , P. R. China
3. School of Mathematics and Statistics, Central South University , Changsha , Hunan 410083 , P. R. China
Abstract
Abstract
In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form
(0.1)
−
Δ
u
+
V
(
x
)
u
=
f
(
x
,
u
)
+
λ
a
(
x
)
∣
u
∣
q
−
2
u
,
x
∈
R
2
,
-\Delta u+V\left(x)u=f\left(x,u)+\lambda a\left(x)| u{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2},
where
λ
>
0
\lambda \gt 0
,
q
∈
(
1
,
2
)
q\in \left(1,2)
,
a
∈
L
2
/
(
2
−
q
)
(
R
2
)
a\in {L}^{2\text{/}\left(2-q)}\left({{\mathbb{R}}}^{2})
,
V
(
x
)
V\left(x)
, and
f
(
x
,
t
)
f\left(x,t)
are 1-periodic with respect to
x
x
, and
f
(
x
,
t
)
f\left(x,t)
has critical exponential growth at
t
=
∞
t=\infty
. By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e.
λ
>
0
\lambda \gt 0
) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.
Cited by
4 articles.
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