Affiliation:
1. Global Graduate School, Anyang University in Korea , Gyeonggi-do 430714 , South Korea
Abstract
Abstract
In this article, we investigate the following Schrödinger equation:
−
Δ
u
−
μ
∣
x
∣
2
u
=
g
(
u
)
in
R
N
,
-\Delta u-\frac{\mu }{{| x| }^{2}}u=g\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},
where
N
≥
3
N\ge 3
,
μ
∣
x
∣
2
\frac{\mu }{{| x| }^{2}}
is called the Hardy potential and
g
g
satisfies Berestycki-Lions conditions. If
0
<
μ
<
(
N
−
2
)
2
4
0\lt \mu \lt \frac{{\left(N-2)}^{2}}{4}
, we will take symmetric mountain pass approaches to prove the existence of infinitely many solutions of this problem.
Reference26 articles.
1. H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, II, Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), 347–375.
2. H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I, Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–345.
3. J. Hirata, N. Ikoma, and K. Tanaka, Nonlinear scalar field equations in RN: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), 253–276.
4. A. Azzollini and A. Pomponio, On the Schrödinger equation in RN under the effect of a general nonlinear term, Indiana Univ. Math. J. 58 (2009), 1361–1378.
5. L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on RN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.