# Existence results for a superlinear singular equation of caffarelli-kohn-nirenberg type

Existence results for a superlinear singular equation of caffarelli-kohn-nirenberg type
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In this paper, using the Mountain Pass Lemma and the LinkingArgument, we prove the existence of nontrivial weak solutions forthe Dirichlet problem for the superlinear equation of Caffarelli-Kohn- Nirenberg type in the case where the parameter L in 0, L2, L2 being the second positive eigenvalue of the quasilinear elliptic equation of Caffarelli-Kohn-Nirenberg type.

Tipo de documento: Artículo - Article

Palabras clave: singular equation, Caffarelli-Kohn-Nirenberg inequality, Mountain Pass Lemma, Linking Argument.

Source: http://www.bdigital.unal.edu.co

## Teaser

Boletı́n de Matemáticas
Nueva Serie, Volumen X No.

2 (2003), pp.

47–58
EXISTENCE RESULTS FOR A SUPERLINEAR SINGULAR
EQUATION OF CAFFARELLI-KOHN-NIRENBERG TYPE
BENJIN XUAN (*)
Abstract.

In this paper, using the Mountain Pass Lemma and the Linking Argument, we prove the existence of nontrivial weak solutions for
the Dirichlet problem for the superlinear equation of Caffarelli-KohnNirenberg type in the case where the parameter λ ∈ (0, λ2 ), λ2 being the
second positive eigenvalue of the quasilinear elliptic equation of CaffarelliKohn-Nirenberg type.
Key Words: singular equation, Caffarelli-Kohn-Nirenberg inequality,
Mountain Pass Lemma, Linking Argument.
2000 Mathematics Subject Classification: 35J60.
1.

Introduction.
In this paper, we investigate the existence of weak solutions for the following Dirichlet problem for the superlinear singular equation of Caffarelli-KohnNirenberg type:
(1.1)
(
−div (|x|−ap |Du|p−2 Du) = λ|x|−(a 1)p c |u|p−2 u |x|−bq f (u), in Ω
u = 0, on ∂Ω,
where Ω ⊂ Rn is an open bounded domain with C 1 boundary and 0 ∈ Ω, 1
np
∗
p n, 0 ≤ a n−p
p , a ≤ b ≤ a 1, q p (a, b) = n−dp , d = 1 a − b ∈ [0, 1],
and c 0.
(*) Benjin Xuan.

Department of Mathematics
University of Science and Technology of China
Universidad Nacional de Colombia
This work is supported by Grants 10071080 and 10101024 from the National Science Foundation of China.
e-mail:wenyuanxbj@yahoo.com.
47
48
BENJIN XUAN
For a = 0, c = p, many results of linking-type for critical points have been
obtained (e.g.

[1, 2, 6] for p = 2, [12] for p 6= 2 and [14] for the case with
indefinite weights).
The starting point of the variational approach to these problems with a ≥ 0 is
the following weighted Sobolev-Hardy inequality due to Caffarelli, Kohn and
Nirenberg [4], which is called the Caffarelli-Kohn-Nirenberg inequality.

Let
1 p n.

For all u ∈ C0∞ (Rn ), there is a constant Ca,b 0 such that
Z
Z
p-q
−bq
q
(1...