## Advances in Differential Equations

- Adv. Differential Equations
- Volume 9, Number 5-6 (2004), 481-508.

### Existence and multiplicity for perturbations of an equation involving a Hardy inequality and the critical Sobolev exponent in the whole of $\Bbb R^N$

B. Abdellaoui, V. Felli, and I. Peral

#### Abstract

In order to obtain solutions to the problem $$ \left\{ \begin{array}{c} -\Delta u=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1},\,\,x\in {{{\mathbb R}}^N}, \\ u>0 \hbox{ in }{{{\mathbb R}}^N}, \mbox{ and }u\in {{\mathcal D}^{1,2}}({{{\mathbb R}}^N}), \end{array} \right. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach a multiplicity of solutions, some hypotheses about the local behavior of $k$ close to the points of maximum are needed.

#### Article information

**Source**

Adv. Differential Equations, Volume 9, Number 5-6 (2004), 481-508.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867933

**Mathematical Reviews number (MathSciNet)**

MR2099969

**Zentralblatt MATH identifier**

1220.35041

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B33: Critical exponents 35D05 35J20: Variational methods for second-order elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47J30: Variational methods [See also 58Exx]

#### Citation

Abdellaoui, B.; Peral, I.; Felli, V. Existence and multiplicity for perturbations of an equation involving a Hardy inequality and the critical Sobolev exponent in the whole of $\Bbb R^N$. Adv. Differential Equations 9 (2004), no. 5-6, 481--508. https://projecteuclid.org/euclid.ade/1355867933