## Differential and Integral Equations

- Differential Integral Equations
- Volume 25, Number 5/6 (2012), 567-578.

### Existence for critical Hénon-type equations

Wei Long and Jianfu Yang

#### Abstract

This paper is concerned with the existence of a nontrivial solution for \begin{equation} -\Delta u=\lambda u+|x|^{\alpha}|u|^{2^*-2}u,\,\,\, {\rm in}\ \ \Omega, \quad u=0,\,\,\, {\rm on} \ \ \partial \Omega, \tag*{(0.1)} \end{equation} where $\lambda > 0$ and $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain. Let $\lambda_k,$ $ k =1,2,\ldots$, be eigenvalues of the operator $-\Delta$; we show for $\lambda_k < \lambda < \lambda_{k+1}$ that problem (0.1) possesses at least a solution and each $\lambda_k$ is a bifurcation point.

#### Article information

**Source**

Differential Integral Equations, Volume 25, Number 5/6 (2012), 567-578.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012680

**Mathematical Reviews number (MathSciNet)**

MR2951742

**Zentralblatt MATH identifier**

1265.35061

**Subjects**

Primary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35J60: Nonlinear elliptic equations 35J61: Semilinear elliptic equations

#### Citation

Long, Wei; Yang, Jianfu. Existence for critical Hénon-type equations. Differential Integral Equations 25 (2012), no. 5/6, 567--578. https://projecteuclid.org/euclid.die/1356012680