## Abstract and Applied Analysis

### A result on the bifurcation from the principal eigenvalue of the $A_p$-Laplacian

#### Abstract

We study the following bifurcation problem in any bounded domain $\Omega$ in $\mathbb{R}^N$: $\begin{cases}A_pu := -\sum^N_{i,j=1}\frac{\partial}{\partial x_i}[(\sum^N_{m,k=1}a_{mk}(x)\frac{\partial u}{\partial x_m}\frac{\partial u}{\partial x_k})^{\frac{p-2}{2}}a_{ij}(x)\frac{\partial u}{\partial x_j}]=\lambda g(x)|u|^{p-2}u + f(x,u,\lambda),u\in W_0^{1,p}(\Omega)\end{cases}$. We prove that the principal eigenvalue $\lambda_1$ of the eigenvalue problem $\begin{cases}A_pu =\lambda g(x)|u|^{p-2}u,u\in W_0^{1,p}(\Omega),\end{cases}$ is a bifurcation point of the problem mentioned above.

#### Article information

Source
Abstr. Appl. Anal., Volume 2, Number 3-4 (1997), 185-195.

Dates
First available in Project Euclid: 14 April 2003

https://projecteuclid.org/euclid.aaa/1050355232

Digital Object Identifier
doi:10.1155/S108533759700033X

Mathematical Reviews number (MathSciNet)
MR1704867

Zentralblatt MATH identifier
0933.35151

Subjects
Drábek, P.; Elkhalil, A.; Touzani, A. A result on the bifurcation from the principal eigenvalue of the $A_p$-Laplacian. Abstr. Appl. Anal. 2 (1997), no. 3-4, 185--195. doi:10.1155/S108533759700033X. https://projecteuclid.org/euclid.aaa/1050355232