# Existence Solution for Class of p-Laplacian Equations

Volume 4, Issue 1, pp 53--59 Publication Date: January 18, 2012
• 527 Downloads
• 599 Views

### Authors

Malihe Bagheri - Department of mathematic, Golestan Institute of Higher Education, Golestan Province, Iran Mahnaz Bagheri - Department of Mathematice, Islamic Azad University, behshar Branch, Iran

### Abstract

We study existence of positive solution of the equation $-\Delta_pu=\lambda|u|^{p-2}u+f(x,u)$ with zero Dirichlet boundary conditions in bounded domain $\Omega\in \mathbb{R}^n$ where $\Delta_p$ denotes the p-laplacian operator defined by $-\Delta_pz=div(|\nabla z|^{p-2}\nabla z); p,\lambda\in \mathbb{R}$ and $p>1$.Our main result establishes the existence of weak solution.

### Keywords

• p-laplacian
• weak solution
• homogenous.

•  35J66
•  35J92
•  35A01

### References

• [1] P. Drábek, Y. X. Huang, Bifurcation problems for the p-Laplacian in $\mathbb{R}^n$, Trans. Amer. Math. Soc., Vol. 349, 171--188 (1997)

• [2] P. Drábek, Y. X. Huang, Multiple positive solutions of quasilinear elliptic equations in $\mathbb{R}^n$, Nonlinear Analysis, Vol. 37, 457--466 (1999)

• [3] S. I. Pohozaev, About one approach to the nonlinear equations, Dokl. Akad. Nauk. (RAC USSR), 241 (1979), 1327--1331

• [4] S. I. Pohozaev, On fibering method for the solution of nonlinear boundary value problems, Trudy Matematicheskogo Instituta imeni VA Steklova, 192 (1990), 140--163

• [5] A. Anane, Simplicité et isolation de la premiere valeur propre du p-laplacien avec poids, Comptes rendus de l'Académie des sciences, 305 (1987), 725--728

• [6] G. Barles , Remarks on uniqueness results of the first eigenvalue of the p-Laplacian, In: Annales de la Faculté des sciences de Toulouse: Mathématiques, 9 (1988), 65---75