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March  2005, 4(1): 9-22. doi: 10.3934/cpaa.2005.4.9

On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian

L. Cherfils 1, and  Y. Il'yasov 2,

1. 

Université de La Rochelle, Laboratoire de Mathématiques et Applications, 17042 La Rochelle cedex, France
2. 

Bashkir State University, Ufa, Frunze 32, Russian Federation

Received  February 2004 Revised  September 2004 Published  December 2004

In this paper, we analyse a family of stationary nonlinear equations with $p\& q$- Laplacian $-\Delta_p u -\Delta_q u=\lambda c(x,u)$ which have a wide spectrum of applications in many areas of science. We introduce a new type of variational principles corresponding to this family of equations. Using this formalism, we exhibit intervals for the scalar parameter $\lambda$ where there exist positive solutions of the considered problems. Furthermore, we prove, in another interval, the nonexistence of nontrivial solutions. These results are different from those of existence and nonexistence for stationary equations with single Laplacian.
Keywords: existence of non-negative solution., $p\& q$-Laplacian, variational principles.
Mathematics Subject Classification: 35J70, 35J65, 47H1.
Citation: L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure & Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9
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