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EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR NONLOCAL -LAPLACIAN PROBLEM

G. A. Afrouzi and M. Mirzapour
Taiwanese Journal of Mathematics
Vol. 18, No. 1 (February 2014), pp. 219-236
Stable URL: http://www.jstor.org/stable/taiwjmath.18.1.219
Page Count: 18
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR NONLOCAL -LAPLACIAN PROBLEM
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Abstract

Abstract In this paper, we study the nonlocal anisotropic p→(x)-Laplacian problem of the following form −∑i=1NMi(∫Ω|∂xiu|pi(x)pi(x)dx)∂xi(|∂xiu|pi(x)−2∂xiu)=f(x,u) in Ω,u=0 on ∂Ω. By means of a direct variational approach and the theory of the anisotropic variable exponent Sobolev space, we obtain the existence and multiplicity of weak energy solutions. Moreover, we get much better results with f in a special form. 2010 Mathematics Subject Classification: 35J62, 35J70, 46E35. Key words and phrases: Anisotropic Sobolev spaces, Variable exponent, Mountain pass theorem, Fountain theorem, Dual Fountain theorem.

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