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Existence and multiplicity of solutions for a p ( x ) -Kirchhoff type problem via variational techniques

A. Mokhtari, Toufik Moussaoui, D. O’Regan (2015)

Archivum Mathematicum

This paper discusses the existence and multiplicity of solutions for a class of p ( x ) -Kirchhoff type problems with Dirichlet boundary data of the following form - a + b Ω 1 p ( x ) | u | p ( x ) d x div ( | u | p ( x ) - 2 u ) = f ( x , u ) , i n Ω u = 0 o n Ω , where Ω is a smooth open subset of N and p C ( Ω ¯ ) with N < p - = inf x Ω p ( x ) p + = sup x Ω p ( x ) < + , a , b are positive constants and f : Ω ¯ × is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

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