Abstract
In this article, we consider a Neumann boundary value problem driven by p ( x )-Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: â Î p ( x ) l u + ÎŽ | u | ζ ( x ) â 2 u = ÎŒ g ( x , u ) + λ f ( x , u , â u ) in Ω , â u â η = 0 on â Ω , where Î p ( x ) l u is the p ( x )-Laplacian-like operator, Ω is a smooth bounded domain in R N , ÎŽ, ÎŒ and λ are three real parameters, p ( x ) , ζ ( x ) â C + ( Ω ⟠), η is the outer unit normal to â Ω and g, f are CarathĂ©odory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.
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