Nonlinear P (x) Contains A Critical Indicator Of The Existence Of -Laplacian Equations

In this paper, we study the following p(x)-Laplacian problem: where Ω is a smooth bounded domain with smooth boundary in Rn. p(x), q(x) are two continuous functions defined is the critical exponent according to the Sobolev embedding, where denote p(x)(?) q(x) the fact that i is the outer normal derivative. On the exponent q(x) we assume that is the critical exponent in the sense that {q(x)=p*(x)}≠(?). Under proper conditions on f and g, applying "Mountain Pass Theorem", we prove the existence of solutions to this p(x)-Laplace operator with nonlinear boundary condition.One of the main motivations is considering a particular and relevant case asso-ciated with problem (1.1) given by where r(x) is a continuous functions defined c proper conditions on f, applying " Dual Fountain Theorem ", we prove multiplicity solutions to this p(x)-Laplace operator with nonlinear boundary condition.