The P(x)-Laplacian Robin Problems And Steklov Eigenvalue Problem On A Bounded Smooth Domain

The purpose of this dissertation is to study the existence and multiplicity of solutions and eigenvalue problem of the following equations on a bounded smooth domain in R^N involving the p(x)-Laplacian of the typewhich is a new and interesting topic.It is a special feature of the present dissertation that there is a term with |u|^p(x)-2u on the boundary in the equations. According to the form of b(x), we consider the Robin problems and Steklov eigenvalue problem respectively.In the case of Robin problems, by the form of the nonlinear term h(x, u), we discuss the four cases, that is, Robin eigenvalue problem, the nonlinear term with a perturbation, the nonlinear term with singular coefficients and the nonlinear term with monotonicity.Another special feature of this dissertation is that in Robin problems there is not a positive term with |u|^p(x)-2u combined with operator -Δ_p(x)u inΩ. Hence, we present a new norm which is equivalent to the usual one.In the cases of Robin eigenvalue problem and Steklov eigenvalue problem, we prove the existence of the infinitely many eigenvalue sequences and present some sufficient conditions for that the spectrum of the corresponding problem is not closed.In the other cases, applying different variational principles, we obtain the existence and multiplicity of solutions or positive solutions of the corresponding problem.