No Flux Boundary Problem For P(x)-Laplacian Equation And Sub-supersolution Theorem

In the paper, we study the existence and eigenvalue problem of p(x)-Laplacian equation with No flux boundary condition in a bounded domainΩ(?) R^N, where No flux boundary condition is given in the following:This problem is a new and interesting topic.When a term of |u|^p(x)-2u is involved in p(x)-Laplacian equation, We apply variation method to obtain the existence of solution and multiplicity of this problem, otherwise, we may use the principle least action to get the existence of solution when f satisfies some appropriate conditions.We prove that p(x)-Laplacian operator with no flux boundary condition has infinity eigenvalues and the first eigenvalue is 0, but to be different with the constant exponent case, generally it is not isolated, i.e. the infimum of all positive eigenvalues is 0.We also consider quasi-linear elliptic equation problem with no flux boundary which has the following form:Because this problem is not usually variational, we can't use variational method. We prove that sub-supersolution theorem of this problem.