Nonuniform Nonresonance of Semilinear Differential Equations

Consider the Dirichlet problem of nonlinear differential equations with the principal part the p-Laplacian. When the nonlinearity satisfies some semilinearity conditions, the usual nonuniform nonresonance conditions are obtained by comparing nonlinear equations with the classical eigenvalues. In this article, we will introduce some weighted eigenvalues. The nonuniform nonresonance conditions, proved in this article using weighted eigenvalues, will improve the usual ones significantly