Resonance Problems For Nonlinear Equation Involving The P-Laplacian

We shall be concerned with the existence of weak solutions for the following five probelmsDirichlet problem:No-flux problem:Neumann problem:Robin problem:Steklov problem:whereâ–³_pu=div(|â–½u|^p-2â–½u)is p-Laplacian, N≥1, 1＜p＜N,Ωis a bounded domain in R^N,(?)u/(?)n denotes the outer normal derivative of u with respect to (?)Ω.When the function f(x, u) is of subcritical growth and satisfies a generalized Landsman-Lazer condition, we obtain the existence of weak solutions to the above equations by using the variational methods (?)λ≥0(λmay reach eigenvaluesλ_k).When the function f(x, u) is of subcritical growth and satisfies a non-Landsman-Lazer condition, we obtain the existence of weak solutions to the above equations by using the Mountain-Pass theorem and the linking theorem for 0≤λ＜λ₂。...