The Study On The Nonsmooth Problems Of Elliptic Equations Involving P(x)-Laplacian

This dissertation investigates some problems for p(x)-Laplacian equations with nonsmooth potential, which include Dirichlet problems, homogeneous Neumann prob-lems and inhomogeneous Neumann problems with indefinite weight. By using the theory of nonsmooth critical point, the solution and multiple solutions of these prob-lems are studied. In chapter 2, we considered the following p(x)-Laplacian Dirichlet problems:where the exponent p(x)∈(?),p(x)＞1,Ω(?)R^N(N＞2) is a bounded domain with smooth boundary, and 2(p^-)²/p⁺＞N with p^-=inf_x∈Ωp(x),p⁺=sup_x∈Ωp(x), besides j(x,ζ) and (?)(x,ζ) are locally Lipschitz function and subdifferential with respectto theζ-variable respectively. Through the Mountain Pass Lemma and Minimax Principle, the solution and multiple solutions of the problem are proved.In chapter 3, the paper considered the following two Neumann problems for p(x)-Laplacian equations with nonsmooth potential: homogeneous Neumann problem and nonhomogeneous Neumann problem with indefinite weightwhereΩ,p(x) mentioned above, also (?), with n being the outward unit normal on (?). In the nonhomogeneous Neumann problem with indefiniteweight, V(x)∈L^∞(Ω) is a sign changing function, andγ₀:W^1,p(x)(Ω)→L^p(x)(?) is the trace operator withγ₀(u)=(?) for all u∈W^1,p(x)(Ω). By using the minimax principle and Weierstrass Theorem, we discussed the existence of solutions of above two Neumann problems.