Some Variational Problems Involving The P(x)-Laplacian

In this doctoral thesis we study the existence of solutions of the following three variational problems with nonstandard growth conditions:where the potential F(x, t)=∫₀^tf (x,s) ds has a suitable oscillating behavior in any neighborhood of the origin(respectively +∞).whereε> 0 is a parameter,Ω(?) R^N is a smooth bounded domian,f∈C((?)×R)and f (x,t) is odd with respect to t,g∈C((?)×R).andwhereΩ(>)R^N is a bounded domain with smooth boundary (?)Ω,ηis the outside unit normal to (?)Ω,λis a positive number,p_i,q∈C ((?)) and q(x) > 1, 2≤p_i(x) < N, for all i∈{1,2,…,N}.We study these variational problems respectively by the variational method and the theory of variable exponent spaces.For the first problem, its working space is W^1,p(x)(R^N).Because of R^N,the embedding isn't compact, so (P.S.) condition don't satisfied. For overcoming this difficulty, some methods have been applied, for example, radical method and compensate-compactness principle. For this problem we apply weight function method to overcome the difficulty. Moreover we consider the existence of multiple solutions of this equation by a general variational principle due to B. Ricceri.For the second problem, the function f (x, t) is odd with respect to t and subcritical growth. We know that this equation possesses the infinitely many solutions under suitable conditions ifε=0.But when adding a perturbed term, that isε> 0, does the equation still possess the infinitely many solutions? or does the equation still possess multiple solutions? We find that if function g∈C ((?)×R),for any j∈N and ifε> 0 small enough, then the equation possesses at least j distinct weak solutions.For the last problem, we consider the existence of a eigenvalue of an anisotropic quasilinear elliptic equation with Neumann boundary in different cases. For an anisotropic operator, it is more complex than for the usual p(x)-Laplacian operator, because it has different role in different space directions. Meanwhile we give some concrete properties of anisotropic operator and a relative embedding theorem involving anisotropic generalized Sobolev space.