Study On Existence Of Multiple Solutions For Several Types Of Variable Exponent Laplacian Problems With Non-smooth Potential

The recent decades witness the increasing interest in the variable exponent prob-lem,which is a branch of nonlinear elliptic partial differential equations.It possesses a solid background in physics and originates from the study on electrorheological fluids and nonlinear elastomechanics.It also has wide applications in different research fields,such as image processing model,stationary thermorheological viscous flows and the mathe-matical description of the processes filtration of an idea barotropic gas through a porous medium.Recently there have been many research works on this topic.However,most of them are restricted to the elliptic PDE's with smooth potential.Actually,the PDE's with nonsmooth potential are preferable in describing the real phenomena.So it makes sense to study the differential inclusion problems involving p(x)-Laplacian with non-smooth potential and a variety of boundary conditions.This thesis studies the existence and multiplicity of the following four p(x)-Laplacian problems with non-smooth potentials.The technical approach relies on the the-ory of variable exponent Lebesgue space Lp(x),variable exponent Sobolev space W1,p(x), variational methods and non-smooth critical point theory.(1)Differential inclusion problem of Dirichlet-type: hereΩ(?)RN is a smooth bounded domain,p∈C(Ω),10,p∈C(RN)with 1p+ and the generalized AR-conditionⅡholds, then the existence of a nontrivial radial solution can be directly deduced by Mountain Pass Theorem.Concerning problem (3), the Poincare inequality is absent due to the periodic bound-ary condition. In order to overcome this difficulty, we make the direct sum decomposition wT1,p(t)=RN(?)V, where V={v∈WT1,p(t):∫0Tv(t)dt=0}. Making use of the non-smooth Linking Theorem we prove that there are at least two nontrivial periodic solutions under the generalized AR-conditionⅠwhich is newly proposed in this work. In the case of generalized AR-conditionⅡ, we can show the existence of at least one nontrivial periodic solution by using generalized Mountain Pass Theorem.As for the last problem, we establish the existence of at least two nontrivial solu-tions by applying the similar methods as mentioned above. The results improve some corresponding works.