Research On The Differential Inclusion Based On The Nonsmooth Critical Point Theory

In this Ph.D. thesis, based on the nonsmooth critical points theorem and Sobolev spaces, we deal with some differential inclusions, which have strong phys-ical background, and derive some interesting results. Furthermore, we also extend some smooth critical points theories to nosmooth case. The paper is divided into six chapters. The main contents are as follows:In chapter 1, we introduce the background and significance for these differen-tial inclusions, and outline the main contents of the dissertation. Some notations and preliminaries are given.In chapter 2, employing the Bartsh-Wang condition to recover a compact embedding theory, and combing the nonsmooth variational technique, we consider the following p(x)-Kirchhoff type problem with a nonsmooth potential where t=()RN1/p(x)(|â–½u|p(x)+V(x)|u|p(x))dx,(?)F(x,u) is the generalized Clarke subdifferential of the function F(x,·). Under weaker hypotheses on the nons-mooth potential at zero (at infinity, respectively), the existence and multiplicity of solutions to problem (P1) are obtained.In chapter 3, the following problem with a subdifferential term and a discon-tinuous perturbation is investigated: where t=(?)Ω1/p(x)|â–½u|p(x)dx.Assuming the existence of an ordered pair of ap-propriately defined upper and lower solutions (?)(x) and (?)(x), by the method of upper-lower solutions, combining penalization techniques, truncations and results from nonlinear and multivalued analysis, we show the existence of solutions in the order interval [(?)(x),(?)(x)] and of extremal solutions in [(?)(x),(?)(x)].In chapter 4, the following quasilinear elliptic problem with a nonsmooth potential in the Orlicz-Sobolev space is concerned: Under suitable conditions, two multiplicity theorems in the Orlicz-Sobolev space are derived. In the first multiplicity theorem, we produce three nontrivial smooth solutions. Two of these solutions have constant sign (One is positive, the other is negative). In the second multiplicity theorem, we derive an unbounded sequence of critical points for problem (P3). Our approach is variational, based on the nonsmooth critical point theory. We also show that C1-local minimizers are also local minimizers in the Orlicz-Sobolev space for a large class of locally Lipschitz functions.In chapter 5, we extend a smooth Ricceri three critical points theorem to non-smooth case. Our approach is based on the non-smooth analysis. As its application, we consider the following p(x)-Laplacian differential inclusion:We show that the problem has at least three nontrivial solutions by our extended nonsmooth three critical points theorem.In chapter 6, we study the following degenerate p(x)-Laplacian differential inclusionwhere Ω is a bounded domain,j1,j2 are nonsmooth and locally Lipschitz functions. We firstly establish a compact embedding W1,p(x)(ω,Ω)â†'â†'Lq(x)(α(x),Ω), under suitable conditions on j1 and j2 we obtain the existence and multiplicity of solu-tions to the degenerate p(x)-Laplacian differential inclusion(P5) by the theories of nonsmooth critical point and the variable exponent Lebesgue-Sobolev spaces.