Existence Of Elliptic Boundary Value Problems With Nonlocal Operators And Solutions To Related Problems

In this paper, we will use the variational method and the theory of critical points to study the existence of solutions for elliptic boundary value problems involving nonlocal operators and related problems.Firstly, in Chapters2and3, we study the following elliptic boundary value problems involving nonlocal operators: where Ω is a bounded domain in RN with the smooth boundary, a(x):Ω→R is a measurable function, f(u),g(u):R→R are continuous functions with the nonlinear term f behaving oscillations at the origin or at+∞. In the case of μ=0and-Lku=(-△)su, by using the idea of constructing a sequence of subsets in X0such that the minimum points of the energy functional on these subsets are actually weak solutions of the problem (I), we show that the problem (I) has infinitely many solutions. In the case of μ≠0, by utilizing suitable truncation techniques and a general variational principle of Ricceri, we prove the existence of arbitrarily many solutions under continuous condition on the perturbation term g. Moreover, we generalize above results to the following general elliptic boundary value problems involving the oscillating nonlinear term and nonlocal operators: where f, g:Ω×R→R are Caratheodory functions.Next, in Chapter4, we consider the following hemivariational inequalities involving nonlocal operators and two parameters: where Ω is a bounded domain in RN with the smooth boundary, λ,μ are two positive param-eters and F(x, u),G(x,u):Ω×R→R are two nonsmooth potentials. Under the different growth condition, by using nonsmooth critical point theory, in the cases of the energy func-tional corresponding to the problem (I) being coercive and not coercive, we respectively prove that, whenever A is big enough and μ is small enough, the problem (Ⅰ) admits at least two nontrivial solutions. Moreover, some properties of solutions are also given. In Chapters5, we are concerned with the existence of solutions of the following class of quasilinear elliptic hemivariational inequalities on unbounded domains Ω∈RN: where P€L1(n),μ1∈L∞(Ω) and j:Ω×R→R is a nonsmooth functional. By using the approximation of bounded domains, the Ky Fan theorem for multivalued mappings and other nonlinear functional analysis techniques, we obtain the existence of solutions of the problem (Ⅲ).Finally, in Chapters6, we discuss a class of fourth order two point boundary value problems: where λ>0is a parameter,f:R→R is a continuous function. We first establish two results on the existence of two nonzero solutions via the Mountain Pass lemma and three critical points theorem of Riccer. Then, based on a general variational principle of Ricceri, we prove that the problem (IV) has infinitely many solution if the nonlinear term f has an suitable oscillation condition. Moreover, we generalize the result on infinitely many solutions to more general boundary value problems where a, c≥0, b≥a, d≥c, b-a≥c/3+d/6.