Existence And Multiplicity Of Solutions For Several Kinds Of Nonlocal P(X)-Laplace Equations

In this thesis, we study the existence and multiplicity of solutions for several kinds of nonlocal p(x)-Laplace problems.Firstly, for the following nonlocal p(x)-Laplace equations with singular coefficients when f{x,u)=a(x)[u]p-2u and f{x,u)=x1-a (.x)[u]p-2u+x2a2(x)[u]r2-2con-sequently, where M:(0,+㏄)�'(0,+㏄) is continuous and bounded, a∈Lr(Ω), a1∈Lr1(Ω), a2∈Lr2(Ω), a{x)>0.a1(x)>0.a1(x)>0, λ1..λ2, are two gener-al constants, need to note that we allow λ1..λ2change signs, by using the theories of weighted variable exponent Lebesgue space, Mountain Pass lemma. Fountain theorem, dual Fountain theorem and straight variational method, we obtain several existence; re-sults of nontrivial global minimal solutions. Mountian Pass type solution and infinite solutions.Secondly, we consider the following nonlocal p(x)-Laplac:e equations with nonlinear Nueinann boundary and the case with singular coefiicients consequently, By using the Sobolev trace embedding theorem. Mountain Pass lemma. Fountain theo-rem and straight variational method, We obtain several existence and multiplicity results of solutions, by using the truncate function method, we obtain the existence of nonnega-tive solution, furthermore, when the right side of this equation has no nonlocal form, we obtain infinite solutions by using Fountain theorem and dual Fountain theorem. For the case with singular coefiicients, we also obtain several existence and multiplicity results of solutions. And at last, we consider the following nonlocal p(x)-Laplace equations in BN and the case with singular coelficients consequently, By using the weighted function method, Mountain Pass lemma and straight variational method, we obtain several existence and multiplicity results of solutions, by using the truncate function method, we obtain the existence of nonnegative solution, by using the genus theorem and straight variational method, we obtain the multiplicity of solutions, furthermore, when f(x, u) has special forms, we obtain infinite solutions by using Foun-tain theorem and dual Fountain theorem. For the case with singular eoeflieicnts, we also obtain several existence and multiplicity results of solutions.This thesis contributes to the nonlinear p(x)-Laplacc equations’ research fields of the following parts: Since the results of nonlocal p(x)-Laplaee equations are rare, for the first time we solved the existence and multiplicity of solutions of singular nonlocal p(x)-Laplace equations with Dirichlet boundary condition; lor the lirst time we solved the existence and nmltiplicity of solutions of nonlocal p(x)-Laplacc equations with nonlinear Nueinann boundary conditions and the ca.se with singular coefficients: for the first time we solved the existence and multiplicity ol solutions of nonlocal p(x)-Laplaee equations in RN and the case with singular coefficients...