Existence Of Multiple Solutions For Kirchhoff Equation With Neumann Boundary Condition

Nonlinear partial differential equations,an important branch of modern mathematics,are widely used in the natural science,physics and engineering axes,and so have great significance and value.A large number of researchers have devoted much attention to these equations for a long time.Moreover,the existence and multiplicity of solutions about Kirchhoff equations which are the most fundamental equations in the partial differential equations have also drawn extensive attention over the past decades.In this paper,we use variational methods,such as the mountain pass theorem,Moser iteration approach,Miranda theorem and quantitative deformation lemma,to obtain the existence of solutions for Kirchhoff problem.The thesis consists of two sections.Chapter 1 is the preface.In Chapter 2,we consider the following Kirchhoff problem with Neumann boundary condition(?)where a,b>0 are two constants,λ≥ 1 is a parameter,Ω is a bounded domain in R3 with smooth boundary δΩ and v denotes the unit outward normal to δΩ.The potential function V and nonlinear term f satisfy the following hypotheses:(V)the potential function V ∈L∞(Ω)\｛0} is nonnegative on Ω;(f1)there exists δ>0 such that f∈C(Ωx[-δ,δ],R)and f(x,t)t-4F(x,t)≥ 0 for all(x,t)∈Ωx[-δ,δ];(f2)there exist p,q ∈(4,6)with p<q such that limt→0f(x,t)/(|t|p-2t)= 0 and limt→o f(x,t)/(|t|9-2t)= ∞ uniformly for x ∈ΩThe conclusions are as follows:Theorem 2.1.1.Under the hypotheses(f1),(f2)and(V),there exists λ1>0 such that for each λ ∈[λ1,∞)the problem(0.2)admits a nontrivial solution uλ1 ∈E and liλ→∞ ||uλ,1||E = 0.If the conditions(f1)and(f2)holds only on one side of point 0,that is,(f1+)there exists δ>0 such that f∈ C(Ω×[0,δ],R)and f(x,t)t-4F(x,t))≥0 for all(x,t)∈Ω x[0,δ];(f2)there exist p,q ∈(4,6)with p<q such that for uniform ax∈Ω,lmt→0+f(x,t)/|t|p-2t=0,limt→0+f(x,t)/|t|q-2t=∞,or(f1-)there exists δ>0 such that f∈C(Ω×[-δ,0],R)and f(x,xt)t-4F(x,t)≥0 for all(x,t)∈ Ω x[-δ,0];(f2-)there exist p,q ∈(4,6)with p<q such that for uniform x∈Ω,lmt→0+f(x,t)/|t|p-2t=0,limt→0+f(x,t)/|t|q-2t=∞,Therefore,we have the following results.Corollary 2.1.2.Under the hypotheses(f1+),(f+2)and(V),there exists λ+>0 such that for each λ∈[λ+,∞)the problem(0.2)admits a nontrivial nonnegative solution uλA,＋∈E and limλ→∞ ||uλ,+||E=0.Corollary 2.1.3.Under the hypotheses(f-1),(f-2)and(V),there exists λ->0 such that for each λ ∈[λ-,00)the problem(0.2)admits a nontrivial nonpositive solution uλ,-∈ E and limλ→∞||uλ,-||E=0.In addition,in order to obtain sign changing solutions,we suppose that(f3)there exists δ>0 such that f∈ C(Ω x[-δ,δ],R)and for each x∈Ω the mapping t→f(x,t)/|t|3 is nondecreasing on[-δ，0)and(0，δ],respectively;(f4)there exist p∈(4,6)and Cp ∈(0,∞)such that limt+o→f(x,t))/(|t|p-2t)= cp.By direct calculation,we have that the hypotheses(f3)and(f4)imply the hypotheses(f1)and(f2).The result of sign changing solutions is as follows.Theorem 2.1.4.Under the hypotheses(f3),(f4)and(V),there exists λ2>0 such that for each λ ∈[λ2,∞)the problem(0.2)has a sign changing solution uλ,2∈Eand limλ→∞||uλ,2||E= 0.