Existence Of Ground-state Solutions To Kirchhoff-type Transmission Problems

Let ? be a bounded domain in?RN(N?2)with a C1,1 boundary(?)be a subdomain of ? with a C1,1 boundary(?).Assume that(?)is connected.Obviously,(?).In this paper,we study the existence of ground-state solutions to the following nonlinear Kirchhoff-type transmission problem via the critical point theory,variational methods and Nehari manifoldwhere ?,? are two positive continuous functions defined on R+:=[0,?),v is the unit outward normal vector to and(?).This problem models the transverse vibrations of a membrane composed of two different materials in ?1 and ?2.The problem is called a transmission problem owing to the boundary conditions u=v and(?).The transmission problem is also studied in physics and biology,for instance,in electromagnetic processes in ferromagnetic media with different dielectric constants,and in the population distribution of subjects living in an environment composed of different ecological media.Due to the particularity of N=1,we can only discuss the radial solution.Therefore,in the third part of this paper,the existence of the radial solution of the transmission problem is studied separately.It is the key point of discussing the transmission problem that establish space struc-ture.In this paper,our analysis is based on the following Sobolev spacewhereAccording to Poincare's inequality,H1?(Q2)is a Banach space under the norm |?v|2,?2 for v?H1?(?2).[To Fu Ma and Jaime Edilberto Munoz Rivera.Positive solutions for a nonlinear nonlocal elliptic transmission problem.Appl.Math.Lett.,16(2):243-248,2003| gave the definition of the norm for the Sobolev space E.They proved that E is complete by using the equivalent norm methods.However,for the convenience of readers,we provide an easier way to understand this fact.This paper consists of five chapters.In Chapter 1,we present the background and latest advance of the research on Kirchhoff-type transmission problem.Moreover,the main results obtained in this thesis are listed.In Chapter 2,we consider the existence of ground-state solutions to the subcritical Kirchhoff-type transmission problem(1).We suppose that the two pairs of functions(?,f)and(?,g)belong to and respectively.Here,we say that a pair of functions(?,f)belongs to AF(U)if(?,f)satisfies the following assumptions:(A0)? is an increasing contimuous function defined on R+ and ?(0)>0;(A1)there exists ??(0,2*/2-1)such that(?(s)-?(0))/s? is decreasing on(0,?),where 2*=2N/(N-2)if N?3 and 2*=? if N=2;(F0)there exist ??(2,2*)and C>0 such that(F1)for each (?) is nondecreasing on(0,?)and nonincreasing on(-?,0),and (?) uniformly in x?U,where ? is defined in(A1);(F2)lims>0 f(x,s)/s=0 uniformly in x?U.In this chapter,let N?2.If(?,f)and(?,g)belong to AF(?1)and AF(?2)respectively,we prove that the problem(1)has a ground-state solution,that is,theorem 2.1.5.In particular,for N=2,we can continue studying the above problem with more weaken conditions.We say that a pair of functions(?,f)belongs to AF'(U)if(?,f)satisfies(A0),(A1),(F'0),(F1),and(F2),where the condition(F'0)is stated as the follows:(F'0)for each ?>0,there exists O?>0 such thatLet N=2.Assume that(?,f)and(?,g)belong to and respectively.Then,we have that the problem(1)has a ground-state solution.This result is actually a supplement of Theorem 2.1.5.In Chapter 3,we devote the existence of ground-state solutions with a critical growth,that is,discuss the existence of ground-state solutions to the following Kirchhoff-type transmission problem:where(?),and v is the unit outward normal vector to(?)?1.From the above transmission problem,if ?>0,then its nonlinearity has critical growth.To discuss the existence of ground-state solutions to the problem(2),we suppose that two pairs of functions(?,f)and(?,g)belong to the set AF,where a pair of functions(?,f)is said to belongs to AF,if(?,f)satisfies(AO),(A1),(F1),(F2),and(F3),where the condition(F3)is the following(F3)f has a quasicritical growth,that is,In this chapter we assume that(?,f),(?,g)?AF.Then there exists ?0>0 such that the problem(2)has a ground-state solution(u?,v?)for all ??[0,?0).Furthermore,it holds that(u?,v?)?(u0,V0)in E as ??0,where(u0,v0)is a ground-state solution to the problem(2)with ?=0,that is,Theorem 3.1.1.In order to obtain a nontrivial solution to the above equation,one should not only seek a critical energy c*ensuring that the energy functional satisfies the(PS)c condition for all c<c*,but also verify that the related mountain pass level or least energy is lower than c*.In this study,we adopt the perturbation method from[Wonjeong Jeong and Jinmyoung Seok.On perturbation of a functional with the mountain pass geometry:applications to the nonlinear Schrodinger-Poisson equations and the nonlinear Klein-Gordon-Maxwell equations.Calc.Var.Partial Differential Equations,49(1-2):649-668,2014]to establish the existence of ground-state solutions to this problem.In fact,Jeong and Seok considered a compact perturbation.Specifically,let I?=I0+J? with J? and J'? being compact.Then it was proved that under some other suitable assumptions on I0 and J?,there exists ?0>0 such that for each(?),the functional I? admits a nonzero critical point that is near a nonzero critical point of I0.However,in this study,because of the presence of the critical terms u5 and v5,we cannot use the Theorem 1 of the above literature to obtain a critical point directly.In this sense,our result supplements to the above literature.In Chapter 4,we study the existence of solutions for the following modified version of Kirchhoff-type transmission problem with a critical growth(?)where (?) v is the unit outward normal vector to and ?,??C1(R+),f,h?C1(R).This system is a modified version of Kirchhoff-type transmission problem because the appearance of nonlocal terms (?).The above problem has two modified version.One is modified version of quasilinear Schrodinger equations,the other is modified version of transmission problem.In this chapter,we will establish the existence of ground-state solutions to Kirchhoff-type transmission problem with more general g and more general perturbation terms ?and ?.To obtain the existence of ground-state solutions to the more general Kirchhoff-type transmission problem(3),we assume that,four pairs of functions(?,g,f),(?,g,h),(?,g,?),and(?,g,?)belong to the set A,where a pair of functions(?,g,f)is said to belongs to A,if(?,g,f)satisfies the following assumptions(A0)??C1(R-)is an increasing function and ?(0)>0;(A1)there exists ??(0,2)such that[?(s)-?(0)]/s? is decreasing on(0,?);(G)g?C1(R,R+)is even with g'(s)?0 for s?R+and g(0)=1;(F'1)f(s)/(g(s)|G(s)|2?G(s))is nondecreasing on(0,?)and nonincreasing on(-?,0),and (?),where (?)is as in(A1).(?)where (?),we call that f has a quasicritical growth;if lf?0,we call that?? has a critical growth;About problem(3),it is worth noting question which is Sobolev critical exponent problem.We know that the critical exponent of problem(2)is 6 which has a significant influence on the properties of the solution.The critical exponent of problem(3)is different for different g and the critical exponent depends on G6.This is an interesting phenomenon.For example,when(?)for s ?R,the critical exponent is 12;when g(s)=s2+1?for s?R,the critical exponent is 18.In this chapter,if(?)with(?)with(?),and(?),our main result is that there exists ?0>0 such that both the problem(3)has a ground-state solution(u?,v?)for all ??[0,?0).Furthermore,it holds that(?)in E as ??0,where(u0,v0)is a ground-state solution to the problem(3)with ?=0.Particularly,let ?2=?,?(s)=1,(?)(?).Then the following equation has a ground-state solution ?? for all ??[0,?0),where q ?(4,12).Furthermore,it holds that(?)where u0 is a ground-state solution to the above problem with ?=0.Moreover,let g(s)=1 and(?).Then by Theorem 4.1.5,we have that the transmission problem(2)also has a ground-state solution,which has been achieved in Chapter 3.Thus,Theorem 4.1.5 could be regarded as a generalization of Theorem 3.1.1.In Chapter 5,we research the existence of the radial solution of the transmission problem.Let(?):(?)be the open ball,the closed ball and the sphere,respectively.We consider the existence of radical solutions to the following Kirchhoff-type transmission problemwhere v is the inner normal to(?)B1,the constants a,c>0,b,d)0,and f?C([0,1]×R),g?C([1,2]×R).We suppose that two functions f and g belong to the set F(0,1)and F(1,2),respectively.Here we say that a functions f belongs to F(U),if f satisfies the following conditions:(f1)for each(?)is nondecreasing on(0,?)and nonincreasing on(-?,0),and(?)uniformly in r?U;(f2)lims?0 f(r,s)/s=0 uniformly in r?U.In this chapter,we first prove that if(u,v)is the solution to the above transmission problem,if and only if(u,v)is the classical solution to the following ordinary differential equations#12In order to obtain the equivalence result of the solution,we first obtain the regularity.That is,if(?),and(u1,v1)is a solution to the system(5),then u1?C2[0,1)and v1?C1[1,2].On this basis,if f and g satisfy the conditions F(0,1)and F(1,2),respectively,then the system(4)has a nontrivial radial solution,that is,Theorem 5.1.1.In particular,when N=1,Theorem 5.1.1 supplements the conclusions of Theorem 2.1.5.