Existence And Multiplicity Of Solutions To Some Kinds Of Differential Equations

The existence and multiplicity of solutions to differential equation are the important content of nonlinear analysis. Partial differential equations posses widespread background. They come from physics, genetic engineering, chemistry, medical science, etc. Recently years, many people studied nonlinear differential equations, especially, nonlinear partial differential equations. For example, by using variational method and critical point theory, many people studied the existence and multiple solutions to ordinary differential or partial differential equations, such as second-order or fourth-order elliptic equations, Shrodinger e-quations, Schrodinger-Poisson systems, Kirchhoff type equations, quasi-linear Schrodinger equations, etc. These study improved deeply the development of nonlinear analysis. In the present paper, by using variational method, critical point theory, Morse theory and the topological degree theory, we consider the existence and multiplicity of solutions to three kinds of differential equations:Schrodinger-Poisson system, Kirchhoff type equation and second-order Sturm-Liouville boundary value problem.This paper consists of four chapters.In Chapter 1, some research background, the research advance of the related work are given. Moreover, the main results obtained in this thesis are listed.In the first section of Chapter 2, we study the following generalized Schrodinger-Poissonwhere q≥ 0 is a parameter, η=±1,f is subcritical and satisfies(f) f ∈ C(R,R+) and there exists c> 0 such that f{t)≤c(|t|+|t|α) for all t ∈ R and some a ∈ (2,4).The nonlinearity g is subcritical, superlinear both at 0 and ∞, satisfying the following conditions(gi) g ∈ C(R,R+) and there exists ci> 0 such that g(t)< c1i(1+|t|p-1) for all t ∈ R and some p ∈ (2,6);(g2) limtâ†'0+g(t)/t=0;(g3) limtâ†'∞g(t)/t=∞.We use the variational method to discuss the existence of position radial solution to the system. Firstly,f is general subcritical satisfying (f). Many references have studied the system with f(t)= t. Under the assumptions of (f), by using the Lax-Milgram theorem, for each u ∈ H1 (R3), the second equation of the system has a unique solution Φ>u ∈ D1,2((R3), substituting Φu to the first equation, then the system can be transformed to one variable equation. Thus, we can obtain the variational structure of the system. Hence the existence of solution to the system can be transformed to the existence of critical point of one variable functional. The nonlinearity g is general, superlinear both at 0 and oo and dosen’t satisfy the following global Ambrosetti-Rabinowitz growth conditions AR) there exists μ> 4 such that 0<μG(t)=f0tg(s)ds≤tg(t) for all t ∈ R.When η=1, the mountain pass geometry of the corresponding functional is not obvious. By using the trick of cut-off and the methods in [30], we obtain that the system has at least one positive radial solution for q≥0 small enough. This part has been published, see [41] (J. Math. Anal. Appl.401 (2013):754-762). When η=-1, although we can obtain the mountain pass geometry, but the boundedness of (PS) sequences can not be obtained by using the standard arguments. We also combine the methods in [30] with Pohozaev identity and obtain that the system has at least one positive radial solution for any q≥O. The main result in this part is as follows.Theorem 2.1.1. If f satisfies (f) and g satisfies (g1i)-(g3). When η= 1, there exists q0> 0 such that, for any q ∈ [0,q0), system (2-1-1) has at least a positive radial solution (u,Φ) ∈ H1(R3) ×D1’2(R3). When η=-1, for any q≥0, system (2-1-1) has at least one positive radial solution (u,Φ) ∈ H1(R3)×D1,2(R3).In the second section of this chapter, we discuss the existence of multiple radial solutions to the following nonhomogeneous general Schrodinger-Poisson systemwhere q≥ 0 is a parameter,f is subcritical satisfying the above condition (f). g is subcritical, superlinear at zero satisfying (g1) and (g2), asymptotical linear or superlinear at oo satisfying(g’3) limtâ†'∞ g(t)/t=l, where 1<l≤∞.h satisfies the following conditions(h1) h ∈ Cl(R3) ∩ L2(R3) is a nonnegative radial function;(h2) |h|2< m with where γs is the embedding coefficient of H1(R3)â†'Ls(R3),s E [2,6], C is a constant depending on g;(h13) (Vh(x),x) ∈ L2(R3) and (â–½h(x),x)≥0, where (·,·■) denotes the usual inner product in R3.By using the variational method, we obtain the existence of at least two radially positive solutions. Firstly, under the assumptions of (hi) and (h2), we can get that for any q≥0, the corresponding functional has a negative local minimum at the neighbour of zero. This minimum can be obtained by means of Ekeland’s variational principle. Since the nonlinearity g is asymptotically linear or superlinear at ∞ and doesn’t satisfy the global (AR) condition, the mountain pass structure of the corresponding functional is not obvious. By introducing a testing function and using the trick of a cut-off function, we obtain the mountain pass structure. Meanwhile, by combing the methods in [30] with Pohozaev type identity, we obtain the boundedness of (PS) sequence for q≥0 small enough. Subsequently, we obtain the existence of the second solution to the system for q≥0O small enough.The main result in this part is as follows.Theorem 2.2.1 If f satisfies (f), g satisfies (g1i), (g2e) and (g3’), h satisfies (hi)-(h3). Then there exists q0> 0 such that, for any q ∈[0,q0), system (2-2-1) has at least two positive radial solutions (u,Φ>) ∈ H1 (R3)×D1,2(R3).In the last section of this chapter, we study the existence and multiple solutions to the following Schrodinger-Poisson system with potential Vwhere λ≥1 is a parameter,f ∈ C(R3×R, R). The potential V satisfies the following conditionsV1i) V ∈C(R3,R+);V2) there exists b> 0 such that 0< meas{x ∈;R3:V(x)≥b}<00, where meas denotes the Lebesgue measure in R3.Conditions (Vi) and (V2) are more general than the following conditions(Vi) V ∈ C(R3, R), there exists M> 0 such that infR3 V≥ M;(V2’) for each b> 0, such that meas{x∈R3:V(x)≤6}<∞.Under the conditions of (V1’) and (V’2), the space {u ∈H1(R3):fR3 Vu2< oo} can be compactly embedded into Ls(R3), s ∈ [2,6) by [10]. In order to obtain the compact results, many references, such as [16,17,42,57], assume that V satisfies (V1’) and (V2’). While V satisfies (V1) and (V2), the compactness of the embedding fails. In order to obtain the compact results, we choose Eλ(λ≥1) as the work space, where Eλ= (E,‖·‖λ),‖u‖λ2= fR3[|â–½u|2+λVu2],u ∈ E. We also need proper assumptions on f. Here, we assume the nonlinearity f is subcritical, superlinear at zero,4-superlinear at ∞0 satisfying the following conditions(f1) f ∈ C(R3×R,R), there exists c> 0 such that |f(x, t)|< c(1+|t|p-1), (x, t) ∈ R3 x R, where p ∈(2,6);(f2) limTâ†'0f(x,t)/t=0 uniformly for x ∈R3;(f3) lim|t|â†'∞f(x,t)t/|t|4 uniformly for x ∈ R3;(f4) for all (x,t) ∈ R3×R,R,F(x,t)= tf(x,t)-4F(x,t)≥-dot2, where 0≤d0< v2-2,v2 is the embedding coefficient of Eλâ†'L2(R3). We can prove that the embedding Eλâ†'L3(R3),s ∈[2,6] is continuous in Lemma 2.3.4.Conditions (f) and (f4) are more general than the following global (AR) conditions, which are in [16](f3’) There exists μ>4 such that(f4/) infx∈R3,|t|=1F(x,t)>0.We apply the mountain pass lemma to discuss the existence and multiplicity of solu-tions to the system. The main result is as follows.Theorem 2.3.1 Assume V satisfy (V1), (V2),f satisfy (f1)-(f4). Then there exists A> 1, such that system (2-3-1) possesses at least one nontrivial solution whenever λ> A. Moreover, if f(x,t) is odd in t, i.e., f satisfies(f5) f(x,t)=-f(x,t), (x,t) ∈R3×R+, then for λ> A, system (2-3-1) possesses infinitely many solutions {(uk,Φk)} satisfyingIn this section, we also discuss the existence of infinity many solutions with negative energy when f satisfies the more general sublinear assumptions than the assumptions in [57](f6) there exist σi ∈(1,2) and functions ηi ∈L2/2-σi(R3,R+),i= 1,2 such that(f7) there exist x0 ∈R3, two sequences {εn},{Mn} and constants c,ε,δ> 0 such that εn> 0, Mn> 0 andConditions (f6) and (f7) were introduced in [66] for fourth-order elliptic equations. Since Schrodinger-Poisson system possesses a nonlocal term Φuu, in order to obtain the compactness under conditions (f6) and (f7), we also assume that V satisfies (V1’) and (V2’). We shall apply the symmetric mountain pass lemma to consider the existence of infinity many solutions with f being odd in t. The main result is as follows.Theorem 2.3.2 Assume V satisfy (V1’), (V2’),f satisfy (f5)-(f7). Then system (2-3-1) possesses infinitely many nontrivial solutions {{uk,Φk)} with λ=1, ukâ†'0,Φk> 0, kâ†' ∞ andIn the third chapter, we consider the existence of multiple solutions to the following nonhomogeneous Kirchhoff type equationwhere a,b are positive constants, p ∈(1,5),h ∈ C1(R3) ∩ L2(R3) satisfies the following conditions(hi) 0≤h(x)= h(|x|) E ∈2(R3) and |h|2≤mp, where γs is the embedding coefficient of H1(rR3<â†'Ls(R3),s ∈[2,6];(h2) (â–½h(x),x) ∈ L2(M3), where (·,·) denotes the usual inner product in R3.Motivated by the methods in [32], which discuss the existence of multiple radial solutions of the nonhomogeneous Schrodinger-Poisson, we apply the variational method to consider the multiple radial solutions to the nonhomogeneous Kirchhoff equation, where p ∈ (1,5). Under the assumption (hi), we obtain the existence of a local minimum of corresponding functional in the neighbor of zero by using Ekeland’s variational principle. Note that the term |t|p-1t is neither 4-superlinear nor satisfies (AR) condition for p E (1,3]. In order to obtain the bounded (PS) sequence, we also use the indirect methods in [30]. Meanwhile, for w ∈ H\{0}, we take a transform of wt(·)=tw(t-2·),t> 0 to construct the mountain pass geometry structure. Where H is the subspace of H1(R3) containing only the radial functions. Finally, the combination of Pohozaev identity with the method in [30] obtains the bounded (PS) sequence. Therefore we obtain the second solution with positive energy. This chapter has been published, see [69] (Abstract and Applied Analysis, 2013, Article ID:806865). The main result in this chapter is as follows.Theorem 3.1.1 Let p ∈ (1,5) and h satisfy (h1) (h2). Then, problem (3-1-1) has at least two nontrivial radial solutions u0 and vo, satisfying J(u0)< 0< J(v0).In the fourth chapter, we discuss the following second-order Sturm-Liouville boundary value problemwhere α,β,γ,δ≥0 with α+β=γ+δ=1,p ∈ C1[0,1],q E C[0,1],p> 0,q≥0 and f ∈ C1([0,1] x R,R). Moreover, we also assume α+γ>0 or q≠0.We discuss the existence of multiple solutions, especially multiple sign-changing so-lutions, when the nonlinear therm f is resonant or cross eigenvalue at 0 and infinity. The nonlinearity f satisfies the following conditions(H1 )f ∈ C1([0,1] x R,R) and tf(x,t)≥ 0 for all (x,t) E [0,1] x R;(H2) ft’(x,0)=λ2n0+1 for some n0≥1 and for x ∈ [0,1]. Moreover, there exists Ï„> 0 such that f (x,t)t≤λ2no+1t2 for all (x,t) E [0,1] x [-Ï„,Ï„];(H3) there exist n1≥1, C1> 0 and a ∈ (0,1) such that lim|t|â†'∞ ff(x, t)/t=λn1 uniformly for x ∈ [0,1], and uniformly for x ∈ [0,1], where F(x, t)=f0tf (x, s)ds for all (x, t) ∈ [0,1]×R;(H4) there exists n1≥1 such that exists uniformly for ∈[0,1] and λ2n1<α(x)<λ2m+i;(H5) there exists b> 0 such that |f(x,t)|< b for all (x,t) ∈ [0,1] x [-bc, be], where c=maxx∈[0,1] e(x),and e is the solution of boundary value problemThe nonlinearity f is resonant at zero about odd eigenvalue (H2) and is either reso-nance (H3) or crosses even eigenvalue at infinity (H4). The assumptions on f are motivated by the assumptions in [39]. By combining the methods of the Morse theory, the topological degree and the fixed point index, we obtain that the problem has only finitely many solu-tions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions. The three kinds of relations play important role in our proof: the connection between the topological degrees in different space, the connection between the topological degree and the fixed point index, the connection of the topological degree and the critical group. This chapter has been published, see [68] (Applied Mathematics and Computation 219 (2012):1061-1072). The main result is the following.Theorem 4.1.1. Suppose that the conditions (H1), (H2), (H5) hold, and one of (H3) and (H4) holds. Then problem (4-1-1) has at least six nontrivial solutions. Moreover, if the problem has only finitely many solutions then, of these solutions, there are two positive solutions, two negative and two sign-changing solutions.