Multiple Solutions And Sign-changing Solutions To Differential Equations

The existence and multiplicity of solutions to partial 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 partial differential equations. For example, by using variational method and critical point theory, many people studied the existence and multiplicity of solutions to many kinds of Schrodinger equations. These study improved deeply the development for nonlinear analysis. In the present paper, by using variational method, Morse theory, critical point theory and the topological degree theory, we consider the existence and multiplicity of solution and sign-changing solution to some kinds of partial differential equations.This paper consists of five 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 Chapter 2, the classical elliptic equation Dirichlet boundary value problem is considered: whereΩ(?) RN is a bounded domain with smooth boundary,f∈C1(Ωx R1, R1) satisfying the subcritical growth condition |ft'(x, t)|≤C(1+|t|p-2), (x, t)∈Ω×R1., where C> 0 is a positive constant, p∈(2,2*),2*= 2N/(N-2) if N≥3; 2*=∞if N=1,2.We combine topological degree, critical group and fixed point index and obtain some connection between them. Under some assumptions, the nonlinear term f is resonant and cross eigenvalue at 0 and∞. We solve the problem that resonant case can not be dealt with by only using the topological degree and fixed point index, for example [1](J. Math. Anal. Appl.314(2006)464-476). We consider the case that f are both resonant at 0 and∞, which has not be studied in many papers. Under resonant case, we remove the bounded condition in [2](Math.Z.233(2000)655-677). Moreover, we still study the case that f is resonant from one side. This resonance is weaker than usual resonant conditions because the existence of limits and growth condition are removed. Then we can obtain the following results:Assume that f is resonant or cross eigenvalue at 0, whil f is resonant, or resonant from one side, or cross eigenvalue at oo, then the above boundary value problem has seven nontrivial solutions; of which two are positive, two are negative and three are sign-changing solutions.When we consider the dimension N=1, we can compute the critical groups of the seventh solution and then obtain the eighth solution. This solution is sign-changing. Thus, we obtain two positive solutions, two negative solutions and four sign-changing solutions under the case N=1. When we consider N=1 and fourth-order equation, the problem is the one in [1]. We not only study the case that f are resonant at 0 and∞, but also we obtain seven nontrivial solutions of which are three sign-changing by compute the critical groups under the case that f is resonant to odd eigenvalue or crosses odd eigenvalue at 0. This case can not be dealt by the methods of [1]. If f is resonant both to even eigenvalue at 0 and∞, we can obtain six nontrivial solutions. This case also can not be studied by the methods of [1]. In [1], they only consider the case that f crosses even eigenvalue at 0 and∞. Thus, we extend the conclusion in [1], and obtain better results.We assume that the following conditions hold:(f1) f(x,t)t≥0, (x,t)∈Ω×R1;(f2) there exists no≥2 withλn0<λn0+1 such that ft' (x,0)=λn0+1, x∈Ω. And there existsδ> 0 such that f(x,t)t≤λn0+1t2, (x, t)∈Ωx [-δ,δ];(f3) there exists b> 0 such that |f(x,t)|<b, (x,t)∈Ω×[-bc,bc], where c maxx∈Ωe(x), and e is the solution of the following boundary value problem:(f4) there exist n1> 1 withλn1-1<λn1 and C1> 0,α∈(0,1) such that (?)f(x,t)/t=λn1, uniformly for x∈Ω, |f(x,t)-λn1t|≤C1(1+|t|α), (x,t)∈ΩxR1, and uniformly for x∈Ω, where F(x, t)=∫0t f(x, s)ds, (x, t)∈Ω×R1; (f5) there exist n1,ε,R>0 such that anduniformly for x x∈Ω; (fθ) there exist n1,ε, R> 0 such thatand =+∞, uniformly for x∈Ω. (f7) there exists no≥2 such thatλn0<λn0+1 andλn0<ft'(x,0)<λn0+1, uniformly for x∈Ω; (f8) there exists n1> 1 withλ2n1<λ2n1+1 such that the limitα(x)= (?)f(x, t)/t exists andλ2n1<α(x)<λ2n1+1, uniformly for x∈Ω.Theorem 2.1.1. Suppose that (f1)-(f3) hold, and one of (f4), (f5), (f6), (f8) holds. Then (2.1.1) has at least seven nontrivial solutions. Of them, two are positive, two arc negative and three are sign-changing.Theorem 2.1.2. Suppose that (f1), (f3) and (f7) hold, and one of (f4), (f5), (f6), (f8) holds. Then (2.1.1) has at least seven nontrivial solutions. Of them, two are positive, two are negative and three are sign-changing.In Chapter 3, we consider the semilinear elliptic equation on RN-Δu+u=f(x,u), u∈H1(RN). Under some conditions, we, by using the flow, give the result of existence of infinitely many sign-changing solutions. Therefore, we improve some results obtained in some papers. In [3](Adv. Math.222(2009)2173-2195), the authors studied infinitely many sign-changing solutions to semilinear elliptic equation on bounded domain. In this chapter, we develop to unbounded domain RN. Moreover, we also improve the results of [4](Comm. Math. Phys.55(1997)149-162), in which they only obtained infinitely many solutions, however, they had not given the signs of these solutions.In this chapter, we assume that the following conditions hold:(A1) there exist p∈(2,2*) and c> 0 such that |f(x,t)|≤c(|t|+|t|p-1), (x,t)∈RN×R1;(A2) there existα> 2, R> 0 such that aF(x,t)≤tf(x,t), (x,t)∈RN×R1, (?) F(x,t)>0, where F(x, t)=∫0t f(x, s)ds,∈RN×R1(A3) limt→0 f(x, t)/t=0 uniformly on RN;(A4) f(gx, t)=f(x, t) for all g∈E O(N) and (x, t)∈RN×R1;(A5) f(x,-t)=-f(x,t) and tf(x,t)≥0 for all (x,t)∈RN×R1.Then we have the main theorem:Theorem 3.1.1. Suppose that (A1)-(A5) hold. Then the equation (3.1.1) has in-finitely many sign-changing solutions.In Chapter 4, we consider the Schrodinger-Poisson system where f∈C(R1,R1). We assume that f satisfies limt→∞f(t)/tp<+∞, and give some results about existence and nonexistence of solution for different A and p by using variational methods. We extend the results in [5](J. Funct. Anal.237(2006)655-674) about the nonlinear term f(u)= up to general nonlinearity f. And we improve the results of [6](Ann. I. Poincare-AN,27(2010)779-791), in which they obtained one positive radial solution for small parameterλ.In fact, we assume the following conditions:(H1) lim supt→+∞f(t)/tp<+∞for some p∈(1,5);(H2) limt→0 f(t)/t= 0;(H3) f(t)t≥4F(t), t∈R1;(H4) there exists q∈(2,5) such that lim inft→+∞f(t)/tq>0;(H5) lirnt→+∞f(t)/t=+∞, where, F(t)=∫0t f(s)ds. Then we have the next results. Theorem 4.1.1. Suppose that (H1) for some p∈(1,2), (H2) and (H5) hold, then there existsλ0> 0 such that system (4.1.1) has at least two positive radial solutions for allλ∈(0,λ0).Theorem 4.1.2. Suppose that (H1)-(H4) for some p∈(3,5) hold, then system (4.1.1) has at least one positive radial solution for allλ> 0.Theorem 4.1.3. Suppose that (H1) for some p∈[2,3], (H2) and (H5) hold, then there existsλ0>0 such that system (4.1.1) has at least one positive radial solutions for allλ∈(0,λ0).Theorem 4.1.4. Suppose that (H1) for some p∈(1,2] and (H2) hold, then there existsλ0> 0 such that system (4.1.1) has no positive radial solutions for allλ∈(A0,∞).The following table sums up the main results of this chapter.In Chapter 5, we study the generalized Kadomtsev-Petviashvili equation: wt+wxxx+(f(w))x=Dx-1wyy, where Dx-1h(x,y)=∫-∞x h(s,y)ds. Under some assumptions, we give the existence of nontrivial ground state solitary wave. We remove the following condition in [7](Appl. Math. Lett.15(2002)35-39)和[8](Minimax Theorems,1996):there exists v0∈Y:={gx:g∈C0∞(R2)} such that lims→+∞F(sv0)/s2→+∞.This is the condition that is usually necessary for studying generalized Kadomtsev-Petviashvili equation. Moreover, the corresponding functional may not satisfy (PS) con-dition or (C) condition.We assume that(B1) f∈C(R1,R1),f(0)=0 for some p∈(3,6),(B2)limt→0f(t)/t=0; (B3) there existsθ≥1 such thatθG(t)≥G(st), t∈R1, s∈[0,1], where G(t)= f(t)t-2F(t).Theorem 5.1.1. Suppose that (B1)-(B3) hold. Then (5.1.1) has a ground state solitary wave.In addition, we study the ground state solitary waves of variable-coefficient general-ized Kadomtsev-Petviashvili: (-uxx+Dx-2uyy+α(x,y)u-f(u))x=0, (5.4.1)where (x,y)∈R2. We assume thatα∈C(R2,R1) and there exitsα,β> 0 such that 0<α≤α(x, y)≤β<∞. Andα(x, y) satisfies the following condition:α(x,y)<α∞=(?)α(x,y)<∞, (x,y)∈R2. (α)Here, we also remove the periodic condition of [9](J. Math. Anal. Appl.361(2010)48-58), in which they studied the variable-coefficient p-Laplace equation.Then we have the next conclusion:Theorem 5.4.1. Suppose that (B1)-(B3) and (a) hold for some p∈(3,4], then (5.4.1) has a ground state solitary wave.