Study Of Solutions To Boundary Value Problems Of The P-Laplace Differential Equations

This thesis is a survey of the recent results in studying the boundary value problems of ordinary differential equations with p-Laplacian operators. We briefly overview the recent situation for studying the problems and also survey the results with concentration on some important equations of second order, some higher order equations, some systems and some boundary value problems of partial differential equations.We introduce the existence of positive solutions, the existence of multiple positive solutions to some boundary value problems of second order equations; introduced the existence of postive soluitons to higher order equations and systems; we also introduced the results to the existence of positive radial solutions to some partial differential equations of second order.The thesis consists of 5 Chapters. In Chapter 1, by reviewing the theory of ordinary differential equations and its development, we introduce the boundary vlaue problems of differential equaitons with p-Laplacian operatorse,introduce the background of the problems and outlined the basic concepts and related theorems.In Chapter 2, we introduce several important theorems including cone stretching and cone-compression fixed-point theorems, fixed-point index of completely continuous integral operators and a generalization of Mawhin continunity Lemma, and also the method in proving the existence of positive solutions to the boundary value problems of second order equations.We first list some necessary and sufficient conditions to the existence of positive solutions to the following boundary value problem with the use of cone-streching and cone compression fixed point theorems.Under the following hypothesis, Hold uniformly for t∈( 0,1),g ,h∈C(0,1). We haveTheorem 1. Assume C),set 0 <α<1, and then Problem (1)has a positive solution if and only if for any Next, we introduce the results for studying the existence of positive solutions to the following problem with the use of fixed point index to completely continuous integral operators, where - (l )= lp - 2 l,p>1,α>0,β≥0,γ>0,δ≥0 and f : [0,1]×R+×R→R+ is continuous, R = ( -∞,+∞),R+=[0,+∞).The main result is the following,Theorem 2. Assume that max ,>0 or f (t ,0,0)≠0. If there exists a d >0,with h ( d)≤(kd)p-1, then Problem (2) has at least one positive solution u *∈C1[0,1] satisfying u *≤d and (u *)'≤kd.At the end of this chapter, we introduce a generalization of Mawhin continuity lemma and show the method in proving the existence of solutions to the following problem whereΦp ( x )=xp- 2 x,p>1,f:[0 ,T]×R2→R is a continuous function,e∈L1 [0 ,T]. We assume that(a).There exist g , h,r∈L1 [0 ,T], such that for any [ ](b).Set fx ( s)= f(s ,x(s),x' (s)) +e(s),there is an M 1 >0,such that for x∈{x∈domM\ kerM:x(t)>M1 ,t∈[o ,T]},∫0 T fx (u )du≠0.(c). For any c∈R,set fc (t )= f(t,c,0)+e(t), there exists an M 2 >0, such that for t∈[0 ,T],if c > M2, one of Tc∫0 T fc (u )du<0and Tc∫0 T fc (u )du>0 holds.We obtain the following existence theorem.Theorem 3. Assume that (a)-(c) hold, if for p >2, 2 p - 2 g1+h1<1,then Problem(3) has at least one solution in domM∩Ω.In Chapter 3, we use Leray-Schauder principle and 3 functional3 fixed point theorems to introduce the existence of positive solutions to the boundary value problems of highter order equations and to some systems.We consider the following boundary value problems of the 4th order whereφp : R→R,φp(u)=up- 2u;f:[0,1]×R2→R,e:[0,1]→R are continuous.The main conclusion is the following theorem.Theorem 4. If there exist funcitonsα,β,γ∈C[0,1] such that then for 0 +0<1p andβ0 <1,Problem(4) has at least one solution in C 1[0,1].To the following bondary value problem where,Φp ( s )=sp- 2 s,p>1is the p-Laplacian operator, a1 (t ),a2(t) may be singular at t=0 or t=1. Ai , Bi are continuous and nondecreasing functions defined on ( -∞,+∞).We make the following hypotheses, (H1) f , g∈C[[0,∞)2 ,[0,∞)]; (H2) Ai , Bi are nondecreasing odd functions defined in ( -∞,+∞), and there exist constants l1 , l2,L1,L2 such that(H3) ai (t) is [ 0,1] a nonnegative measu r a ble function,which may be singular at t =0,1,and satisfies that 0 <∫01 ai (t)dt<∞,i=1,2.It is easy to know from (H3) that there is a constantμ∈[0,12], such that Then, we haveTheorem 5. Assume (H1)(H2)(H3),and there is a constantμ∈---0,12---,a , b,c>0 with 0 < a φp (bξ1 )g(u,v)<φp(bξ2)0≤t≤1,0≤u+v≤bμ, (H6) f (u ,v)<φp (aη1 )g(u,v)<φp(aη2)0≤t≤1,0≤u+v≤a. Then Problem (5)-(6) has at least 3 positive solutions ( x1 ,y1), ( x 2 ,y2),(x3,y3) satisfying In chapter 4, we study the following problem whereφp( s) is the p-Laplacian operator , i.e.,φp( s )= sp- 2 s,p>1, Assume that (H1) f is nonnegative and continuous on [0 , +∞); (H2) There is a sequence { }t+1 < t, t1<12,l→im∞t=t-≥0,li→ma(t)=∞,i=1,2,,Tih e miain thteoriem is tti and 0 <∫01 a (t)dt<∞.Theorem 6. Assume (H1),(H2),picking { } (t k+1 ,tk), k=1,2,, picking { }rk k=1 and { }R kk=1 satisfying R k +1 <θkrk 0,α(s 2)s is monotonically increasing and an odd mapping homotopic to R.We assume to the functionsφ,f,g the following, ( )A1 For each constant c >0,there is a positive number A( c)>0,such thatφ-1 ( cz )≥A(c)φ-1(z) for all z≥0,and A( c)→∞as c→∞,(conversely, there is a constant B ( c)= A(11c)>0, such thatφ-1 ( cz )≥B(c)φ-1(z) for z≥0, and B ( c)→0 as c→0+). ( A2 ) f, g:[0 ,∞) 2→R is continuous and nondecreasing,and there is a constant M >0, such that f (u , v)≥-M2, g (u , v)≥-M2 for all The main result is the followingTheorem 7. Assume ( )A1 - ( )A4 ,then forλsuitably large,Problem(8)has a positive radial solution...