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1 ENTIRE BLOW-UP SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS AND SYSTEMS THESIS Jesse D. Peteson, Second Lieutenant, USAF AFIT/GAM/ENC/8-2 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wight-Patteson Ai Foce Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

2 The views expessed in this thesis ae those of the autho and do not eflect the official policy o position of the United States Ai Foce, Depatment of Defense, o the United States Govenment.

3 AFIT/GAM/ENC/8-2 ENTIRE BLOW-UP SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS AND SYSTEMS THESIS Pesented to the Faculty Depatment of Mathematics and Statistics Gaduate School of Engineeing and Management Ai Foce Institute of Technology Ai Univesity Ai Education and Taining Command In Patial Fulfillment of the Requiements fo the Degee of Maste of Science in Applied Mathematics Jesse D. Peteson, BS Second Lieutenant, USAF Mach 28 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

4 AFIT/GAM/ENC/8-2 ENTIRE BLOW-UP SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS AND SYSTEMS Jesse D. Peteson, BS Second Lieutenant, USAF Appoved: D. Aihua W. Wood Thesis Adviso D. Alan V. Lai Committee Membe D. Matthew Ficus Committee Membe Date Date Date

5 AFIT/GAM/ENC/8-9 Abstact We examine two poblems concening semilinea elliptic equations. We conside α β single equations of the fom Δ u = p( x) u + q( x) u fo < α β 1 and systems Δ u = p( x ) f( v), Δ v= q( x ) g( u), both in Euclidean n -space, n 3. These types of poblems aise in steady state diffusion, the electic potential of some bodies, subsonic motion of gases, and contol theoy. Fo the single equation case, we pesent sufficient conditions on p and q to guaantee existence of nonnegative bounded solutions on the entie space. We also give altenative conditions that ensue existence of nonnegative adial solutions blowing up at infinity. Similaly, fo systems, we povide conditions on p, q, f, and g that guaantee existence of nonnegative solutions on the entie space. The main equiement fo f and g will be closely elated to a gowth equiement nown as the Kelle-Osseman condition. Futhe, we demonstate the existence of solutions blowing up at infinity and descibe a set of initial conditions that would geneate such solutions. Lastly, we examine seveal specific examples numeically to gaphically demonstate the esults of ou analysis. iv

6 Acnowledgments Fist and foemost, than you to my family and fiends. I geatly appeciate all the suppot you have given me. I cedit you completely fo eeping me sane duing the peiods of seemingly endless studying. I especially than my wife. The suppot you have given me, both local and long distance, has been immeasuable. To D. Flint and D. Schaal at South Daota State Univesity, I than you fo convincing me to tae on an additional undegaduate degee in mathematics. I cetainly would not be hee today without you guidance duing my undegaduate wo. I extend my deepest thans to D. Aihua Wood, my thesis adviso. I have leaned much about elliptic theoy and analysis, and I geatly appeciate you mentoship. You consistently challenge me and desie me to be at my best. I also than D. Matthew Ficus and D. Alan Lai. The advice you have povided, both fo this thesis and in the classoom, has been invaluable. I owe any success at AFIT to you. Than you all! Jesse D. Peteson. v

7 Table of Contents Page Abstact... iv Acnowledgments... v Table of Contents... vi List of Figues... vii List of Tables... viii I Intoduction II Bacgound Single Equations Systems Peliminaies III Single Equations Peliminaies Main Results Examples IV Systems Peliminaies Main Results Examples V Conclusion Summay Futhe Wo Bibliogaphy... Bib 1 Vita... Vita 1 vi

8 List of Figues Page Figue 3-1. Numeical Solutions of (3.32) fo Vaious Cental Values Figue 3-2. Numeical Solutions of (3.34) fo Vaious Cental Values Figue 3-3. Numeical Solutions of (3.35) fo Vaious Cental Values Figue 4-1. Entie Solution Existence Region fo Cental Values of (1.2) Figue 4-2. Numeical Solutions of (4.4) fo Vaying Cental Values Figue 4-3. Numeical Solutions of (4.4) fo Cental Values Nea Bounday Figue 4-4. Entie Solution Existence Region fo Cental Values of (4.4) Figue 4-5. Non-monotonic Function f of System (4.41) Figue 4-6. Numeical Solutions of (4.41) fo Cental Values Nea Bounday Figue 4-7. Numeical Solutions of (4.41) fo Vaying Cental Values Figue 4-8. Existence Region fo Cental Values of (4.41) Figue 4-9. Solutions Showing Set of Cental Values fo (4.41) is Not Convex vii

9 List of Tables Page Table 2-1. Existence of Solutions fo Multiple Tem Single Equation (1.1) Table 5-1. Existence of Solutions fo System (1.2) viii

10 ENTIRE BLOW-UP SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS AND SYSTEMS I. Intoduction Ou wo is divided into two pats. Fist, we examine the semilinea elliptic equation u p( x) u α β Δ = + q( x) u, x R n (1.1) fo n 3. This poblem was the focus of the thesis by Smith [26] in which multiple existence and nonexistence esults wee obtained. We examine this poblem only biefly in an effot to close a few emaining gaps. The aguments we pesent ae simila to those used in [26] and the wos efeenced theein. examine The second pat of ou study is dedicated to semilinea elliptic systems. Namely, we Δ u = p( x ) f( v), Δ v = q( x ) g( u), x n Ω R (1.2) whee again n 3. Hee we ae esticting ou poblem to the adial case. Seveal authos have studied this system fo monotonic f and g. We will conside a moe geneal system whee f and g may be non-monotonic. To ou nowledge, no esults exist fo system (1.2) when f and g ae not monotone. This genealization pesents seveal unique challenges that we must addess. 1-1

11 Fo both single equations and systems, we ae pimaily concened with poving the existence of lage solutions. A solution to (1.1) is lage if ux ( ) as x Ω. If n Ω= R, we equie ux ( ) as x. The latte case is called an entie lage solution. Similaly, a lage solution to system (1.2) is a solution ( uv, ) such that u( x ) and v( x ) as x Ω, and an entie lage solution is one such that u( x ) and v( x ) as x. The aguments developed in ou analysis and the mathematical foundations of elliptic theoy may be applied to multiple poblems in a wide vaiety of technical fields. Elliptic equations simila to those we analyze hee ae elated to steady-state eaction-diffusion, subsonic fluid flows, electic potentials of some bodies, and contol theoy. Fo example, [17] descibes a geneal stochastic diffusion pocess with feedbac contols. The contols ae to be designed so that the state of the system is constained to some egion. Finding optimal contols is then shown to be equivalent to finding lage solutions fo a second ode semilinea elliptic equation. As anothe example, [8] models the steady state of non-linea heat conduction though a 2-component mixtue with a system of semilinea elliptic equations simila to the systems we study hee These ae just two examples of application. The mathematical methods and ideas we use and develop hee ae applicable in many scientific and engineeing disciplines. Befoe we discuss peliminaies fo ou wo, let us fist examine esults obtained by othes in this field. 1-2

12 II. Bacgound poblem Biebebach [3], in 1916, was the fist to study lage solutions to the semilinea elliptic Δ u= f( u), x Ω (2.1) u whee f ( u) = e. Since that time, many authos have studied elated poblems fo single equations and systems. In this section we will pesent only the most ecent and/o most elevant accomplishments which have led to ou study. 2.1 Single Equations In 1957, Kelle [11] and Osseman [21] established necessay and sufficient conditions fo the existence of solutions to (2.1) on bounded domains in solutions exist on Ω if and only if f satisfies n. They showed that lage 1/2 t f () s ds dt <. (2.2) 1 This equiement is sometimes efeed to as the Kelle-Osseman condition and continues to be significant in cuent studies. Indeed, we will equie seveal of ou functions to satisfy this Kelle-Osseman condition. Bandle and Macus [2] late examined the equation Δ u= p( x) f( u) (2.3) 2-1

13 fo f non-deceasing on [, ) and poved the existence of positive lage solutions povided the function f satisfies (2.2) and p is continuous and stictly positive on Ω. Lai [12] showed the same esults hold fo (2.3) when p is allowed to vanish on lage pats of Ω, including its bounday. Lai, Poano, and Wood [16] elaxed the monotonicity condition on f by equiing that thee exist some nonnegative, nondeceasing, Hölde continuous function g and positive constants M and s such that g() s f() s Mg() s fo all s s. (2.4) We note that ou wo on systems is a simila genealization; howeve, the aguments equied fo ou analysis ae vey diffeent fom those used in the single equation case. Many authos have examined moe specific foms of (2.3). The equation Δ u = p( x) u γ (2.5) has been of paticula inteest. Cheng and Ni [4] consideed the supelinea case ( γ > 1) and poved that (2.5) has lage solutions on bounded domains povided p is stictly positive on Ω. Lai and Wood [14] genealized this to allow p to vanish on lage potions of Ω including its bounday. They also showed the existence of an entie lage solution to (2.5) povided that max p( x) d <. (2.6) x= This is a weae condition compaed to the equiements in [4]. 2-2

14 Fewe esults ae nown fo the sublinea case (< γ 1) of (2.5). In [15], Lai and Wood poved that entie lage adial solutions of (2.5) exist if and only if p () d =. (2.7) They also demonstated that fo a bounded domain Ω, (2.5) has no positive lage solution when p is continuous in Ω. In addition to lage solutions, Lai and Wood consideed entie bounded solutions. They poved entie bounded solutions of (2.5) exist when (2.7) holds and p is locally Hölde continuous. Also, nonnegative, entie bounded solutions do not exist fo (2.5) when min p( x) d =. (2.8) x= Smith [26] then consideed (1.1), which is a multi-tem adaptation of the single tem equation (2.5). Fo the supelinea (1 < α β) and mixed ( < α 1 < β) cases, esults compaable to those fo single equations in [14] wee obtained. In the sublinea case, Smith [26] poved the existence of entie lage adial solutions and showed the existence of entie bounded solutions fo the nonadial poblem. Howeve, both of these poofs equie < α β < 1. The poofs do not hold when β = 1. Table 2-1 is a summay of the esults fom [26]. Ou wo on single equations emoves condition (f) fom the table, theeby closing the gap fo β =

15 Table 2-1. Existence of Solutions fo Multiple Tem Single Equation (1.1) u p( x) u β Δ = + q( x) u Supelinea/Mixed < α β, β > 1 Sublinea < α β 1 Sublinea < α β 1 Bounded Domain n Ω Entie Domain n Ω= Bounded Domain n Ω Entie Domain n Ω = Entie Domain n Ω= Lage Solutions/ Requiements Yes a Yes a,b No a) p, q ae c-positive (see Definition 4-2) b) max p( x) d < and x= max q( x) d < x= Yes c,d,f c) adial case only: p( x) = p( x) and qx ( ) = q( x) d) e) max p( x) d = o x= min p( x) d = o x= f) β 1 max q( x) d = x= min q( x) d = x= Bounded Solutions Yes b,f No e 2.2 Systems Next we shall discuss the bacgound of semilinea elliptic systems of the type given in (1.2). While systems ae a natual extension of single equations and occu in many of the same aeas of application, not all the methods employed to study them cay ove. Fo instance, we do not have a meaningful maximum pinciple fo systems. Additionally, the baie method, which we pesent late, is a common method fo poving the existence of solutions fo single equations. Howeve, a simila method fo systems exists only when athe estictive conditions ae placed on the gowths of the given functions. Fo example, if we conside systems built fom equations compaable to (2.1), we have Δ u = f( u, v), Δ v = g( u, v). 2-4

16 A baie method simila to that fo single equations exists only when f is monotonic in v, and g is monotonic in u. Still, this aea has been well studied by seveal authos. Most elated to ou study, ae the wos of Lai and Wood [13], Cistea and Radelescu [6], and Peng and Song [22]. Lai and Wood [13] examined the system ( ) ( ) α Δ u = p x v, β Δ v = q x u, x n Ω R (2.9) whee n 3. They poved the existence of entie nonnegative solutions and chaacteized the set of cental values, S + +, fo these solutions. By cental values, we mean ( u(), v ()) whee uv, ae solutions of (2.9). Lai and Wood poved S is closed, bounded, and convex. They also futhe geometically chaacteized this set by descibing bounds fo S. Finally, they poved that lage solutions exist fo cental values that lie in the closue of {( ab, ) S: ab, }. We note that fo αβ>, 1, they equied tp() t <, tq() t <. (2.1) Both Peng and Song [22] and Cistea and Radulescu [6] genealized thei wo to show compaable esults fo system (1.2) whee f, g C[, ) ae nonnegative monotonic functions. The appoaches taen in each pape wee slightly diffeent, and each pai of authos equied diffeent additional constaints on f and g. In [6] seveal cases wee consideed. When 2-5

17 gcf ( ( t)) lim = fo all c >, t t (2.11) system (1.2) has entie solutions, all of which ae bounded when (2.1) holds. Futhe, entie solutions exist and ae lage when neithe inequality in (2.1) hold. Both papes then consideed (1.2) whee the nonnegative functions pq, C[, ) satisfy (2.1), and functions f, g C[, ) ae nondeceasing, satisfy the Kelle-Osseman condition (2.2), and f() = g() =, f() s >, g() s > fo s >. (2.12) In [22], these functions wee futhe equied to be convex, while in [6], thee wee additional conditions f g 1, C [, ) and f() s lim inf = σ >. (2.13) s gs () In these cases, both pais of authos showed existence of entie lage solutions and chaacteized the set of cental values fo which such solutions exist. Ou esults on systems futhe genealize this poblem by allowing f and g to be nonmonotonic. Instead, we will equie the function G, given by { } Gs ( ) = min min f( t),min gt ( ), s, (2.14) s t s t to satisfy the Kelle-Osseman condition in (2.2). Note this automatically implies f and g must satisfy this condition as well. 2-6

18 systems. Now we discuss peliminay theoy that will be necessay fo both single equations and 2.3 Peliminaies Fo eadability, we shall eseve the tems lemma, theoem, and coollay fo esults which we justify in ou wo. Any esult taen diectly fom anothe efeence shall be labeled as Peliminay X-X. The Azela-Ascoli Theoem is ou main tool fo examining systems; howeve, it is also needed fo a single equation esult. Hee we give the Azela-Ascoli Theoem and seveal necessay definitions. Definition 2-1 A subset K of a nomed space X is called pecompact (sometimes called elatively compact o conditionally compact) if its closue K is compact in the nom topology of X. Peliminay 2-1 (Theoem 1.34 of [1]) (Azela-Ascoli Theoem) Let Ω be a bounded domain in n. A subset K of C( Ω ) is pecompact in C( Ω ) if (i) Thee exists M such that φ( x) M fo evey φ K and x Ω. That is, K is unifomly bounded. (ii) Fo evey ε >, thee exists δ > such that φ( x) φ( y) < ε fo all φ K, xy Ω,, and x y < δ. That is, K is equicontinuous. 2-7

19 To apply the Azela-Ascoli Theoem, we will often show that a sequence of functions is both unifomly bounded and equicontinuous. We will need to accomplish this fo seveal diffeent sequences thoughout ou aguments. Howeve, these sequences will have similaities, and ceating the following lemma will geatly steamline ou wo in late sections. This lemma will allow us to conclude a paticula sequence is both unifomly bounded and equicontinuous only by showing unifom boundedness. Lemma 2-1 Let { u } be a sequence of functions of the fom t 1 1 ( ) n n = + ( ) ( ( )), [, ]. (2.15) u a t s p s f v s dsdt R whee a +, p, f C[, ) ae nonnegative, and { v } is an abitay sequence of nonnegative continuous functions on [, R ]. If the sequence { v } is unifomly bounded on [, R ], then { u } is equicontinuous on [, R ]. Poof. Since { } [, R]. Then we have v is unifomly bounded, thee exists M such that v () M fo all and all 2-8

20 () = () ( ()) u s p s f v s ds p() s f( v ()) s ds t M ps ()max( f()) t ds t M max( f( t)) p( s) ds C <. R Also, we clealy have u ( ). Thus, { u } is equicontinuous on [, R ], and ou poof is complete. These ae the basic ideas needed fo both the single equation and system cases. Now we tun ou attention to peliminaies equied exclusively fo ou single equation aguments. 2-9

21 III. Single Equations 3.1 Peliminaies We begin by pesenting a esult on the baie method o uppe-lowe solution appoach. This method is well-nown fo equations on bounded egions (see Theoem of Sattinge [24]). Howeve, we wish to apply the baie method on the unbounded domain n. We theefoe use the following esult (Peliminay 3-1) fom Shae [25]. We also povide definitions elated to this esult. Definition 3-1 Let f be a function defined on an open set n Ω. Fo α 1 <, we say that f is Hölde continuous with exponent α, witten f exists a nonnegative constant C such that C α on Ω, if fo evey xy Ω,, thee f ( x) f( y) C x y α. When α = 1, we say f is Lipschitz continuous. Futhe, we say that f is locally Hölde (Lipschitz) continuous on n Ω when fo evey x Ω, thee exists an open ball B( x, ) such that f esticted to B( x, ) is Hölde (Lipschitz) continuous. Peliminay 3-1 (Lemma 3 of [25]) (Lemma on Baie Method) Let u1, u2: n, u ( x) u ( x) fo all 1 2 n x be such that Lu + f ( x, u ), 1 1 Lu + f ( x, u ),

22 whee f is locally Hölde continuous in ( x, u ) and locally Lipschitz in u, and L is an elliptic opeato of second ode. Then thee exists a solution u of Lu + f ( x, u) = with u1 u u2. The Laplacian is a second ode elliptic opeato, and theefoe Peliminay 3-1 applies diectly to equation (1.1). Definition 3-2 Given Peliminay 3-1, we call u 1 an uppe solution and u 2 a lowe solution to Lu + f ( x, u) =. As mentioned with the Azela-Ascoli Theoem, we will often be woing with sequences of functions. In many of ou poofs, we will be attempting to put bounds on these functions. One useful inequality fo doing so is Gonwall s Inequality o Gonwall s Lemma. Peliminay 3-2 (Theoem of [1]) (Gonwall s Inequality) Let t t t 1. Let ψ () t and φ () t be continuous functions such that ψ () t and t φ() t K ψ()() s φ s ds + t holds fo t t t 1 and K a constant. Then t φ() t Kexp ψ() s ds t 3-2

23 fo t t t Main Results We now pesent ou main esults fo single equations. Ou fist esult is an extension of Theoem 22 of [26] which equied < α β < 1. We adopt a simila agument to obtain esults fo < α β 1. Theoem 3-1 Suppose p and q ae nonnegative locally Hölde continuous in n and max p( x) d <, x = max q( x) d <. (3.1) x = Then (1.1) has a nonnegative nontivial entie bounded solution in <. n if α β 1 Poof. Since [26] shows the case fo < α β < 1, we will assume < α β = 1. Define x = { p x q x } θ () = max (), (). (3.2) We will constuct uppe and lowe solutions fo (1.1) by examining the equation n Δ z = θ( )( z α + z β ) = θ( )( z α + z), = x. (3.3) Since this equation is adial, we can ewite the Laplacian in adial fom. Doing so, equation (3.3) is equivalent to the odinay diffeential equation 3-3

24 n 1 α z () + z () = θ ()( z + z), [, ). (3.4) Multiplying each side of (3.4) by n 1 yields N 1 () + () = ( () ) = ()( + ). α z z z θ z z Integating twice, we obtain t α () = + θ ()( + ), (3.5) z c t s s z z dsdt whee c= z() is ou cental value. Theefoe a solution to (3.3) is any fixed point of the opeato T : C[, ) C[, ) defined by t α () ()( ),. (3.6) Tz = c + t s θ s z + z dsdt Note the integation in this opeato implies a fixed point z C[, ) is in classical solution to (3.3). We will show thee exists such a fixed point Tz bounded. Let z () c = fo all, and define the sequence C 2 [, ), maing z a = z and that z is t α = 1 = + θ 1+ 1 (3.7) z () Tz () c t s ()( s z z ) dsdt fo = 1, 2,. Induction shows this sequence is inceasing. Clealy 3-4

25 t α = + θ + = 1. z () c c t s ()( s z z ) dsdt z () Then, assuming z z + 1, we have t α + 1() = + θ ()( + ) z c t s s z z dsdt c + t s θ ()( s z + z ) dsdt = z + 2 t (). α Hence { z } is inceasing. We now show { z } is unifomly bounded. Integating, (3.7) becomes t α () = + θ ()( 1+ 1) z c t s s z z dsdt t α c + t s θ ()( s z + z ) dsdt t= t 2 n α 2 n α θ θ n t= 1 1 = c+ t s ()( s z + z ) ds t t ()( t z + z ) dt. 2 n 2 (3.8) By L Hopital s ule, we have t t 2 n α α n 2 lim t s θ( s)( z z) ds lim s θ( s)( z z) ds/ t t + = t + = lim t θ ( t)( z + z ) /( n 2) t t α n 3 = + 2 α lim t θ ( t)( z z) /( n 2) t =. (3.9) Thus (3.8) and (3.9) imply 3-5

26 t= t 1 2 n α 1 2 n α ( ) + θ( )( + ) θ( )( + ) 2 n 2 n t= z c t s s z z ds t t t z z dt 2 n α t α θ θ n 1 = c+ s ( s)( z + z ) ds ( t)( z + z ) dt 2 n 2 1 α 2 n α = c+ tθ( t)( z + z) dt s θ( s)( z + z) ds n 2 1 c+ n 2 tθ ( t)( z α + z ) dt. (3.1) We will conside the domain of z in two intevals. Notice that α () = θ ()( 1+ 1) z s z z ds. (3.11) Thus, we may choose such that z () 1 fo [, ], z ( ) 1 fo [, ). (3.12) It is possible that = o =. Indeed, since z (), we see fom (3.12) and (3.5) that if ou cental value c > 1, then =. If c < 1, then eithe = with z ( ) 1 fo all o < with z( ) = 1. We ae attempting to bound { z }, and theefoe = is tivial. Theefoe if we conside <, and split ou domain, (3.1) becomes 3-6

27 1 α 1 α + θ + + θ + n 2 n 2 z () c t ()( t z z ) dt t ()( t z z ) dt 2 1 c+ tθ() t dt+ tθ()(2 t z ) dt n 2 n c+ tθ() t dt+ tθ() t zdt n 2 n 2 2 = R+ tθ () t zdt n 2 whee R is a constant because (3.1) implies tθ () t dt <. (3.13) Finally, Gonwall s inequality (Peliminay 3-2) yields 2 z() R+ tθ () t zdt n 2 2 R exp tθ ( t) dt n 2 2 R exp tθ ( t) dt n 2 = M < which is finite due to (3.13). We have shown that { z } is a unifomly bounded monotonic sequence. Theefoe it conveges pointwise to some z. Futhe, since { z } is in the fom of (2.15), Lemma 2-1 implies that { z } is also equicontinuous. Pointwise convegence and equicontinuity imply unifom convegence, and thus z C 2 [, ). We have a fixed point of 3-7

28 (3.6) and bounded solution of (3.3). Finally, we show this function z and its bound M fom uppe and lowe solutions to (1.1). Let u1 M z u2. Clealy, so Δ u = pu + qu, α Δ u = θ ( u + u ) pu + qu, α α Δu ( pu + qu ), α Δu ( pu + qu ). α Hence, thee exists a positive nontivial entie bounded solution u of (1.1) such that M = u1 u u2 = z by the Baie Method (Peliminay 3-1). when β = 1. We will also conside Theoem 21 in [26] fo < α β 1. This again fills the gap fo Theoem 3-2 Suppose α β 1 < and that px ( ) = p( x ) C( ) and ( ) qx ( ) = q x C( ) such that p and q ae nonnegative. Then equation (1.1) has an entie lage positive solution if and only if p () d = o q () d =. (3.14) 3-8

29 Poof. Again, since Smith [26] poved this esult fo < α β < 1, we will conside < α β = 1. An agument fo necessity identical to that in Theoem 21 of [26] will wo fo β = 1. We ae left to show sufficiency. Radial solutions of (1.1) satisfy the odinay diffeential equation n 1 α β u () + u () = p( x ) u + q( x ) u. (3.15) As in Theoem 3-1, it follows that (3.15) has nonnegative solutions if the opeato T : C[, ) C[, ) defined by t α ( ) (3.16) Tu() = c + t s p() s u + q() s u dsdt has a fixed point in C[, ) whee u() = c is ou cental value. We begin by showing that (3.16) has a fixed point in C[, R ] fo abitay < R <. Simila to ou pevious poof, we let u = c and define the sequence α ( ), (3.17) u = Tu () = c + t s p() s u + + q() s u dsdt 1 t = 1, 2,. The same induction agument fom Theoem 3-1 shows the sequence { u } is nondeceasing. We will now pove that { u } in unifomly bounded and equicontinuous on R. Define 3-9

30 { p s q s } θ () s = max (), (). (3.18) Integating as we did in (3.8) and (3.1), we have 1 α θ + n 2 u () c s ()( s u u ) ds, R. (3.19) Also simila to the pevious poof, we have u, so we may choose such that u ( ) 1 fo [, ], u ( ) 1 fo [, R]. (3.2) We ae attempting to obtain a bound, and thus we need only conside finite and Splitting ou integal fom (3.19) into two pieces, we have by (3.2). 1 α θ + n 2 u () c s ()( s u u ) ds 1 α 1 α = c+ sθ()( s u + u) ds+ sθ()( s u + u) ds n 2 n c+ sθ() s ds+ sθ()(2 s u ) ds n 2 n 2 = c+ h() s ds+ h() s u ds whee hs () = 2 sθ ()/( s n 2). Since { u } is an inceasing sequence of inceasing functions, we must have + 1 fo all =,1,. Thus, fo all which yields 3-1

31 u+ 1() c+ h() s ds+ h() s uds c+ h() s ds+ h() s u ds = W + h() s u ds. (3.21) We now use induction to show u + 1() W exp h() s ds. (3.22) Clealy we have. u = c c+ h() s ds exp h() s ds = W exp h() s ds Now, assume that (3.22) is tue fo abitay. Then fo + 1, (3.21) implies, u W + + h() s u () s ds 1 s W + h() s W exp h() t dt ds s = W + W exp h( t) dt = W + W exp h( s) ds W = W exp h( s) ds. 3-11

32 Thus, (3.22) is tue by induction. Finally, since each u is inceasing, we aive at R u () W exp h() t dt W exp h() t dt M R, R. (3.23) That is, { u } is unifomly bounded on [, R]. Notice that ou sequence is of the same fom as (2.15). Theefoe { u } is also equicontinuous on [, R] by Lemma 2-1. Since { u } is a monotonic, unifomly bounded, equicontinuous sequence of functions on [, R ], u [, R], and we have a fixed point of (3.16) in C[, R ]. u unifomly. That is Tu = u fo Next, we extend this esult to show that (3.16) has a fixed point in C[, ). We do so using a diagonal agument simila to the agument in Theoem 1 of [15]. Define the sequence of fixed points { w } by Tw = w on [, ], w C[, ]. (3.24) Resticted on [,1], { w } is bounded by M 1 as given in (3.23). Using Lemma 2-1, we can also show it is equicontinuous on this inteval. Thus, the Azela-Ascoli Theoem (Peliminay 2-1) 1 implies that thee exists a subsequence, call it { w }, that conveges unifomly on [,1]. Let w 1 v unifomly on [,1] as. (3.25) 1 1 Similaly, the sequence { w } is bounded and equicontinuous on the inteval [,2]. Hence, it 2 must contain a convegent subsequence { w } that conveges unifomly on [,2]. Let 3-12

33 w 2 v unifomly on [,2] as. (3.26) Note that { w} { w} { w} = 2. This implies v2 = v1 on [,1]. Continuing, we have and a sequence { v } such that n 1 { w} { w} { w} = n v ( ) C[, ] = 1,2, v ( ) = v ( ) fo [,1] 1 v ( ) = v ( ) fo [,2] 2 v ( ) = v ( ) fo [, 1]. 1 (3.27) Thus, we obtain a sequence { v } that conveges to v on [, ) satisfying v () = v() if. (3.28) This convegence is unifom on bounded sets, implying v C[, ). We theefoe have ou fixed point Tv = v of (3.16) in C[, ), and equation (1.1) has an entie adial solution. We lastly must show that this solution is lage. We note the agument of Theoem 1 in [15] demonstates that (3.14) implies t t s ( ps ( ) + qs ( )) dsdt=. (3.29) Ou solution satisfies 3-13

34 t α () () () u = c+ t s psu + qsu dsdt α { } [ ] c+ min u (), u() t s p( s) + q( s) dsdt α { } [ ] = c+ min u (), u() t s p( s) + q( s) dsdt. t t (3.3) Theefoe, letting, (3.29) and (3.3) imply lim u ( ) =, and we have a lage solution. This concludes ou poof. α The condition β = 1 emained an open poblem because compaisons between u ( ) β and u ( ) ae quite diffeent depending whethe o not u ( ) < 1 o u ( ) > 1. To esolve this issue, we simply split the domains of ou functions into two intevals and caied out the compaisons sepaately. 3.3 Examples In this section we pesent seveal examples of equations that satisfy the estictions given in ou analysis. We also solve seveal examples numeically. Note that Theoem 3-2 only applies to adial equations while Theoem 3-1 applies to moe geneal nonadial poblems. Fo simplicity, all ou examples will be adial. Examining Theoem 3-1, we must choose functions p, q that satisfy (3.1). That is, 2 equation (1.1) must contain nonnegative functions pq, C[, ) that decay at faste than 1/ s as s. Fo example, ( 1 ) α ( 2) 3 4 β Δ u = x + u + x + u o 2 x α x β u u 3 u Δ = + (3.31) 3-14

35 ae two vey simple equations that meet this equiement. Accoding to ou analysis, fo < α β 1, entie bounded solutions exist fo equations of this fom. Choose α = 1/2, β = 1, and conside 1/2 2 x x u Δ = u + 3 u. (3.32) Using a Runge-Kutta algoithm, we will solve this poblem numeically. Figue 3-1 shows these solutions fo a vaiety of cental values as initial conditions. Indeed, we have bounded entie solutions. Figue 3-1. Numeical Solutions of (3.32) fo Vaious Cental Values Next, we conside examples fo Theoem 3-2. We now must have p, g satisfy (3.14). A few simple examples would be 3-15

36 2 u x u α β Δ = + x u, 2 x x u u α β Δ = + 3 u, ( ) ( 1 ) α ( 1 ) ( 1/ 1 ) α ( 1/ ( 1 )) β Δ u = + x u + + x u, o 2 2 β Δ u = + x u + + x u. (3.33) Again, we will choose to examine some of these equations numeically. Conside 1/2 ( 1/ ( 1 )) ( 1/ ( 1 )) Δ u = + x u + + x u. (3.34) We examine a vaiety of cental values between and 1, and plot ou numeical solutions in Figue 3-2. Figue 3-2. Numeical Solutions of (3.34) fo Vaious Cental Values We show esults fo x 1 and clealy obseve solutions gow quicly. We then show the same solutions in x 5. As we examine lage values fo x, ou solutions continue to be defined. This is exactly what we would expect; ou analysis guaantees these solutions ae entie and lage. We will examine anothe inteesting equation. Conside ( 1 ) α ( 1 ) 2 2 β Δ u = + x u + + x u. (3.35) 3-16

37 Note fo this example p() q() 1/(1 ) 2 = = + satisfies (3.14) even though p = q = + =. The integand decays to zeo, but the value of the lim ( ) lim ( ) lim (1/(1 2 ) ) integal in (3.14) still appoaches infinity as. Solutions ae shown fo a vaiety of cental values in Figue 3-3. Figue 3-3. Numeical Solutions of (3.35) fo Vaious Cental Values Ou solutions gow much slowe. Fom the plot, it is not clea that solutions ae lage. Howeve, ou analysis guaantees that evey solution goes to infinity as x. 3-17

38 IV. Systems We now conside semilinea elliptic systems as given in (1.2). As with ou section on single equations, we begin by pesenting peliminay mateial essential fo ou study of systems. 4.1 Peliminaies We fist povide seveal definitions. We define a set of cental values, what it means fo a function to be cicumfeentially positive, and we constuct impotant functions. Definition 4-1 (Defined in Theoem 1 of [13]) Given system (1.2), we define the set of cental values S as + + S {( a, b) : u() = a, v() = b, and ( u, v) is an entie solution to (1.2)}. (4.1) Definition 4-2 (Condition A in [12]) A function p is said to be cicumfeentially positive (cpositive) on a domain Ω if fo any x Ω satisfying px ( ) =, thee exists a domain Ω such that x Ω, Ω Ω, and px ( ) > fo all x Ω. Definition 4-3 Given system (1.2), we define { } { } ( x ) = p( x ) q( x ) ( x ) = p( x ) q( x ) ψ min,, φ max,. (4.2) ψ We biefly note that these functions have some useful popeties. Namely, ( x ) p( x ) φ ( x ), ψ ( x ) q( x ) φ ( x ) and φψ, ae both nonnegative and c- 4-1

39 positive when pq, ae nonnegative and not identically zeo at infinity. Also, we note that when pq, satisfy (2.1), we have ( ) tψ() t dt tφ() t dt t p() t + q() t dt = tp() t dt + tq() t dt <. Thus ψ, φ also satisfy (2.1). Definition 4-4 Given system (1.2), we define { } s t s t { } Gs ( ) = min min f( t),min gt ( ), s, H ( s) = max max f ( t), max g( t), s. t s t s (4.3) These functions have some vey impotant chaacteistics as well. We demonstate these popeties in the following simple lemma. Lemma 4-1 Let f, g C[, ) satisfy (2.12), and suppose G, shown in (4.3), satisfies the Kelle-Osseman condition (2.2). Then G and H as given in (4.3) ae continuous, nondeceasing, and satisfy Gs () f() s Hs (), Gs () gs () Hs (). (4.4) Futhe, G and H satisfy (2.2) and (2.12). 4-2

40 Poof. It is clea G and H ae continuous, nondeceasing, and satisfy (2.12) fom definition. It is also clea that Gs () f() s Hs (), Gs () gs () Hs (). (4.5) Function G satisfies the Kelle-Osseman condition (2.2) by hypothesis. Then, since Gs () Hs (), we have 1 1 1/2 t 1/2 t H () s ds dt G() s ds dt <. Thus H satisfies the Kelle-Osseman condition, and ou poof is complete. As we develop an agument fo the existence of entie lage solutions fo system (1.2), we will fist show that solutions exist on bounded domains fo cental values outside of S (given in (4.1)). In ode to pove this, we will utilize a vesion of Schaude Fixed Point Theoem. Lemma 4-2 is the vesion we will apply and is a vey simple consequence of this theoem. Peliminay 4-1 (Coollay 11.2 of [9]) (Coollay of Schaude s Fixed Point Theoem) Let G be a closed convex set in a Banach space B, and let T be a continuous tansfomation of G into itself such that the image TG is pecompact. Then T has a fixed point. Definition 4-5 A linea tansfomation between two Banach spaces is called compact (sometimes called completely continuous) if the images of bounded sets ae pecompact. 4-3

41 Peliminay 4-2 (Theoem of [19]) Any compact linea tansfomation between Banach spaces is continuous. Lemma 4-2 Let G be a closed, convex, and bounded set in a Banach space B, and let T be a compact tansfomation that maps G into itself. Then T has a fixed point. Poof. This follows diectly fom Peliminay 4-1, Definition 4-5, and Peliminay 4-2. Afte using Lemma 4-2 to establish solutions on bounded domains, we will show that any bounded solution on such a domain may be extended to include a lage domain. We accomplish this extension with Catheodoy s Theoem. We now povide this theoem and elated definitions. Definition 4-6 Conside the equation x = f(, t x) (4.6) with initial condition x() τ = ξ. We say that φ is a solution to (4.6) in the extended sense on some inteval I if φ () t = f(, t φ()) t fo all t I except on a set of Lebesgue-measue zeo and φ() τ = ξ. Peliminay 4-3 (Theoem 1.1 of Chapte 2 of [7]) (Caatheodoy s Theoem) Conside the equation given in (4.6). Let ab, be constants, and define the ectangle 4-4

42 {(, ): τ, ξ } R = tx t a x b whee (, τξ ) is a fixed point in the (, tx ) plane. Let f be defined on R. Suppose f is measuable in t fo each fixed x, and continuous in x fo each fixed t. If thee exists a Lebesgue-integable function m on the inteval t τ a such that f (, tx) mt (), (, tx) R, then thee exists a solution φ of (4.6) in the extended sense on some inteval t τ β, ( β > ) satisfying φ() τ = ξ. Note that if f is continuous on R, then φ is a solution in the taditional sense. That is φ () t is continuous and φ () t = f(, t φ()) t fo all t ( τ β, τ + β ). We ae now eady to pesent ou main esults fo systems. 4.2 Main Results The theoems and lemmas pesented in this section build off one anothe. We begin by poving existence of entie solutions and wo towad showing the existence of entie lage solutions. n Theoem 4-1 Assume pq, C( ) ae nonnegative, not identically zeo at infinity, and satisfy (2.1). Let f, g C[, ) satisfy (2.12), and let G, given in (4.3), satisfy the Kelle-Osseman condition (2.2). Then system (1.2) has infinitely many entie nonnegative solutions. Poof. Radial solutions of system (1.2) ae solutions to the odinay diffeential system 4-5

43 n 1 u () + u () = p() f(()), v, n 1 v () + v () = qgu () (()),. (4.7) It follows that solutions to (1.2) ae fixed points of the opeato Tuv (, ) : C[, ) C[, ) C[, ) C[, ) defined by Tu ( ( ), v ( )) t a + t s p() s f (()) v s dsdt,, = t 1 n n 1 b + t s q s g u s dsdt () (()), (4.8) whee u() = a and v() = b ae the cental values fo the system. We will begin by establishing a fixed point of (4.8) in C[, R] C[, R] fo abitay R >. Define u () = a and v () = b fo all, and conside the sequence t a t s p s f v 1 s dsdt = 1 1 = t 1 n n 1 b + t s q s g u 1 s dsdt ( u, v ) T( u, v ) + () ( ()),, () ( ()), (4.9) fo = 1, 2. Conside the single equation u given by ( ) n Δ u = φ x H( u), x, (4.1) u as x 4-6

44 whee φ and H ae defined in (4.2) and (4.3) espectively. Recall that φ is nonnegative, c- positive, and satisfies (2.1). By Lemma 4-1, H satisfies the Kelle-Osseman condition (2.2). These conditions guaantee such a u exists by Theoem 2 of [12]. Futhe, this adial solution satisfies t ( ) = () + ( ) ( ( )), (4.11) u u t s φ shus dsdt whee u () >. Note this solution is non-deceasing. Indeed () = () ( ()) d u s φ shus ds d. (4.12) We shall choose ou cental values a and b such that a u() and b u(). Next, we use induction to show each individual sequence in (4.9) is unifomly bounded on bounded sets. Specifically, we will show max{ u ( ), v ( )} u( R) fo all and all [, R]. Since H and u ae non-deceasing functions and ab, u(), we have t 1 = + u () a t s p() s f ( v ()) s dsdt t () φ( ) ( ) u + t s s H b dsdt t () φ( ) ( ( )) u + t s s H u s dsdt = u (),. and similaly v 1 () u (). Now, assume u, v u. This implies 4-7

45 t + 1() = + () ( ()) u a t s p s f v s dsdt t () φ( ) ( ( )) u + t s s H v s dsdt t () φ( ) ( ( )) u + t s s H u s dsdt = u (). We similaly obtain v + 1 () u(), so u, v u( ) by induction. Again, since u is nondeceasing, we have the bound u (), v () u() u( R). (4.13) M fo all 1 and all [, R]. We now M < because u is entie. Then, since each individual sequence { u },{ v } is of the fom (2.15), Lemma 2-1 guaantees that { u },{ v } ae each equicontinuous on [, R ]. We have two sequences, { u } and { v }, that ae unifomly bounded and equicontinuous on [, R ]. By the Azela-Ascoli Theoem (Peliminay 2-1) thee exists a u C[, R] and a subsequence { u } such that j u j u unifomly on [, R ]. If we conside the coesponding subsequence { v } { v }, the Azela-Ascoli Theoem again implies thee exists a v C[, R] and a subsequence of { v }, call it { v }, such that j j ji v ji v unifomly on [, R ]. 4-8

46 Theefoe we have ( u, v ) ( u, v) unifomly on [, R] [, R]. j i j i Hence, ( uv, ) is a fixed point of (4.8) in C[, R] C[, R]. Next, we extend this esult to show T has a fixed point in C[, ) C[,. ) We poceed with a diagonal agument simila to that in Theoem 3-2. Howeve, in this case, we ae consideing fixed points and convegence in C[, ) C[, ) athe than C[, ). The idea is the same. Let {( w, z )} be a sequence of fixed points defined by Tw (, z) = ( w, z) on [, ], ( w, z ) C[, ] C[, ]. (4.14) fo = 1, 2,. As ealie, we may show that both { w } and { z } ae bounded and equicontinuous on [,1]. Thus by applying the Azela-Ascoli Theoem to each sequence 1 1 sepaately, we can deive {( w, z )} contains a convegent subsequence, {( w, z )}, that conveges unifomly on [,1] [,1]. Let unifomly on [,1] [,1] as. 1 1 ( w, z) ( u1, v1) 1 1 Liewise, the subsequences { w } and { z } ae each bounded and equicontinuous on [,2] so thee exists a subsequence {( w, z )} of {( w, z )} such that unifomly on [,2] [,2] as. 2 2 ( w, z) ( u2, v2) 4-9

47 Notice that {( w, z )} {( w, z )} {( w, z )} so ( u2, v2) = ( u1, v1) on [,1] [,1]. = 2 Continuing, we obtain a sequence {( u, v )} such that ( u, v ) C[, ] C[, ] = 1, 2, ( u ( ), v ( )) = ( u ( ), v ( )) fo [,1] 1 1 ( u ( ), v ( )) = ( u ( ), v ( )) fo [, 2] 2 2 ( u ( ), v ( )) = ( u ( ), v ( )) fo [, 1] 1 1 Thus ( u, v ) conveges to ( uv, ) which satisfies ((),()) u v = ( u(), v()) if. (4.15) The convegence is unifom on bounded sets, and thus ( u ( ), v ( )) C[, ) C[, ) is a fixed point of (4.8) and an entie solution to (1.2). We chose ou cental values < a u() and < b u() abitaily whee u is defined in (4.1). Theefoe [, u()] [, u()] is a subset of ou set of cental values S given in (4.1). We conclude (1.2) has an infinitely many entie solutions. Note we did not use the function G fom in (4.3) diectly in this esult. Rathe, since G satisfied the Kelle-Osseman condition (2.2), we wee guaanteed H satisfied this condition accoding to Lemma 4-1. We then used H in ou agument. The function G need not satisfy (2.2) to show existence of entie solutions. In fact, we only need f o g to satisfy the Kelle- Osseman condition, which we pove in the following coollay. 4-1

48 n Coollay 4-1 Assume pq, C( ), ae nonnegative, not identically zeo at infinity, and satisfy (2.1). Let f, g C[, ) satisfy (2.12). Also, suppose f o g satisfies the Kelle-Osseman condition (2.2). Then system (1.2) has infinitely many entie nonnegative solutions. Poof. Without loss of geneality, suppose f satisfies (2.2). Then H as defined in (4.3) still satisfies the Kelle-Osseman condition because 1 1 1/2 t 1/2 t H () s ds dt f () s ds dt <. In addition, H clealy must still satisfy (2.12). Ou agument then poceeds exactly lie the poof of Theoem 4-1. While we may have the existence of entie solutions to system (1.2) when G does not satisfy (2.2), it is necessay fo G to satisfy the Kelle-Osseman condition as we chaacteize ou set of cental values. We examine this set next. Theoem 4-2 Given the hypotheses in Theoem 4-1, the set of cental values S, given in (4.1), is closed and bounded. Poof. To show S is bounded, we poceed in a manne compaable to Cistea and Radulescu [6]. Fo contadiction, suppose S is unbounded. Conside the equation 4-11

49 n ( x) G( / 2), x, Δ η = ψ η η as x (4.16) whee ψ and G ae given in (4.2) and (4.3). Recall ψ is nonnegative, c-positive, and satisfies (2.1). Also, G satisfies (2.2), so an entie lage solution η exists fo equation (4.16) by Theoem 2 in [12]. Since we ae assuming S is unbounded, we can find cental values ab, S such that a+ b> η(). Notice that u () + v () f (()) v Gv (()) G if v () u (), 2 u () + v () gu ( ( )) Gu ( ( )) G if u ( ) v ( ). 2 (4.17) Then using (4.17), we have Δ ( u+ v) = p () f() v + qgu () () ψ ()( f() v + g()) u u () + v () ψ () G. 2 (4.18) Conside some closed finite ball B(, R ). We will use the maximum pinciple to show u+ v η in this ball. Since η and u+ v ae adial, conside these equations in thei elated odinay diffeential equation fom. Define h () (1 ) 2 1/2 = +, and suppose fo contadiction that u+ v> η at some point in [, R ]. Let ε > be small enough such that [ u v η εh ] [, ] max ( + )( ) ( ) ( ) >. R 4-12

50 Let [, R] be the point whee this maximum occus so ( u+ v)( ) η( ) εh( ) >. Theefoe at, by (4.18) and ou assumption that u ( ) + v ( ) > η( ), we have Δ (( u+ v)( ) η( ) εh( )) u ( ) + v ( ) ψ( ) G ψ( ) G( η/2) Δεh( ) 2 u ( ) + v ( ) = ψ( ) G G( η/2) εh( ) 2 Δ εδh ( ) >, (4.19) whee the last inequality holds because Δ h () < fo all [, R]. This may be seen by diect calculation, and is also a specific fom of Lemma of Poano [23]. We have a contadiction, and thus u+ v η in B(, R ). (4.2) Howeve, this now contadicts ou oiginal assumption that ab, wee chosen such that u() + v() = a+ b> η(). Theefoe, we conclude that S is bounded. Now we pove S is closed by showing S contains its bounday. Let ( a, b) S. Then thee exists some ball centeed at ( a, b ) with adius 1/ such that B(( a, b),1/ ) S. Fo each 1, we denote the abitay point ( a, b) S B(( a, b),1/ ). Note {( a, b )} ( a, b ) as. Then, we define the sequence 4-13

51 t a + t s p() s f ( v ()) s dsdt,, ( u( ), v( )) = t b + t s q( s) g( u ( s)) dsdt, (4.21) whee each ( u, v ) is an entie solution to (1.2). These solutions exist since each cental value ( a, b ) B(( a, b ),1/ ) S S. We now show {( u, v )} has a convegent subsequence on C[, ) C[,. ) Simila to ou agument fo Theoem 4-1, we fist demonstate that the sequence has a convegent subsequence on C[, R] C[, R] fo abitay R, and then we extend to C[, ) C[,. ) We must mae seveal mino adjustments to achieve ou bounds fo this new sequence, but the idea is compaable. Fo each = 1, 2, (4.2) gives us u + v η fo [, R] whee η is defined in (4.16). Since η (), we have u () + v () η( R) < fo R (4.22) whee η ( R) is finite because η is entie. Thus { u },{ v } ae each unifomly bounded on [, R ]. Lemma 2-1 implies { u },{ v } ae equicontinuous on [, R ]. As in the pevious poof, we may use the Azela-Ascoli Theoem to show a convegent subsequence of (4.21) exists on n C[, R] C[, R]. This gives us a solution to system (1.2) in B(, R) with cental values ( a, b ) S. Since R is abitay, we can use the same diagonal agument fom Theoem 4-1 to show (4.21) has a convegent subsequence on C[, ) C[, ). Theefoe we have the solution 4-14

52 u a t s p s f v s dsdt v b t s q s g u s dsdt t () = + () (()),, t ( ) = + ( ) ( ( )), whee ( a, b) S. Hence ( a, b) S implying S is closed. This completes ou poof. Thus fa, we have established the existence of entie solutions to system (1.2) and chaacteized the set of cental values S as closed and bounded. Fo monotonic and convex functions f and g, Peng and Song [22] futhe chaacteized S as convex. We cannot show this fo ou non-monotone poblem. In fact, we will povide an example late fo which S (found numeically) appeas non-convex. Still, Theoem 4-1 and Theoem 4-2 povide a limited geometic desciption of S which we pesent as the following coollay. Coollay 4-2 Given the hypotheses in Theoem 4-1, ou set of cental values S given in (4.1), satisfies T1 S T2 whee T1 = {( a, b) : a u(), b u()} and T2 = {( a, b) : a+ b η()}. The functions u and η ae given in (4.1), and (4.16) espectively. See Figue

53 Figue 4-1. Entie Solution Existence Region fo Cental Values of (1.2) Poof. In Theoem 4-1, we poved the existence of entie solutions to (1.2) by constucting T1 S. In Theoem 4-2, and we demonstated S to be bounded by constucting T2 S. Note that we dew S in Figue 4-1 as connected, but we have not shown this to be the case. Indeed, we now much less about ou set of cental values as compaed to monotonic poblems. Howeve, the fact that S is closed and bounded is enough to show the existence of entie lage solutions. Befoe we poceed, we will need to pove seveal mino lemmas. 4-16

54 Lemma 4-3 Unde the hypotheses of Theoem 4-1 and fo any (, cd) (, cd) + + such that S and c d, system (1.2) has a solution in some ball B(, ρ ), whee < ρ <. Poof. Let ( cd, ) be defined as above. We wish to find a adial solution to (1.2) in B(, ρ ), ρ >. This solution is a fixed point of the opeato T : C[, ρ] C[, ρ] C[, ρ] C[, ρ] defined by ( ˆ ˆ ) T( u( ), v( )) u( ), v( ) t 1 n n 1 c + t s p s f v s dsdt () (()), ρ, t + d t s q( s) g( u( s)) dsdt, ρ. (4.23) We will show fo ρ sufficiently small, a fixed point, and theeby a solution exists. We accomplish this using the vesion of Schaude s Fixed Point Theoem given in Lemma 4-2. Fist, we establish a subset X C[, ρ] C[, ρ] that satisfies the necessay hypotheses of this lemma. Note that fo ρ >, C[, ρ] C[, ρ] is a Banach space with nom ( ) ( uv, ) = max u, v whee u = sup u( ). Define the subset X C[, ρ] C[, ρ] by ρ {(, ) [, ] [, ]: (, ) (, ) min{, } X = uv C ρ C ρ uv cd cd, whee cd, ae ou given cental values. Since this is a closed ball in a Banach space, X is closed, bounded, and convex We tun ou attention to the opeato T given in (4.23). We show T is a compact opeato. Let Y be an abitay bounded set given by 4-17

55 {(, ) [, ] [, ]: (, ) } Y = u v C ρ C ρ u v M fo some M >. Then, suppose ( u, v ) is an abitay sequence in Y, and conside the image of this sequence Tu (, v) = ( uˆ, vˆ ). Clealy t ˆ () = + () ( ()) u c t s p s f v s dsdt c+ t s φ() s H( v ()) s dsdt t ρ t c + t s φ() s H ( M ) dsdt = M whee φ and H ae given in (4.2) and (4.3), and M is constant. Similaly v () M. Thus, u ˆ and v ˆ ae unifomly bounded. Futhe, since u ( ) < M and v ( ) < M fo all and all ρ, Lemma 2-1 implies that u ˆ and v ˆ ae also equicontinuous on ρ. Finally, using the Azela-Ascoli Theoem as befoe, thee exists a unifomly convegent subsequence ( uˆ, vˆ ) ( uˆ, vˆ) whee ( uv ˆ, ˆ) C[, ρ] C[, ρ]. This implies that the image of an abitay j j bounded set unde T, is (sequentially) compact. That is, T is a compact opeato. ( uv, ) Lastly, we must show that T maps elements of X bac into X. Again, tae any X. Since X is bounded, suppose ( uv, ) we have the estimate Q. By integating as in (3.8) and (3.1), 4-18

56 t t t s p() s f(()) v s dsdt t s φ() s H(()) v s dsdt t (()) φ() H v t s s dsdt HQ ( ) tφ( tdt ) ρ HQ ( ) ρ φ( tdt ) whee φ and H ae given in (4.2) and (4.3). Similaly, we may show t t s qsgus () (()) dsdt HQ ( ) ρ φ() tdt. ρ Since φ ( ) is well defined fo [, ) and cd>, by hypothesis, we can choose ρ > small enough so that ρ HQ ( ) ρ φ ( tdt ) < min{ cd, }. Doing so, fo ( uv, ) X, we have ( ˆ ˆ ) T( u( ), v( )) = u( ), v( ) whee c uˆ( ) t = c + t s p() s f (()) v s dsdt c+ H( Q) ρ φ( t) dt ρ c+ min{ c, d}, and similaly d vˆ( ) d + min{, c d} fo all ρ. Thus ( uv ˆ, ˆ) X. We have shown X to be closed, convex, and bounded. Also, we have poven the opeato T in (4.23) is compact, and we have shown fo any x X, Tx X. It follows fom 4-19

57 Lemma 4-2 that (4.23) has a fixed point. Hence system (1.2) has a solution in B(, ρ ). Futhe, we chose (, cd) S implying ρ <. Now that we have established the existence of solutions fo any pai of positive cental values, we will show fo any such (, cd) S, a lage solution exists on a finite domain. Lemma 4-4 Given the hypotheses of Theoem 4-1, let (,) uv be a solution to (1.2) with cental values (, cd) S, c d, and define the set R = { > : thee exists a solution of (1.2) in B(, ) such that ( u(), v()) = ( c, d)}. (4.24) sol Let R cd, be given as Rcd, = sup Rsol. (4.25) Then lim u ( ) = = lim v ( ). Rcd, Rcd, Poof. Tae (, cd) S, c d. By Lemma 4-3, Rsol, and since (, cd) S, R, <. Let ( uv, ) be a solution of (1.2) in B(, R, ) with cental values ( cd, ). Since u and v, cd cd lim u ( ) R cd, and lim v ( ) exist (possibly infinity). Fo contadiction, suppose R cd, lim u ( ) = A<. R cd, This implies 4-2

58 t lim ( ) = lim + ( ) ( ( )) Rcd, R cd, v d t s qsgus dsdt Rcd, t d + t s q() s H (()) u s dsdt Rcd, t d + H( A) t s q( s) dsdt = B <. Thus ur (, ) = Aand vr (, ) = Bae well defined. We now conside system (1.2) ove the cd cd inteval Rcd, Rcd, + ε, whee ε >. Again, ou equations ae adial, so we may integate system (1.2) to obtain u A t s p s f v s dsdt R R v B t s q s g u s dsdt R R t () = + () (()), cd, cd, + ε, Rcd, t () = + () (()), cd, cd, + ε. Rcd, (4.26) This is equivalent to the poblem u = s p s f v s ds u R = A R R + v = B + t s q s g u s dsdt R R + ( ) ( ) ( ( )), ( cd, ), cd, cd, ε, t ( ) ( ) ( ( )), cd, cd, ε. Rcd, (4.27) Substituting v () into ou equation fo u (), we get the initial value poblem 4-21

59 () () (()) u = s p s f v s ds s t = s p() s f B+ t y q( y) g(( u y)) dydt ds R cd, Rcd, Fu (, ()) (4.28) whee Rcd, Rcd, + ε and ur ( cd, ) = A. If we show this initial value poblem has a solution on some inteval R, R, + β ( < β ε), then (4.27) and (4.26) will have a solution as cd cd well. This would imply that system (1.2) has a solution on R, + β, contadicting the cd definition of R cd, in (4.25). We use Caatheodoy s Theoem shown in Peliminay 4-3. Define the ectangle { (, ):, cd, ε } R = u R u A A (4.29) whee ε >, vaiables, we see A= lim u( ), and R, is given in (4.25). Teating and u as two independent R cd, cd s t Fu (, ) = s psf () B+ t y qygudydt ( ) () ds R cd, Rcd, s t = s p() s f B + g() u t y q( y) dydt ds. R cd, Rcd, (4.3) Recall f, g, p, q ae all defined on [, ), and conside Fu (, ) on [ Rcd, ρ, Rcd, + ρ] [,2 A]. Let N = max g( u), and choose ρ small enough so that u 2A 4-22