journal of differential equations 148, 4764 (1998) article no. DE983456 First Variations of Principal Eigenvalues with Respect to the Domain and Point-Wise Growth of Positive Solutions for Problems Where Bifurcation from Infinity Occurs J. Lo pez-go mez Departamento de Matema tica Aplicada, Universidad Complutense, 284-Madrid, Spain and J. C. Sabina de Lis Departamento de Ana lisis Matema tico, Universidad de La Laguna, 38271-La Laguna (Tenerife), Spain Received May 8, 1997; revised January 15, 1998 In this paper the first variation of the principal eigenvalue of &2 in with respect to a general family of holomorphic perturbations of is analyzed. Then, the results from this analysis are used to ascertain the point-wise growth to infinity of the positive solutions of a class of sublinear elliptic boundary value problems with vanishing coefficients at the value of the parameter where bifurcation from infinity occurs. 1998 Academic Press 1. ITRODUCTIO In this wor we consider the following family of eigenvalue problems &2u=* u in, u= on, (1.1) where is a smooth bounded domain of R, 1, whose boundary possesses a finite number of connected components, 2 is the Laplace operator, and, &, is a family of smooth domains obtained from by the action of an holomorphic family of C 2 -diffeomorphisms T :. It is well nown that the lowest eigenvalue _ 1 [ ] of (1.1) is the unique eigenvalue to a positive eigenfunction, denoted by. (x), that _ 1 [ ]is simple and that it varies continuously with, [5]. In fact, the continuous dependence of _ 1 [ ] is valid for rather general elliptic operators, general domains and general perturbations from them (cf. [1, 2, 11]), though it may fail for eumann boundary conditions (cf. [5]). Moreover, in Section 47 22-39698 25. Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
48 LO PEZ-GO MEZ AD SABIA DE LIS VII.6.5 of [1] it was shown that the eigenvalues *() of (1.1) are holomorphic in near = when the domain is obtained from by a transformation of the form T (x)=x+r(x) x #, where R(x) is a smooth vector valued function defined in an open set containing the closure. The transformed eigenfunctions (x) :=.(T (x)), x #, are also holomorphic in. Here.( y), y #, is an arbitrary eigenfunction associated to *(). Although the process used in determining the power series in for the eigenvalues and eigenprojections is, in general, rather complicated (cf. Remar 4.18 on p. 46 of [1]), there are several inds of formulas for the first variation of the principal eigenvalue _ 1 [ ], d d _ 1[ ] =. Among them are the following: (6.31) on p. 422 of [1], which is of an abstract nature thus not useful for our purposes herein; and the formula on p. 275 of [13]. amely, 2 d d _ 1[ ] = =& (R, n) \. ds, (1.2) n + where. has been normalized by &. & L 2 ( )=1, and n is the outward unit normal on. Formula (1.2) generalizes to (5.1.1) of [7], found for the special case of two-dimensional domains. Besides its intrinsic interest, formula (1.2) has proven to be pivotal in the problem of analyzing the pointwise growth of the positive solutions of a class of sublinear boundary value problems where bifurcation from infinity occurs due to the presence of vanishing coefficients in the model, this being the problem from which our interest in ascertaining the first variation of the principal eigenvalue comes. Thus, in this paper, our attention will be focused on both problems. In Section 2 we extend some of the results of [13] to cover the case of general holomorphic families T of the form T =I+R+O( 2 ) as, (1.3) under minimal regularity requirements on the perturbed domains (C 1, instead of C 3 ), and show in particular that (1.2) remains valid by means of a direct striing proof. In Sections 3 and 4 we analyze the point-wise growth mentioned above. We should point out that (1.2) can not be
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 49 obtained straight ahead from the analysis of [9] and [15], where the corresponding variation of the Green functions was analyzed, nor from the abstract analysis of [16], since the differentiation of the principal eigenvalue entails the differentiation of the minimum of the variational functional with respect to the domain rather than the differentiation of the functional itself. To describe our results, consider the problem &2u=*u&a(x)u r in, u =, (1.4) where is a smooth bounded domain of R, r>1, * # R is regarded as a real parameter, and a # C( ) is a non-negative weight function, a{, such that D := [x # : a(x)>]//. It will be assumed in addition that D possesses a finite number of connected components, that D consists of a finite number of smooth connected pieces, and that the open set :="D is connected. Under these assumptions it is nown that (1.4) possesses a positive solution if, and only if, _ 1 []<*<_ 1 [ ], and that it is unique if it exists (cf. [3, 6, 12, 14]). Moreover, if we denote it by % *, then lim &% * &, =. * A _ 1 [ ] In Section 4 we use the theory of Sections 2, 3 to analyze the point-wise growth of % * as * A _ 1 [ ]. Our results show that lim % * (x)= for all x #, (1.5) * A _ 1 [ ] uniformly in compact subsets of, and that if a(x) decays sufficiently fast as x approaches D, then lim % * (x)= for all x #D. (1.6) * A _ 1 [ ] For instance, if a(x) is of class C 1 near D, then (1.6) holds (cf. Theorem 4.3 and Remar 4.2).
5 LO PEZ-GO MEZ AD SABIA DE LIS The uniform growth to infinity of % * on any compact subset of is in strong contrast with its behavior in D, where the growth of % * is controlled by the positive solution of the corresponding problem in D subject to infinity boundary conditions. In particular, lim % * (x)< for each x # D. (1.7) * A _ 1 [ ] To prove (1.6) we use some subsolutions built from the principal eigenfunctions of some auxiliary boundary value problems in a family of domains obtained from by an holomorphic family of C 2 -diffeomorphisms T satisfying (1.3). The proof of (1.7), being beyond the scope of this wor, will be given in [8]. This paper is organized as follows. Section 2 contains a short self-contained proof of (1.2). In Section 3 we show that the domain perturbations required for the proofs of (1.5) and (1.6) can be obtained by an holomorphic family of diffeomorphisms. In Section 4 we use the results of Sections 2, 3 to prove (1.5) and (1.6). 2. HOLOMORPHIC DEPEDECE AD FIRST VARIATIOS OF PRICIPAL EIGEVALUES Throughout this section the domain is assumed to be of class C 1. First, we will use the theory of Chapter VII of [1] to show that if, &, is obtained from by a holomorphic family of C 2 -diffeomorphisms T, &, then the principal eigenvalue _ 1 [ ] is real holomorphic in. Then we shall find (dd)(_ 1 [ ]) =. Assume that =T ( ), where T : is a family of C 2 -diffeomorphisms that can be expressed in the form T (x)=x+ : n=1 with R (n) # C 2 ( ; R ) for each n1, and n R (n) (x) x #, (2.1) lim [&R (n) &, +&D x R (n) &, +&D 2 x R(n) &, ] 1n <+. (2.2) n Here, we have denoted &D x R(n) &, := sup &D x R(n) (x)&, x # 2.
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 51 In the sequel, for any H # C 2 (, R ), D x H(x) v will stand for the linear action of D x H(x) onv # R, and D 2 H(x)[v x 1, v 2 ] for the bilinear action of D 2 H(x) on (v x 1, v 2 )#R _R. To homogenize the notation we set R () :=I. Thans to (2.2), the series D T x (x)= : n= n D x R(n) (x), (2.3) is absolutely convergent either in C 1 ( ; R 2 ) when =1 or in C( ; R 3 ) if =2. To avoid the difficulty that the underlying Hilbert space depends on, we transform problem (1.1) into a problem in. Let y denote the spatial variable in, and. the principal eigenfunction associated with _ 1 [ ], &. Setting it is easily seen that y=t (x), (x)=. (T (x)), x #, & :, l=1 (D y h, D y h l ) 2 & : x x l l=1 2 y h l x l =_ 1 [ ] in, = on, (2.4) where the function coefficients h i are given by T &1 ( y) :=(h 1 ( y, ),..., h ( y, )), y=t (x), x #. We now analyze the dependence on of the differential operator L(x, D x, ) defined by the left-hand side of (2.4), i.e., It is easily seen that L(x, D x, ) :=& :, l=1 & : l=1 2 (D y h, D y h l ) x x l 2 y h l x l, y=t (x). D y T &1 ( y) y=t (x)=i&d x R (x)+o( 2 ), (2.5) where the corresponding series is absolutely convergent in C 1 ( ; R 2 ). Moreover, D 2 y T &1 ( y) y=t (x)[},}]=&d 2 x R (x)[},}]+o( 2 ), (2.6) where the corresponding series is absolutely convergent in C( ; R 3 ). ow, we are ready to analyze the dependence in of the coefficients of
52 LO PEZ-GO MEZ AD SABIA DE LIS L(x, D x, ). Since the mapping D y h i ( y, ) y=t (x) is the i th component of D y T &1 ( y) y=t (x), (2.5) implies that D y h i ( y, ) y=t (x) can be expressed into a series of powers of which is absolutely convergent in C 1 ( ; R ) for &. Thus, the coefficients (D y h, D y h l ) y=t (x), 1, ln, are real analytic in for &. As for the coefficients 2 y h l, observe that they are the trace of D 2 h y l( y, ) y=t (x), the l th component of D 2 T &1 y ( y) y=t (x), and that, thans to (2.6), D 2 h y l( y, ) y=t (x) can be developped as a power series in that is absolutely convergent in C( ; R 2 ). Thus, the corresponding series for 2 y h l ( y, ) y=t (x) is also absolutely convergent in C( ; R) for &. Therefore, if L(x, D x, ), &, is regarded as a family of closed operators with common domain D(L)=H 1( ) & H 2 ( ) and values in L 2 ( ), then this family is real holomorphic of type (A) in in the sense of Kato (cf. [1, Chapter VII, Section 2]). Indeed, for all u # L 2 ( ) and v # H 1 ( ) & H 2 ( ), the L 2 -product ul(},d x, ) v is real holomorphic in for &. Therefore, we find from Theorems 1.7, 1.8 of Chapter VII, Section 1.3, of [1] that _ 1 [ ] and are real holomorphic in for &. In particular, these features mae rigorous the following analysis of the coefficients of (2.4) up to the first order in. Thans to (2.5), we obtain D y h (T (x))=e &D x R (x)+o(2 ), x #, (2.7) where e is the th vector of the canonical basis of R and R (x) is the th component of R (x), 1. Thus, \R (D y h, D y h l ) = l & x l Moreover, it readily follows from (2.7) that 2 y h l (T (x))=&2 x R l (x)+o(2 ), (x)+ R l (x) ). (2.8) x ++O(2 where i =1 if =i and i = if {i. Substituting this relation together with (2.8) into (2.4) gives & :, l=1_ l& \R + R l x l x +& ow, setting 2 x x l + : l=1 2 x R l x l =_ 1 [ ] +O( 2 ). (2.9) =. +. 1 +O( 2 ), _ 1 [ ]=_ 1 [ ]+* 1 +O( 2 ), (2.1)
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 53 where. is the principal eigenfunction associated with _ 1 [ ] normalized so that. 2 =1, substituting (2.1) into (2.9), dividing the resulting relation by, and passing to the limit as gives &2. 1 + :, l=1\ R + R l x l x + 2. x x l + : l=1 2 x R l. x l =_ 1 [ ]. 1 +* 1., (2.11) where we have used &2. =_ 1 [].. ow, the Fredholm alternative applied to (2.11) provides us with the following value for * 1 : * 1 = :, l=1. \ R Integrating by parts gives Hence, + R l x l x + 2.. + : x x l. 2 x R l=1 x. (2.12) l l R. * 1 =&2 :. R + :, l=1 x l x x l. 2. =1 x + : R, l=1 x \. 2 x l+. R. * 1 =&2 :. R &_ 1 [ ] :, l=1 x l x x l. 2 =1 x + : R, l=1 x \ A further integration by parts gives R. &2 :., l=1 x l x x l. 2 x l+. (2.13) =&2 (R, {. )({., n) ds&2_ 1 [ ] (R, {. ).. + : R 2.. (2.14), l=1 x l x l x
54 LO PEZ-GO MEZ AD SABIA DE LIS Similarly, and : R, l=1 x \. 2 x l+ = {. 2 (R, n) ds&2 :. R 2., (2.15), l=1 x l x l x &_ 1 [ ] : R. 2 =1 x =2_ 1. (R, {. ), (2.16) since. = on. ow, substituting (2.14)(2.16) into (2.12) yields * 1 =&2 (R, {. )({., n) ds+ {. 2 (R, n) ds. (2.17) On the other hand, since for each x # {. (x)=(. n)(x) n(x), (2.17) reduces to 2 * 1 =& (R, n) \. ds, (2.18) n + which is the wanted value. The previous features can be summarized into the following result. Theorem 2.1. Let /R be a bounded domain of class C 1 and, &, a perturbed family of domains from given by a family of C 2 -diffeomorphisms T satisfying (2.1) and (2.2). Let _ 1 [ ] denote the principal eigenvalue of, &. Then, the family of eigenvalue problems (1.1) is real holomorphic in and the first variation of the principal eigenvalue * 1 := (dd)(_ 1 [ ]) = is given by (2.18). Remar 2.1. If R (z)=p(z) n(z), for some non-negative function p{, then we find from (2.18) that * 1 =& p(. n) 2 ds<. Therefore, in this case _ 1 [ ] decreases linearly to _ 1 [ ]as a. 3. A IMPORTAT EXAMPLE In this section we restrict ourselves to dealing with a particular class of perturbations of which are pivotal for the analysis carried out in Section 4. First, we introduce the family of perturbed domains. Then, we prove that can be obtained from by a holomorphic family of diffeomorphims.
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 55 Let 1 j,1jp, be the components of and n j,1jp, the outward pointing unit normal associated to every 1 j / (cf. [4]). ow, given m1, mp, we pic up m arbitrary components of, say 1 i, 1m, and for each > small enough we consider the domain m :=. =1 [x # R " : d(x, 1 i )<]. (3.1) Observe that is obtained from by enlarging it by an amount just in the direction of the outward unit normals n i of the preselected group 1 i1,..., 1 im of components of, while remains unchanged with respect to the remaining ones. The most simple example is given by the following perturbation := [x # R :1+<&x&<R], <<1, of the annulus :=[x # R :1<&x&<R] where R>1. A less elementary example can be obtained from := >. p i=1 B R i (x i ), where /R is an arbitrary bounded smooth domain, p1, x j #, 1 jp, are p arbitrary points of, and R j >, 1 jp, are p positive real numbers such that B R i (x i ):=[x # R : &x&x i &R i ]/, 1j p. In this case, the perturbation of is defined by choosing 1i 1 <}}}< i m p, # (, min 1m R i ) and taing :=" p B i=1 R i & i (x i ), where i = if i # [i 1,..., i m ] and i = if not. In the proof of Theorem 4.3 of Section 4 it will be seen where our interest into these class of perturbations comes from. In the rest of this section we will prove that they fit into the scenario of Section 2, though a little more regularity on will be required for this. As a consequence from this suplementary regularity it will be seen that, moreover, _ 1 [ ] decays linearly as a. It should be also observed that the linear behavior of the principal eigenvalue is by no means evident even for the simplest case of the annulus when proceeding to the direct analysis of the problem in terms of the underlying Bessel functions. Theorem 3.1. Assume that is a bounded domain of R of class C 3. If is given by (3.1), then for each > sufficiently small, there exists a mapping T : R, such that
56 LO PEZ-GO MEZ AD SABIA DE LIS (i) T # C 2 ( ; R ) and T : is a bijection. (ii) The family T is real holomorphic in for &, in the sense that (2.1) and (2.2) are satisfied. (iii) R 1i =n i if i # [i 1,..., i m ], whereas R 1i = if i [i 1,..., i m ]. Proof. Since the components 1 i of are compact surfaces of class C 3 in R, they are orientable and possess well-defined C 2 outward unit normal fields n i =n i (x) [4]. Let = > be sufficiently small so that the = -neighborhoods of 1 i, A i := [x # R : d(x, 1 i )<= ], 1i p, be pair-wise disjoint. It is well-nown that for each 1ip, = can be chosen sufficiently small so that A i be a tubular neighborhood of 1 i (cf. [4, II.7]). This means that for every x # A i there exist unique z # 1 i and &= <{<= such that x=z&{n i (z). (3.2) Thus, by reducing once more =, if necessary, the implicit function theorem shows the existence of two unique mappings { i # C 2 (A i ; R),? i # C 2 (A i ; 1 i ), 1ip, such that x=? i (x)&{ i (x) n i (? i (x)), (3.3) for every x # A i and each 1ip. ext, introduce {^ i # C 2 ( ; R) as the extension of { i to by { i (x)== if d(x, 1 i )=. Let n^ i # C 2 ( ; R ) be any regular extension of the vector field n i (? i (x)) to the whole of and consider any function ` # C 3 ([, ); [, )) satisfying `()=1, `({) `({)< for { #[,= 2), and `({)= for {= 2. It is easily seen that for each 1ip the mapping H i (x) :=`({^ i(x)) n^ i(x) x #, is of class C 2, satisfies H i (x)= if d(x, 1 i )= 2, and H i (x)=`({ i (x))_ n i (? i (x)) if d(x, 1 i )<= 2. In particular, H i 1l = li n i, where li =1 if l=i and li = if l=% i. Setting R (x) :=H i1 (x)+}}}+h im (x), it will now be shown how the family T :=I+R satisfies all the requirements of the statement. It is rather clear that they are locally invertible
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 57 mappings of class C 2 satisfying (2.1) and (2.2). So, it suffices to prove that each of them defines a bijection from onto. Since R vanishes in V := {x # : d(x, 1 i ) = 2 for each 1m =, the restriction of T to V equals the identity mapping in V. Moreover, setting U i := {x # : d(x, 1 i ) = 2=, then, either T (x)=x for all x # U i,ifi{i,1m, or T (x)=x+`({ i (x)) n i (? i (x)) for all x # U i, if i=i for some 1m. Thus, from the coordinate representation (3.3), it is easily seen that T also defines a diffeomorphism from U i onto U i _ [x # R " : d(x, 1 i )], when 1m. This completes the proof. ow, from Theorems 2.1, 3.1 the following result is obtained. Theorem 3.2. If is given by (3.1), then the family of eigenvalue problems (1.1) is real holomorphic in and m d d (_ 1[ ]) = =& : =1 1 i\. K 2 n i+ ds<. (3.4) 4. POIT-WISE DIVERGECE OF POSITIVE SOLUTIOS In this section we analyze the behavior of the positive solutions to the following nonlinear boundary value problem &2u=*u&a(x) u r in, u =, (4.1) where is a bounded domain of R of class C 3, r>1, * # R is regarded as a real parameter, and a # C() is a non-negative weight function, a=%, satisfying H1. The open set D :=[x # : a(x)>]
58 LO PEZ-GO MEZ AD SABIA DE LIS satisfies D / and it possesses a finite number of connected components D 1,..., D l, such that D i & D j=< if i{ j. Thus, D=D 1 _ }}} _ D l. H2. Every connected component D i is a bounded domain of class C 3. H3. The open set defined by :="D is connected. Remar 4.1. Under these assumptions the subdomain is of class C 3 and = _ D consists of a finite number of components (at least l+1). Since D /, this number is greater than one even in the simplest possible case when D is connected. This explains why we are interested in domains whose boundary exhibits more than one component. ote that the one-dimensional version (=1) of (4.1) only requires (H1). In the sequel we will focus our attention on the case 2. The analysis and results can easily be adapted to cover the one-dimensional situation as well. The main features concerning the existence, uniqueness, and dependence on * of the positive solutions of (4.1) are summarized in the following result. The existence and uniqueness were found in [3] and [14]. The dependence on * of the positive solutions as well as the validity of the result for general second-order elliptic operators, not necessarily selfadjoint, was analyzed in [6] and [12]. Theorem 4.1. if, The problem (4.1) possesses a positive solution if, and only _ 1 []<*<_ 1 [ ]. (4.2) Moreover, if such a solution exists, then it is unique, and if we denote it by % *, then % * # W 2, p () & C 1+: ( ) for each p>, where :=1&p, and C 1+: ( ):=[u # C 1+: () :u =]. Furthermore, lim &% * &, =, * a _ 1 [] lim &% * &, =. (4.3) * A _ 1 [ ] In addition, the mapping * % * from _ 1 []<*<_ 1 [ ] into C 1+: ( ) is differentiable, point-wise increasing and v=(d% * d*)#w 2, p () & C 1+: ( ). In this section our goal is to analyze the point-wise growth of % * as * A _ 1 [ ], where, according to Theorem 4.1, bifurcation to positive solutions from infinity occurs. Our first result reads as follows. Theorem 4.2. Let K be a compact subset of. Then lim % * = and lim * A _ 1 [ ] * A _ 1 [ ] d% * = uniformly in K. d*
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 59 Proof. Differentiating (4.1) with respect to * gives (&2+ra% * r&1 &*) d% * d* =% * in and (d% * d*)= on. In we have a= and hence, (&2&*) d% * d* =% *. Let. be the principal eigenfunction associated with _ 1 [ ]. Fix and consider c> such that * 1 #(_ 1 [], _ 1 [ ]) % *1 >c. 1 in. Then, thans to Theorem 4.1, for each * #(* 1, _ 1 [ ]) we have % * >% *1 >c. in. Moreover, if * #(* 1, _ 1 [ ]) then the operator &2&* satisfies the strong maximum principle in and hence, d% * c d* >c(&2&*)&1. = _ 1 [ ]&*. in. Since. is bounded away from zero in K, lim * A _ 1 [ ] In addition we also get d% * = uniformly in K. d* # 1[ ]&* 1 % * (x)>% *1 (x)+log \ 1 [ ]&* + for x # K, where #=c inf K.. Thus, both % * and % * diverge to infinity uniformly in K when * A _ 1 [ ]. This completes the proof. K The following result complements Theorem 4.1, providing us with a sufficient condition on the weight function a(x) so that lim % * (x)= for each x #D= ". (4.4) * A _ 1 [ ]
6 LO PEZ-GO MEZ AD SABIA DE LIS Theorem 4.3. Assume that the weight function a=a(x) is of class C 1 in some neighbourhood of the boundary D of its support. Then (4.4) is satisfied uniformly on ". Remar 4.2. Observe that, as a consequence of our assumption, the following holds: {a(x)=, for every x #D. (4.5) evertheless, as it will be seen from the proof of Theorem 4.3, such a restriction can be substantially relaxed by assuming, for instance, that a(x)=o(dist(x, D)) as dist(x, D) +. Proof. by For > small enough, let be the perturbation of defined := _ [x # D : d(x, D)<]. By Theorem 3.2, is a holomorphic perturbation from of the form (3.1) and hence Theorem 3.2 implies where _ 1 [ ]=_ 1 [ ]+* 1 +O( 2 ) as a, (4.6) * 1 <. (4.7) Let. be the principal eigenfunction associated with _ 1 [ ], normalized so that Pic up * satisfying &. &, =1. _ 1 [ ]<_ 1 [ 2 ]<*<_ 1 [ ], (4.8) and consider the function u # C( ) defined by u ( y)= {C. ( y) for y #, for y. where C> is an amplitude constant to be chosen later. It is easily seen that u provides us with a wea subsolution of (4.1) if, and only if, a( y) C r&1. r&1 ( y)*&_ 1 [ ] for all y #. (4.9)
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 61 By (4.8) a sufficient condition for (4.9) is the following a( y) C r&1. r&1 ( y)_ 1 [ 2 ]&_ 1 [ ] for all y #. For y # we have a( y)=. Therefore, u is a subsolution of (4.1) provided a( y) C r&1. r&1 ( y)_ 1 [ 2 ]&_ 1 [ ] for all y # ". (4.1) ow, the choice of the suitable amplitude C=C() requires the analysis of the decay order as of the several quantities involved in (4.1). amely, _ 1 [ 2 ]&_ 1 [ ], sup " a( y) and sup ". ( y). Firstly, thans to (4.6) we have, _ 1 [ 2 ]&_ 1 [ ]=&* 1 2 +O(2 ) as a. (4.11) Moreover, sup " a( y)=o() as. Indeed, for each y # " there exists a unique? i ( y)#d i, for some 1il, such that y&? i ( y) (cf. the proof of Theorem 3.1), and hence, a( y) =a( y)&a(? i( y)) = 1 ({a(ty+(1&t)? i ( y)), y&? i ( y)) dt. Therefore, we find from (4.5) that Finally, lim a a( y) sup =. (4.12) " sup ". ( y)tc as, (4.13) for some positive constant C. To prove (4.13) we proceed separately in each of the regions W i :=[x # D i : dist(x, D i )]. For each > small enough there exists y # W i such that sup Wi. ( y)=. ( y ). We claim that y #D i. Indeed, observe that = in the proof of Theorem 3.1 can be chosen so that. =. (x) in (2.1) satisfy (cf. (3.3)) ({. (? i (x)&{ i (x) n i ), n i ) <, for each x #, dist(x, D i )=, where n i =n i (? i (x)) and 1il. From (2.1) this implies that increases in the direction of &n i, i.e., for {
62 LO PEZ-GO MEZ AD SABIA DE LIS increasing in (3.2). Taing into account. ( y)= (T &1 ( y)), this shows the claim above. Let x be such that y =T (x ). Since y #D i, we have that x =z & n i (z ), where z =? i (x ) (cf. (3.3)). Moreover, since. ( y )= (x ) and (z )=, we find that. ( y )= (x )=& 1 ({ (z &tn i (z )), n i (z ))) dt. (4.14) If we now let we obtain, modulus some subsequence, that z x i for some x i #D i. Therefore, taing limits in (4.14) as we find that sup W i. ( y)=. ( y )=&. n i (x i) +o(), as. Hence, the constant C giving the behavior (4.13) of sup ". ( y) is given by the maximum of the numbers &(. n i )(x i), 1il. Thus, it follows from (4.1), that u is a subsolution of (4.1) if we tae 1 C=C() := sup ". ( y){ _ 1(r&1) 1[ 2 ]&_ 1 [ ] sup " a( y) = 1 = sup ". ( y){ * 1(r&1) 12+O(). (4.15) sup " a( y)= It follows from (4.12), (4.13), and (4.15) that This implies that lim C()=+. lim u (x)=+ uniformly on each compact subset of. To complete the proof of the theorem it suffices to show that lim u (x)=+ on D. This follows from the fact that, inf. ( y)tc 1 y #D
BOUDARY PERTURBATIO AD POIT-WISE GROWTH 63 as for some positive constant C 1. Indeed, the same arguments as in the proof of the estimate (4.13) lead to inf. ( y)=&. (x^ y #D n i) +o(), i as, for some x^ i #D i, and each 1il. So, the constant C 1 is given by Therefore we finally get C 1 = min 1il{ &. (x^ i) = n. i u ( y)c()(c 1 &= 1 ) for each y #D, and = 1 > small. Thus lim u =+ uniformly on D. It should be observed that the problem (4.1) always exhibits arbitrarily large supersolutions in the interior of the positive cone (cf. [6, Lemma 3.4] and [12, Lemma 3.2]). Since (4.1) admits a unique solution % * we obtain u ( y)% * ( y) y # for each _ 1 [ 2 ]<*<_ 1 [ ]. Therefore, the growth to infinity of u leads to the corresponding behavior for % *, and the proof of Theorem 4.3 is completed. K ACKOWLEDGMETS The authors wish to than to DGICYT of Spain for research support under grants DGICYT PB93-465 and DGES PB96-621. REFERECES 1. J. M. Arrieta, Elliptic equations, principal eigenvalue and dependence on the domain, Comm. Partial Differential Equations 21 (1996), 971991. 2. I. Babusa and R. Vyborny, Continuous dependence of eigenvalues on the domain, Czech. Math. J. 15 (1965), 169178. 3. H. Brezis and L. Oswald, Remars on sublinear elliptic equations, onlinear Anal. 1 (1986), 5564. 4. M. do Carmo, ``Differential Geometry of Curves and Surfaces,'' Prentice Hall Internationals, Englewood Cliffs, J, 1976. 5. R. Courant and D. Hilbert, ``Methods of Mathematical Physics,'' Vol. I, Wiley, ew Yor, 1962.
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