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Instability of nonnegative solutions for a class of semilinear elliptic boundary value problems
JOURNAL OF COMPUTATIONAL AND APPLIEDMATHEMATICS
ELSEVIER
Journal of Computational and Applied Mathematics 88 (1998) 125-128
Instability of nonnegative solutions for a class of semilinear elliptic boundary value problems C. Maya,
R. Shivaji *
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, M S 39762-5921, United States Received 1 September 1997
Abstract We consider the boundary value problem - A u ( x ) = 2f(u(x)),
Bu(x)=O,
x E f2,
xE~(2,
where f2 is a bounded region in ~" with smooth boundary Bu(x) = cth(x)u + (1 - ct)Ou/~n where c~E [0, 1], h : ~I2 --~ ~+ with h = 1 when ~ = 1,)~ > 0 , f is a smooth function such that f " ( u ) > 0 for u > O, f ( u ) < 0 for u E (0,fl) and f ( u ) > 0 for u > fl for some /3 > 0. We provide a simple proof to establish that every non-trivial nonnegative solution is unstable. © 1998 Elsevier Science B.V. All rights reserved.
Keywords: Instability; Nonnegative solutions A M S classification: 35B35; 35J65
I. Introduction In this p a p e r w e c o n s i d e r the s t a b i l i t y o f n o n - t r i v i a l n o n n e g a t i v e s o l u t i o n s to the s e m i l i n e a r e l l i p t i c boundary value problem
- Au(x)=2f(u(x)), B u ( x ) = 0,
xEf2,
x E 00,
(1.1) (1.2)
w h e r e f2 is a b o u n d e d r e g i o n in Nn w i t h s m o o t h b o u n d a r y , B u ( x ) = e h ( x ) u ( x ) + (1 - e ) O u / O n w h e r e E [0, 1] is a c o n s t a n t , h : ~f2 --~ R + is a s m o o t h f u n c t i o n w i t h h = 1 w h e n e = 1, i.e., the b o u n d a r y * Corresponding author. E-mail: [email protected] 0377-0427/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S 03 7 7 - 0 4 2 7 ( 9 7 ) 0 0 2 0 9 - 4
126
C. Maya, R. Shivaji/ Journal of Computational and Applied Mathematics 88 (1998) 125-128
condition may be of Dirichlet, Neumann or mixed type, 2 > 0 is a constant and f is a smooth function satisfying
f(u)
for uE(O, fl)
and
f(u)>O
for u > / 3 for some f l > O ,
(1.3)
and
f"(u) > 0
for u > 0.
(1.4)
Recall that a solution of (1.1)-(1.2) is stable in the maximum norm if given any ~ > 0 there exists a 6 > 0 such that if ]IK0(x)- U(x)l]~ < 6 then ][u(x,t)- U(x)I[~ < ~ for all t~>0, where u satisfies the initial value problem ut = Au + 2 f ( u ) for x E ~2, t > 0, Bu = 0 for x E c3~2, t ~> 0 and u(x,O) =K0(x) for x E O. We prove Theorem 1.1. Every nontrivial nonnegative solution o f (1.1)-(1.2) is unstable. We shall prove the instability of a non-trivial nonnegative solution u by showing that the principal eigenvalue /Zl, of the equation linearized about u is negative; the instability of u then follows from the well-known principle of linearized stability. In [1], instability of such solutions was proven when f " ( u ) > 0 for u > 0, f ( 0 ) < 0 (semipositone) and i f ( u ) > 0 for u > 0. In this paper we extend this result to include the case when f is not monotone, and also include the case when f ( 0 ) = 0 (in which case our assumptions imply that i f ( 0 ) < 0). The instability of positive solutions for the case when f ( 0 ) = 0, i f ( 0 ) t> 0 and f " ( u ) > 0 for u > 0 is well known. See [5] where the authors extend the result in [1] and prove our result by using subsuper solutions arguments to first prove that /~ ~< 0 and then using the linearized equation about u to show that ~ 7~0. To do so they analyze and make use of the properties of the functions G ( u ) = - t f ( u ) + f ( t u ) for t > 0, u > 0 and L(u) = u f ' ( u ) - f ( u ) . The purpose of this paper is to prove Theorem 1.1 directly by analyzing the linearized equation. We overcome the difficulty of f being nonmonotone, by re-writing f as the sum of a monotone function and a linear function involving f ( 0 ) and i f ( 0 ) . By doing so we arrive at a much simpler proof clearly indicating the role of f ( 0 ) in establishing the instability result. For existence results of nonnegative solutions for (1.1)-(1.2) satisfying (1.3)-(1.4) see [2, 3].
2. Proof of Theorem 1.1. Let 9(u(x))-= f ( u ( x ) ) - f(O) + ]f'(O)iu(x). Then 9(0) = 0, 9'(u) = i f ( u ) + If'(0)], f " ( u ) > 0 for u > 0 and, therefore, 9'(u) > 0 for u > 0 and 9(u) > 0 for u > 0. Now, (1.1)-(1.2) can be rewritten as
--Au(x) = 2{9(u(x)) + f ( O ) -- [f'(O)lu(x)}, Bu(x ) = O,
x E Of2.
x E O,
O"(u) =
(2.1) (2.2)
C. Maya, R. Shivaji/Journal of Computational and Applied Mathematics 88 (1998) 125-128
127
Let u0 be any nonnegative solution of (2.1)-(2.2). Then the linearized equation about Uo(X) is
-A4,(x) - 2{g'(uo(x)) - If'(o)l}¢(x) =/~ep(x), x E f2,
(2.3)
Bdp(x) = O, x C dO.
(2.4)
Let /q be the principal eigenvalue and let ~(x)( >~O) be a corresponding eigenfunction. We calculate (2.1)g'(uo)~(x)- (2.3)g(u0) and then integrating over f2 yields
2 {(-Auo)g'(uo)tP(x) - (--A ~(x))g(uo) -- 2f(O)g'(uo)~(x) + ,~lf'(O)luo(x)g'(uoW(x) -
21/'(0)1 q'(x)g(uo)} dx
= - ~ l f~ ~(x)g(uo)dx.
(2.5)
But by Green's first identity
( dUo
£ ( - Auo )g' (uo )tP(x ) dx = Jo V'(g'(uo)~P(x))~7Uo(X) dx - f~o g'(uo)~(s) k,-~n J as = f~ g"(UoW(X)l VUol 2 dx
+ £ d(uo)(W'Vuo)dx -
(dUo~
f g'(uo)~'(s)\ dn ,I ds
(2.6)
and
f (AT(x))g(uo) dx = -
7(g(uo))V~(x)dx +
o g(u°) ~n
ds
= - fog'(Uo)(V'UoV'~)dx + L g(uo) (~n ) ds.
(2.7)
By using (2.6)-(2.7) in (2.5) we get
£
£
dx
- 2 f ( O ) fo g'(uo)~(x) dx + 21f'(O)l
L{g'(uo)uo(x) - g(uo)} ~(x) dx
(duo~ ~ d,. + fo{g(uo)(~n)-¢(Uo)~(s)~dnjj
(2.8)
We notice that when ~ = 1 (then h = l ) we have u 0 = 0 for sEdf2 and, therefore, g(uo)=0 for s E dr2 and also we have ~ = 0 for s E dr2. Hence,
L {g(Uo)(Tn ) --g'(uo)~(s)(dUo) \ dnj } ds=O
(2.9)
C Maya, R. Shivaji/Journalof Computationaland AppliedMathematics88 (1998) 125-128
128
and when ~ ¢ 1, we have
f~ {g(uo)(~f~) -g'(Uo)~'(s)\ ~n/}
f
ds = J,9l [ (1 - 0~)J [uog'(uo) - g(uo)] ds.
But e i> 0, h > 0, T >~ 0 for s E 0£2 and uog'(uo) Therefore, if e ¢ 1
-
g(uo)
> 0 for Uo > 0.
(Ouo
fo~ { g(uo) ( ~n ) - g'(uo)ql(s) k c~nj } ds ~ O. Also, since
g"(Uo) >
(2.10)
0 for Uo > 0, w e get
f g"(uo)'e(x)lVuol 2 dx > 0.
(2.1 1 )
Thus, using (2.9)-(2.11) in (2.8) we obtain
(-#~) ~ T(x)g(uo)dx > - 2 f ( 0 ) L g'(uo)T(x)dx
+Alf'(O)l j~ {g'(uo)uo - g(uo)} T(x) dx.
(2.12)
Now by using (1.3)-(1.4) and the fact that uog'(uo) - g(uo) > 0 for Uo > 0 in (2.12) it is easy to see that ( - l q ) f~ T(x)g(uo) dx > O. But T > 0 for x E (2 and g(uo) > 0 for Uo > 0 and hence, #1 < 0 and the result follows (see [4]). References [1] K.J. Brown, R. Shivaji, Instability of nonnegative solutions for a class of semipositone problems, Proc. AMS 112 (1) (1991) 121-124. [2] A. Castro, J. Cossio, J.M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain. J. Math., to appear. E3] A. Castro, R. Shivaji, Semipositone problems, semigroups of linear and nonlinear operators and applications, in: J.A. Goldstein, G. Goldstein (Eds.), Nonlinear Operators and Applications, Kluwer Academic Publishers, New York, 1993, pp. 109-119 (invited review paper). [4] D.H. Sattinger, Monotone method in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (11) (1972) 979-1000. [5] A. Tertikas, Stability and instability of positive solutions of semi-positone problems, Proc. AMS 114 (4) (1992) 1035-1040.
ELSEVIER
Journal of Computational and Applied Mathematics 88 (1998) 125-128
Instability of nonnegative solutions for a class of semilinear elliptic boundary value problems C. Maya,
R. Shivaji *
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, M S 39762-5921, United States Received 1 September 1997
Abstract We consider the boundary value problem - A u ( x ) = 2f(u(x)),
Bu(x)=O,
x E f2,
xE~(2,
where f2 is a bounded region in ~" with smooth boundary Bu(x) = cth(x)u + (1 - ct)Ou/~n where c~E [0, 1], h : ~I2 --~ ~+ with h = 1 when ~ = 1,)~ > 0 , f is a smooth function such that f " ( u ) > 0 for u > O, f ( u ) < 0 for u E (0,fl) and f ( u ) > 0 for u > fl for some /3 > 0. We provide a simple proof to establish that every non-trivial nonnegative solution is unstable. © 1998 Elsevier Science B.V. All rights reserved.
Keywords: Instability; Nonnegative solutions A M S classification: 35B35; 35J65
I. Introduction In this p a p e r w e c o n s i d e r the s t a b i l i t y o f n o n - t r i v i a l n o n n e g a t i v e s o l u t i o n s to the s e m i l i n e a r e l l i p t i c boundary value problem
- Au(x)=2f(u(x)), B u ( x ) = 0,
xEf2,
x E 00,
(1.1) (1.2)
w h e r e f2 is a b o u n d e d r e g i o n in Nn w i t h s m o o t h b o u n d a r y , B u ( x ) = e h ( x ) u ( x ) + (1 - e ) O u / O n w h e r e E [0, 1] is a c o n s t a n t , h : ~f2 --~ R + is a s m o o t h f u n c t i o n w i t h h = 1 w h e n e = 1, i.e., the b o u n d a r y * Corresponding author. E-mail: [email protected] 0377-0427/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S 03 7 7 - 0 4 2 7 ( 9 7 ) 0 0 2 0 9 - 4
126
C. Maya, R. Shivaji/ Journal of Computational and Applied Mathematics 88 (1998) 125-128
condition may be of Dirichlet, Neumann or mixed type, 2 > 0 is a constant and f is a smooth function satisfying
f(u)
for uE(O, fl)
and
f(u)>O
for u > / 3 for some f l > O ,
(1.3)
and
f"(u) > 0
for u > 0.
(1.4)
Recall that a solution of (1.1)-(1.2) is stable in the maximum norm if given any ~ > 0 there exists a 6 > 0 such that if ]IK0(x)- U(x)l]~ < 6 then ][u(x,t)- U(x)I[~ < ~ for all t~>0, where u satisfies the initial value problem ut = Au + 2 f ( u ) for x E ~2, t > 0, Bu = 0 for x E c3~2, t ~> 0 and u(x,O) =K0(x) for x E O. We prove Theorem 1.1. Every nontrivial nonnegative solution o f (1.1)-(1.2) is unstable. We shall prove the instability of a non-trivial nonnegative solution u by showing that the principal eigenvalue /Zl, of the equation linearized about u is negative; the instability of u then follows from the well-known principle of linearized stability. In [1], instability of such solutions was proven when f " ( u ) > 0 for u > 0, f ( 0 ) < 0 (semipositone) and i f ( u ) > 0 for u > 0. In this paper we extend this result to include the case when f is not monotone, and also include the case when f ( 0 ) = 0 (in which case our assumptions imply that i f ( 0 ) < 0). The instability of positive solutions for the case when f ( 0 ) = 0, i f ( 0 ) t> 0 and f " ( u ) > 0 for u > 0 is well known. See [5] where the authors extend the result in [1] and prove our result by using subsuper solutions arguments to first prove that /~ ~< 0 and then using the linearized equation about u to show that ~ 7~0. To do so they analyze and make use of the properties of the functions G ( u ) = - t f ( u ) + f ( t u ) for t > 0, u > 0 and L(u) = u f ' ( u ) - f ( u ) . The purpose of this paper is to prove Theorem 1.1 directly by analyzing the linearized equation. We overcome the difficulty of f being nonmonotone, by re-writing f as the sum of a monotone function and a linear function involving f ( 0 ) and i f ( 0 ) . By doing so we arrive at a much simpler proof clearly indicating the role of f ( 0 ) in establishing the instability result. For existence results of nonnegative solutions for (1.1)-(1.2) satisfying (1.3)-(1.4) see [2, 3].
2. Proof of Theorem 1.1. Let 9(u(x))-= f ( u ( x ) ) - f(O) + ]f'(O)iu(x). Then 9(0) = 0, 9'(u) = i f ( u ) + If'(0)], f " ( u ) > 0 for u > 0 and, therefore, 9'(u) > 0 for u > 0 and 9(u) > 0 for u > 0. Now, (1.1)-(1.2) can be rewritten as
--Au(x) = 2{9(u(x)) + f ( O ) -- [f'(O)lu(x)}, Bu(x ) = O,
x E Of2.
x E O,
O"(u) =
(2.1) (2.2)
C. Maya, R. Shivaji/Journal of Computational and Applied Mathematics 88 (1998) 125-128
127
Let u0 be any nonnegative solution of (2.1)-(2.2). Then the linearized equation about Uo(X) is
-A4,(x) - 2{g'(uo(x)) - If'(o)l}¢(x) =/~ep(x), x E f2,
(2.3)
Bdp(x) = O, x C dO.
(2.4)
Let /q be the principal eigenvalue and let ~(x)( >~O) be a corresponding eigenfunction. We calculate (2.1)g'(uo)~(x)- (2.3)g(u0) and then integrating over f2 yields
2 {(-Auo)g'(uo)tP(x) - (--A ~(x))g(uo) -- 2f(O)g'(uo)~(x) + ,~lf'(O)luo(x)g'(uoW(x) -
21/'(0)1 q'(x)g(uo)} dx
= - ~ l f~ ~(x)g(uo)dx.
(2.5)
But by Green's first identity
( dUo
£ ( - Auo )g' (uo )tP(x ) dx = Jo V'(g'(uo)~P(x))~7Uo(X) dx - f~o g'(uo)~(s) k,-~n J as = f~ g"(UoW(X)l VUol 2 dx
+ £ d(uo)(W'Vuo)dx -
(dUo~
f g'(uo)~'(s)\ dn ,I ds
(2.6)
and
f (AT(x))g(uo) dx = -
7(g(uo))V~(x)dx +
o g(u°) ~n
ds
= - fog'(Uo)(V'UoV'~)dx + L g(uo) (~n ) ds.
(2.7)
By using (2.6)-(2.7) in (2.5) we get
£
£
dx
- 2 f ( O ) fo g'(uo)~(x) dx + 21f'(O)l
L{g'(uo)uo(x) - g(uo)} ~(x) dx
(duo~ ~ d,. + fo{g(uo)(~n)-¢(Uo)~(s)~dnjj
(2.8)
We notice that when ~ = 1 (then h = l ) we have u 0 = 0 for sEdf2 and, therefore, g(uo)=0 for s E dr2 and also we have ~ = 0 for s E dr2. Hence,
L {g(Uo)(Tn ) --g'(uo)~(s)(dUo) \ dnj } ds=O
(2.9)
C Maya, R. Shivaji/Journalof Computationaland AppliedMathematics88 (1998) 125-128
128
and when ~ ¢ 1, we have
f~ {g(uo)(~f~) -g'(Uo)~'(s)\ ~n/}
f
ds = J,9l [ (1 - 0~)J [uog'(uo) - g(uo)] ds.
But e i> 0, h > 0, T >~ 0 for s E 0£2 and uog'(uo) Therefore, if e ¢ 1
-
g(uo)
> 0 for Uo > 0.
(Ouo
fo~ { g(uo) ( ~n ) - g'(uo)ql(s) k c~nj } ds ~ O. Also, since
g"(Uo) >
(2.10)
0 for Uo > 0, w e get
f g"(uo)'e(x)lVuol 2 dx > 0.
(2.1 1 )
Thus, using (2.9)-(2.11) in (2.8) we obtain
(-#~) ~ T(x)g(uo)dx > - 2 f ( 0 ) L g'(uo)T(x)dx
+Alf'(O)l j~ {g'(uo)uo - g(uo)} T(x) dx.
(2.12)
Now by using (1.3)-(1.4) and the fact that uog'(uo) - g(uo) > 0 for Uo > 0 in (2.12) it is easy to see that ( - l q ) f~ T(x)g(uo) dx > O. But T > 0 for x E (2 and g(uo) > 0 for Uo > 0 and hence, #1 < 0 and the result follows (see [4]). References [1] K.J. Brown, R. Shivaji, Instability of nonnegative solutions for a class of semipositone problems, Proc. AMS 112 (1) (1991) 121-124. [2] A. Castro, J. Cossio, J.M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain. J. Math., to appear. E3] A. Castro, R. Shivaji, Semipositone problems, semigroups of linear and nonlinear operators and applications, in: J.A. Goldstein, G. Goldstein (Eds.), Nonlinear Operators and Applications, Kluwer Academic Publishers, New York, 1993, pp. 109-119 (invited review paper). [4] D.H. Sattinger, Monotone method in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (11) (1972) 979-1000. [5] A. Tertikas, Stability and instability of positive solutions of semi-positone problems, Proc. AMS 114 (4) (1992) 1035-1040.