Nonexistence of positive solutions of quasilinear differential equations

Mathl. Comput. Modelling Vol. 26, No. 5, pp. 1-8, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/97 $17.00 + 0.00

PII: s0895-7177(97)00154-1

Nonexistence of Positive Solutions of Quasilinear Differential Equations HUEI-LIN HONG Department of Mathematics, National Central University Chung-Li, 32054Taiwan, R.O.C. Fu-HSIANG WONG Department of Mathematics and Science, National Taipei Teacher’s College 134, Ho-Ping E. Road, Sec. 2, Taipei 10659, Taiwan, R.O.C. [email protected] CHEH-CHIH YEH Department of Mathematics, National Central University Chung-Li, 32054 Taiwan, R.O.C. YehQwangwei.math.ncu.edu.tw

(Received and accepted July 1997) Abstract-Consider

the following quasilinear differential equation: ( IU’(t)lp-2 u’(t))’ + f(t, u(t)) = 0,

a < t < b,

p > 1,

(E)

subject to one of the following boundary conditions: u(a) = u(b) = 0,

(J=I)

u(a) = u’(b) = 0, u’(a)

=

W2)

u(b) = 0.

WC31

Under suitable conditions on f(t, u), we obtain the nonexistence (BCi), i = 1,2,3.

Keywords-Positive

solution, Nonexistence, p-Laplacian problems. 1.

In this paper, we shall consider

?I@))

to one of the following

INTRODUCTION

the quasilinear

(IU’WIp-2 subject

of solutions of (E) with respect to

differential

+ f(t,u(t))

boundary

= 0,

equation a
p>l,

(El

conditions: u(a)

=

u(b) = 0,

WA)

u(a) = u’(b) = 0,

W2)

u’(a) = u(b) = 0,

(BC3)

Tywet

1

by

d&-W

2

H.-L.HONG

where p > 1 is a constant, f E C( [a, b] xlIp, (0,oo)). in the study of non-Newtonian fluids, nonlinear

et al.

The problems considered here have application elasticity, and reaction-diffusion. It has been

studied by many authors, such as Agarwal and O’Regan [1,2], Anane [3,4], Hirano (51, Naito [6], Narukawa and Swzuki [7], and O’Regan [8], ( see also references therein). They established several existence results. Further, Wang and Gao [9] considered a quasilinear elliptic boundary value problem

of the form -A+

= F(U) + h(z),

in 0,

where F E C((-co, co), (0, oo)), R is a bounded domain and A,, denotes the pLaplacian A,u := div(lVulp-2)Vu), the following THEOREM

two interesting

A.

results

PW

on as2,

u = 4,

concerning

in RN (N 2 1) with smooth

boundary,

with p > 1 a constant. They obtained the nonexistence of positive solutions.

Suppose

where p* is defined by $

:= (P-;‘“)‘-““(P;‘;“)

(P-:+6)p-’

(P - 1)(6 - 1)

Here p > N, 6 > ((p - l)N)/(p open ball Bo contained in R.

-(p-1+6)

P-l

(P-N)6--(p-l)N p-1+6’p-1+6

- N), 0 is the beta function,

)

and ro is the radius

r;.

of the largest

Then, the boundary value problem (BVP) has no positive solutions in W19p(fl) n Loo(Q) for any given h(z) E L”(R) with h(z) 2 0 a.e. in R and for any given q5 E W1-(‘~P)J’(ds2) nLW(as2) with rj(x) 2 0 a.e. on dR. THEOREM

B. Suppose

00 I0

udu

**

F2/(~-1)(4

’ CL

’

where p** is defined by

** :I-1

P,’

(2~=~))(p-l)‘(p-1)~~,(p_1).

Here, TO is the same as in Theorem A. Then the boundary value problem (BVP) has no positive solutions in IGP(s2) fl LOO(R) for any given h(z) E LOO(R) with h(z) > 0 a.e. in R and for any given q5 E W’-(‘~p)~p(lX2) f~ LW(dR) with 4(z) > 0 a.e. on 82. Wang and Gao [9] discussed the boundary value problem (BVP) on a ball contained in R. The purpose of this paper is to improve Theorems A and B. Our results also can be used on’an annulus domain. In Section 2, we will state and prove our main results. In Section 3, we will state some corollaries. 2. THEOREM

1. Suppose

MAIN

RESULTS

f E C( [a, b] x W, (0, co)) and

(Cl)

Quasilinear Differential Equations

where fo(u) := mintE[,,b] f(t, U) and p1 is defined by

CL1 :=

(;)p-l+* (“p’;“) (p-~+~)‘-18(p_~+~‘pPT~6)-(p-~+~)(b_a

Here S is a positive constant such that 6 > 1. Then the boundary value problem (BVG) with boundary condition (BCi)) has no positive solutions for each i = 1,2,3. PROOF.

Suppose that u(t) is a positive solution of (BVPl), (p - 1) Juy2

u” = --j(t, IL),

(i.e., (E)

thus, it satisfies in (a,b).

(1)

Since ~(a) = u(b) = 0, it is clear that there exists a unique to E (a,b) such that ~‘(to)l = 0, M = ?.&(tO)= ma&+$] u(t), U’(t) > 0 in (a, to), u’(t) < 0 in (to, b). Now for any given (5 > 1, multiplying (1) by Iu’16 and integrating it, we obtain =-P-l+6 IUyt)lp-‘+6

P-l

,p-1+6 _ p_l

=

“,--:”

[[IdrILL

1 ‘---;‘(b-.)l-’

[rf’+,u)u’dr]’

(by6>

[l;jj;“(s)ds]6

l),

where we have used the Hijlder inequality with one negative exponent and the equality holds if and only if t = to. Therefore, by (2), for t # to,

,u’)

[(-)

#‘(s)ds]

-6’(p-1+6)

> (’ ,-;

“) 1’(p-1+6) (b _

a)U-6),Cp-l+6).

Integrating the above inequality over [a, to] and [to, b], respectively, we have -6/(p-1+6)

-6/(P-1+6)

dv 1 (b _ a)(l-6)l(p--1+6)(t0

_ a)

and

1

-6/(P-1+6)

-6/(p--1+6)

dt =

fo”6(s) ds

ly’-l+6)

> (” 4;

dv

(b _

a)(l-6)/(p-l+6)(b

_

to).

A combination of the above two inequalities yields

J (J M

0

-6/(p-1+6)

M

v

f;‘6

(s)

ds

dv>;

(P~~:b)“‘p-1+6’(b_n)Yi~~-~+~)

(3)

4

H.-L. HONG et al.

Next, condition (Cl) implies that

up-’5 Afo(u),

for21 E [O,oo),

i.e., u(P-1)/6

A'/6f,1/6(u)

<

for 21E [O,oo).

9

Integrating the above inequality over [v, M], we get b p-1+6

@P1+6)/6

_ v(P-l+s)/6)

5

Al/6

1”

f;/$j

ds,

i.e.,

x A”(P-‘+6)

M(P-1+6)/6

_

-6/b-1+6)

+1+6)/6

(

>

Integrating it over [0, M], we obtain

I” (lMf;‘6(s)ds)-6’ dv (‘-1+6i

=

(”- i ’“)“(p-l+a) A’/(P-l+6) lM (j@p-1+6)/g _ (p_i +J -a’(p-l+a) A~/(P-I+~) lM + (1 _ (;J@-1+6)/6)

=

(p

_

f + 6)b-1”(pm1+6i

A1/(p-1+6)

=

(p

_

y + 6)(pM1”(pm1+6)

A11(P-1+6) ~‘~6/(~-1+6)-1(~

=

(p_~+~)~p~1~‘~p~1+6~~~~~~-~+6~~(p_~+~,

I

where

fl

v(P_l+6)i6~

~-~P-WP-~+~~~~

It follows from (3) and (4) that

1’

F-1(1

-6h-l+6)

dv

_ E~(p_l~~~p_1+6~_l

ppy:d),

- t)V-’ dt,

dv

_ cJ_6/(p_1+6)

is the usual b&a function

.8&Y> =

i.e.,

1’

-~/CP--1+6)

for 2, y > 0.

~

~

(4)

Quasilinear Differential Equations which contradicts solutions.

condition

(Cl).

the boundary value problem (BVP1) has no positive

Therefore,

The proof of cases (BVP2)

5

and (BVP3)

are similar to that of (BVPl),

so we omit the

details. that f E C([a, b] x IR, (0,~))

THEOREM 2. Suppose

and

where ~2 is defined by ‘plkl)

and fo is defined in Theorem 1. Then, the problem (BVPi) i = 1,2,3.

PROOF. Suppose that u(t) is a positive solution of (BVPl),

has no positive

solutions for each

thus, it also satisfies (1).

Since

u(a) = u(b) = 0, it is clear that there exists a unique to E (a, b), such that u’(to) = 0, hil = u[to) =

maxtE[+l u(t), u’(t) > 0 in (alto), u’(t) < 0 in (to, b) and u(t) > 0 on (a, b). Multiplying 111) by (~‘l-(p-~)/~

and integrating it, we get ,4P-w

=

5

sgn (to - t)

2 i (sgn(to

-t)

x (sgn(t,-t)[’

2

;

Ito _

s

t to f (7, u) (~‘l-(‘-‘)‘~ [

dT

12/(P+1)dr)(pt1)‘2

f-2/(p-1)(T,u)[u’I

tl(p+1)/2( lt” f

-2,(P-l+,

dr)lp-1)‘2

+‘d7)

--(p-1)‘2

,

where we have used the Hijlder equality with one negative exponent. Hence,

(5)

Next, we separate the proof into the following two cases.

CASE 1. Suppose that to > (a + b)/2. Integrating (5) over [a, to] and using integration by parts, we get

I

s

to 1u’J tto f-2’(p-1)(y_+idTdt a

= u(t)

stto

f- 2’(P-1)(qu)u’dT

to

to +

Ia

uf-2’(p-1)(qu)u’d~

sa

6 CASE 2.

H.-L. HONG et al.

Suppose that to 5 (a + b)/2. Integrating

(5) over [to, b] and using integration by parts,

we obtain

J,f Jtto lu’l

f-2’(p-1)(7-,u)u’dTdt

= -u(t)

Hence,

J 0

M

2)2’(p-1)(p-_l) -2/+1)(s) ds2 ( 1. sfo

(bg’(p-1)

.

(6)

The remainder can be proven by using (6), and adopting a similar proof to that of Theorem 1. We omit the details here.

3. APPLICATIONS The problems under consideration pLaplace equation

arise in studies of radially symmetric

Apu+f*(z,u)=o,

sxWN,

Ra,
solutions of the

0%)

~11

with one of the following sets of boundary conditions:

au

on IzI = &,

and

U = 0,

on 1x1= Ri,

(IV)

21= 0,

on lzl=&,

and

du dr =O,

on 121= Ri,

(BC;)

?.b= 0,

on 121= Re,

and

u = 0,

on 1x1= Ri.

(BG)

z=,

0

Here, f * E C(i2 x W, (0, oo)), r = 1~1,and g d enotes differentiation in the radial direction. A radial symmetric solution u = u(r), T = 121,of (El) satisfies the ordinary differential equation ‘+N-1 f

(Iu’Ip-221’)

ju’Ip-2 u’ +

With the change of variables, t = dp- N)h-l) an equation of the form of (E), where &P-N)/(P-l),

f *(r, u) = 0,

&
(7)

(for p # IV) or t = log r (for p = IV), (7) becomes

R~-N)/(P-‘) >

&.P-N)/(P-‘1,

,

@P-N)/(P-1)

and

f&4

= IsI’

@‘-Wp-Wf*

(

+4/b-N),

u 9 >

forP

# N

Quasilinear Differential Equations Ol-

a = log&,

b = log RI,

and

f(t,u)

= eNtf*

(et,u)

and boundary condition (BCf ) become (BCi) for each i = 1,2,3. to the following quasilinear elliptic problem:

u(x) > 0,

COROLLARY 1. Suppose

p++---:“)

(J32)

do,

the following

two corollaries.

by

(P-~‘“)“-‘((P~~~~~)(~(P;:)N)~-’ 6

-(p-1+6)

P-l

( p-1+S’p-1+6

x

on

that g E C(Q x R, (0, co)) and

where go(u) := infzER g(x, u), & is defined

XP

Hence, we can apply our results

in 52,

A,u + g(x, U) = 0, where 0 = {x E RN 1 Ro < 1x1 < RI}, and obtain

for p = N,

,

)

b((~-“)6-(~-1)N)l((p--1)(6--1))

_

a((~--n)6-_(~-l)N)/((p-l)(6-l))

1-6

(

>

)I

p-l+6

bp-W/(P-1)

_

a(~--NM~--l)

Here, p > N 2 1, 6 is a positive number so that 6 > ((p - l)N)/(p value problem (E&) has no positive solutions in W11P(s2) n L”(0).

- N).

PROOF. Suppose that G(z) is a solution of (Ez) for p > N > 1. Then, the argument in proving the weak maximum principle in [lo]. Let us consider the quasilinear elliptic problem of the form

It is clear that Q(Z) E 0 is a lower solution Therefore, using the result in [ll], we see that

=

on

0,

(E3)

aa.

a.e. E R.

Note that the solution V(X) should be of radial symmetry by the results for all z E 0. Then, w(z) must satisfy (7) with the boundary condition Similarly,

1, we can obtain

result.

that g E C(s2 x R, (0,oo))

where p > 1, N 1 1, and ~5 is defined cl; :=min

{

and

by

(;)(p+‘)/(@) Jdd”+Rl)i2 ( fh+Rl)/2

(;>

dT)

and go is defined

as in Corollary

in W’vP(n)

n Lm(R).

1.

Then,

the boundary

dT)

(N-I),(~+I)

( ;)(p+l)/(p--lJ l;+Rl),2(l&+Rl),2(;) (N--l)~b+l) solutions

in [12], i.e, V(X) = v(lzl), (BC1) or (BC3). Hence,

the conclusion.

we have the following

COROLLARY 2. Suppose

ii(z) L 0 in Sz, by using

of (E3) and fi(x) is an upper solution of (E3), (Es) has a solution w E W”+(0), such that,

0 5 V(X) I G(z),

by Theorem

the boundary

in 0,

A,u + g(z, U) = 0, u(x)

Then,

(p+l)‘(p-l)dt,

(p+l)l(p-l)

value problem

(J$)

dt}

,

has no positive

H.-L. HONG et al.

8

REFERENCES 1.

R.P. Agarwal and D. OXegan,

Some new existence results for differential and integral equations,

Nonl.

Anal. TM&A (to appear). 2. 3.

R.P. Agarwal and D. O’Regan, Positive solutions to superlinear singular boundary value problems, Jour. Computational and Appl. Math. (to appear). A. Anane, Simplicitb et isolation de la prem&e valeur propre du pLapla.cien avec poids, C. R. Acad. Sci. Paris 305, 725-728 (1987).

4. A. Anane, J&ude des valeurs propres et de la esonance pour l’operaeur plaplacien,

Th&e de Doctorate, Universite Libre de Bruxelles, (1988). 5. N. Hirano, Multiple solutions for quasilinear elliptic equations, Nonlinear Anal. TM&A 15, 626-638 (1990). 6. Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Di&rential and Zntegml Equations 8 (7), 1813-1822 (1995). 7. K. Narukawa and T. Suzuki, Nonlinear eigenvalue problem for a modified capillary surface equation, finkc. Ekva. 37, 81-100 (1994). 8. D. O’Regan, Some general existence principles and results for (4(~‘))’ = qf(t, y, y’), 0 < t < 1, SIAM J. Math. Anal. 24 (3), 648-668 (1993). 9. J. Wang and W. Gao, Nonexistence of solutions to a quasilinear elliptic problem, Nonl. Anal. TM&A 28 (3), 533-538 (1997). 10. D. Gilbarg and N.S. Trudinger, Elliptic Partial Equations of Second Order, Springer, Berlin, (1983). 11. J. Deuel and P. Hess, A criterion for the existence of solutions of nonlinear elliptic boundary value problems, Proc. Royal Sot. Edinburgh 74, 49-54 (1975). 12. B. Gidas, W.M. Ni and L. Nirenberg, Symmetries and related properties via the maximum principle, Comm. Math. Phys. 68, 209-243 (1979).