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Positive solutions for Dirichlet problems associated to semilinear elliptic equations with singular nonlinearity
Nonlinear Amdysis, Theory, Printed in Great Britain.
Methods & Applicalions,
Vol. 21, No. 3, pp. 181-190,
1993. 0
0362-546X/93 $6.00 + .OO 1993 Pergamon Press Ltd
POSITIVE SOLUTIONS FOR DIRICHLET PROBLEMS ASSOCIATED TO SEMILINEAR ELLIPTIC EQUATIONS WITH SINGULAR NONLINEARITY GUI SHANGBIN Department (Received
of Mathematics, 1 March
Lanzhou
1991; received
Key words and phrases: solution.
University,
Lanzhou,
in revised form
8 January
Semilinear
elliptic equation,
Gansu
730000,
People’s
Republic
1993; received for publication
singular
nonlinearity,
Dirichlet
of China
1 March 1993)
problem,
positive
1. INTRODUCTION
IN THIS PAPER
we are
going to study the existence of the solution of the following problem Lu
I
+ f(x,
u = 0,
24, Du)
= 0,
on X&
u > 0,
in 0,
(1.1) (1.2)
where Q is a bounded domain in R” with smooth boundary, L is a second order uniformly elliptic linear operator, and&x, u, c) is a C’ function defined on Sz x (0, +co) x R” which may be singular at u = 0 with respect to the second variable U. Existence of a positive solution for the problem (1.1) and (1.2) is an interesting subject of study. A great number of results have been obtained in the case wheref(x, U, <) is defined and of C’ class on Q x (-00, +a~) x R”, both for bounded domains Q and unbounded domains Q, all concerning either the uniqueness of solutions or the existence of multiple solutions (cf. [l-3] for instance). Some results in the case wheref(x, U, 0 is defined on Q x (-co, +m) x R” and is continuous but not of C’ class or even not continuous with respect to the second variable at u = 0, have also been obtained (cf. [4], for instance). Recently, some attention has been paid to the case wheref(x, u, <) is only defined on fl x (0, +CCJ)x R” and has no limit as u -+ O+, that is, the nonlinearity is singular (cf. [5-131, for instance). However, all the discussions concerning singular nonlinearity are greatly limited at the point that the problems considered have very special nonlinear termf(x, u, c) as well as a very special domain Q. In particular, only nonlinear terms of the form f(]xl , u), as well as domains which are balls or annuli, are considered if the domain fi in the problem (1.1) and (1.2) is required to be bounded. In the present paper, we are going to apply the upper and lower solutions method, combined with the perturbation method, to derive a general existence result for positive solutions of problem (1 .l) and (1.2) with singular nonlinearity. Our restriction on the nonlinear term f(x, U, <), as well as the domain Sz, is very weak and the result obtained here can, thus, be used to deal with a wide class of nonlinear elliptic boundary value problems with singular nonlinearity which may have convection terms. In the following section we state our main result and give some applications of it. The proof of the main result is left until Section 3. 181
182
CU!
2.
THE
MAIN
%lANGBlN
RESULT
AND
ITS APPLICATIONS
Our basic assumptions are as follows: (A,) n is a bounded domain in R" with smooth boundary; (AJ L = Cy,j= 1a,(X)(d'/dXi 8X,> + I;=1bi(X)(d/dXi), where (0 c a < l), and there is a positive constant L such that i i,j=
aij(xlyiYj
2
vy E
VXEsz,
LlYl2~
au, bi are
all
R";
in
P(a)
(2.1)
1
(A,) for any E > 0, the restriction of the function f(x, u, [) on 0 x [E, +oo) x R" has an extension&(x, u, 4) on a x (-00, +oo) x R" which satisfies the conditions in [2], i.e. for any fixed (u, r) E (-a~, +a) x R",.j,(x, u,t)belongs to C=(Q) as a function of x, andf,(x, u, r) is differentiable with respect to the variables u and r, and df,/au, af,/lgj (j = 1,2, . . . , n) are continuous in 0 x (-00, +w) x R"; (AJ apart from the condition (A&, the functionf(x, u, c) satisfies the following conditions: (i) there is a constant p < 1 such that for sufficiently small u > 0 and any r E R" sufficiently near to the origin the following holds f(x, u, 0 2 AUP, (ii) thereexistconstantsp’<
l,O
J-(x, u, 0 s B, + &UP + &l&r +
vxen;
l,Osr<2ands<
&uSlrl’,
VXEQ,
(2.2) 1 -rsuchthat
v u E (0,+a),
v[eR". (2.3)
Here A, Bjdenote different positive constants. The main result is the following theorem. THEOREM1. Under the above assumptions, the problem (1.1) and (1.2) has at least one solution which belongs to C2’,(Q) fl C(n). This theorem will be proved in the following section. We now give some applications of it. Let us first consider the following problem
Lu + u(x)uP - b(x)uP’(l + IDul2)q = 0, u = 0,
on
in Sz,
an.
(2.4)
(2.5)
Here, as well as below, n denotes the bounded domain with smooth boundary, elliptic operator as given in the condition (A,).
and L is an
THEOREM2. Suppose that a(x), b(x) E Ca(@ (0< (Y< 1) and 0 < Cl 5 u(x) I
c,,
0 s b(x) I
c,,
VXEC-L
If p < 1, p’ > p and q I 1, then the problem (2.4) and (2.5) has at least one positive solution U(X) E c2+yn)
n c(Q).
183
Semilinear elliptic equations
Proof. The conditions
(Ai)-
are obviously satisfied. We have
a(x)zP - b(x)u@(l + l
- 2c2uP’-p) 1 &.P, VI<1 5 1,
VUE(0.(4(1CtiC2))l’(il’-il)).
and a(x
- b(x)LP’(l + lr1*)4 I c*lP,
VXE!&
v U E (0, +oo),
v
Thus, by theorem 1 we see that the problem (2.4) and (2.5) has at least one positive solution U(X)E c*+U(Q) n C@). Remark.
If p c 0, then the nonlinearity in (2.4) is singular. If furthermore, p 5 -1, then the nonlinearity in (2.4) is singular and nonintegrable. The same remark can be made for the following results. Next let us consider the following problem Lu + a(x
u = 0, THEOREM 3.
+ b(x)(l + IDu12)q - g(x)up’ - h(x)IDulq’ = 0,
in L2,
on an
(2.6) (2.7)
Suppose that a(x), b(x), g(x), h(x) E C”(n) and
0 < C, 5 a(x) 5 C,,
c, I b(x) 5 c,,
0 I h(x) I c,,
Olg(x)lC*,
VXEsz.
If p < 1,O < q < 3, p’ > p and 2q < q’ I 2, then the problem (2.6) and (2.7) has at least one positive solution U(X)E C*+Ol(Q)fl C@). Proof. Obviously, the conditions (A,)-(A,) a(x
are satisfied by this problem. Moreover, we have
+ b(x)(l + l<12)q- g(x)up’ - /z(x)l~lq’
L cr2.P + c,l
- c*up’-P) + lrl*q(ci - c*l~l9’-*9)
and a(x
+ b(x)(l
+ /
v 24E (0, +oo),
VEER”,
which means that the condition (AJ is also satisfied. Hence, by applying theorem 1 we get our conclusion.
184
GUI
SHANGBW
Our next application shows that in some cases the solution guaranteed unique. Consider the following problem Lu f m(x)uP(l + ]Dz@)Q = 0, u = 0,
by theorem
in Sz,
1 is (2.8)
on XX
(2.9)
THEOREM4. Suppose that m(x) E C”(Q) and 0 < C, I m(x) I C,. If either (i) qlOandp
f(x, u90 5
vu>o,
VXEQ
f(x, u, 0 2 2qc1up, c2up,
v ItI 5 1,
VXEQ
vu>o,
v
vxen,
vu>o,
v
and under condition (ii) we have f(x, u, 0 2 CIUP, f(& u, 0 5
c2up
+
vu>o,
VXEQ,
c2~plt12q,
v
Thus, by a direct application of theorem 1, we get the conclusion. To end this section, we give an example that can be solved exactly. Consider the following singular boundary value problem (2.10)
-R < x < R,
u”(x) + u(x)-P = 0 , i u(+R) = 0,
(2.11)
where p > 1 and R > 0. According to theorem 4, we know that this problem has a unique positive solution. We now construct its solution. One may easily verify that for any m > 0, the integral y&P
_
&-P
) -l/2
*
s0 is finite, and the function g(m) defined by this integral is monotone Moreover, the following holds lim g(m) = 0, m+o+
increasing in (0, +oo).
lim g(m) = +=J. ?Pl++CG
Thus, for any A > 0 there exists exactly one positive number, which we denote as m(R), such that g(m(R)) = R. Set fR(u)
=
’ (s’-~ - r~~(R)‘-~)-“~ds, 0
v u E (0, m(R)].
Semilinear
185
elliptic equations
It is obvious that the function fR(u)is monotone increasing in (0, m(R)], and Jim+ fR(u)= 0,
fR(m(R))= R. We now define a function uR(x) on [-R, +R] as follows
uR(x)
=
fi’(R + 4,
if x E (-R, 01,
fi’(R - X),
if x E [0, +R), ifx=
1 0,
&R.
One may easily verify that this function belongs to C”(-R, +R) n C[-R, +R] and it is the unique solution of the problem (2.10) and (2.11). A similar method can be used to construct the unique solution of the following boundary value problems U”(X)+ CzP( 1 + &(x)2)4 = 0, u(kR)
-R
< x < +R,
= 0,
where C is a positive constant and p, q are as in theorem 4. We leave it to the interested reader. 3. THE
PROOF
OF THE
MAIN
RESULT
This section is arranged to prove theorem 1. The method of proof used here can be outlined as follows: first, we prove that for sufficiently small E > 0, the perturbed problem
u > E,
f(x, 2.4, Du) = 0,
Lu +
in Sz,
on afi
i U = E,
(3.1) (3.2)
has a solution u = u, by using the upper and lower solutions method; secondly, we prove that there is a sequence Ej > 0 (j = 1,2, . . .) converging to zero, such that the corresponding sequence of functions ucj (j = 1,2, . . .) converges to a solution of the problem (1.1) and (1.2). By the Krein-Rutman theorem, we know that under the conditions (A,) and (AZ), there exist a function 4(x) E C2+,(fi) tl Cl@) and a positive number p such that Uj
=
U(x)
= -M(x),
where n denotes the outward-pointing LEMMA
1. Under
the assumptions
in Sz,
4(x) > 0,
unit normal vector on XL
(A1)-(A4),
there
exist
constants
E* > 0, c > 0, A4 > 0 and
v E
(0, I), such that for any E E (0, e*], the problem (3.1) and (3.2) has a solution u = u, which belongs to C2+,(Q) fl C(a) and satisfies the following inequalities VXEQ.
c+(x) + E I U,(X) I M(4(x) + E)“,
(3.3)
Proof. From the assumption (AJ (i) we see that there exist constants a > 0 and b > 0 such that the inequality (2.2) holds when 0 < u 5 a and I<[ I b. Set E,, = $a, and take v E (0, 1) such that 2 1 -p”l
(
v 5 min -
2-r -r--s
>’
186
GUI SHANGB~
For 0 < E c .eO, let ii,(x) = &J(X) + E and h,(x) = M($(x) + a)“, where c and M are positive constants to be determined. We now prove that ii,(x) and i;,(x) are lower and upper solutions of the problem (3.1) and (3.2), respectively, if c and E are sufficiently small and A4 is sufficiently large. and let Mr = maxlD$(x)l. Let MO = yy$d~(~~l,
Then if 0 < c I min(a/2M,,
xek=l
have
we
VXEf-2,
l~~,(x)I5 b,
E < ii,(x) 5 a,
b/M,),
and, thus, by applying the inequality (2.2) we get Lii, + f-(x, ii,, Ix,)
2 -pop
+ A(&
r -p(&
+ &)p
+ E) + A(@
+ &)p
= (a$ + &)p(A - p(cc$ +
&y-p) El1-P 1,
Therefore,
in Q.
by setting
and c* = min &, (
$,
&(;~‘op)).
we see that the inequality Lii, + f(x, ii,, lx,)
2 0
holds in a if 0 c c I c* and 0 c E 5 E*. This, combined with the fact that @Jan = E, means that ii, is a lower solution of (3.1) and (3.2) when 0 < c d c* and 0 < E I a*. Next we notice that D&(x) = vM(+(x) + &)“-%#J(X), and L;,(X)
=
=
V(V
-V(l
l)M($(X)
-
-
+ &)‘-’
V)M(+(X)
i
i,j=
+ &)“-’
&j(X)
84(X) x
1
I
%4x)
aij(X) x 1
I
-Lv(l
-
v)M(9(x)
+ &)“-21Dr$(x)12
5
-Av(l
-
v)M(+(x)
+ &pg(x),
x I
-
+ vM($(x)
+
Ey+‘L~(X)
J
Mx)
i
i,j=
84(X) x
-
PVMHX)
J
/wA4(4(x)
where
g(x)= ImJ(x)12 + *(I r.- v)w2*
+ Ey-+qX)
+ dY-%(X)
187
Semilinear elliptic equations
Thus, by applying the inequality (2.3) we get LB, + f(x, ii,, DC,) I - Av(l - v)M(+(x) + @‘-*g(x) + B, + B,MP’(+(x)
+ E)“@
+ B, v*Mq(4(x) + &)(y-l)qlDqb(x)lq + B, ~‘M’+~(c$(x) + &)+1)r+vs/D4(x)lr = M(c$(x) + c)‘-*[+(l + B,M-“-P”($(X) + B, ,fM-(1-r-s)
- v)g(x) + B,M-‘(4(x)
+ e)*-’
+ E)2-d’-P’) + B2v~M-‘1-4)(~(x)
+ e)*-W’(‘-4)jD+(x)14
(4(x) + ~)(*-~)-“(‘-~-~)lD~(x)l~].
Notice that the properties of the eigenfunction
4 guarantee that ming(x) > 0. Moreover, the xsii assumptions on the constants v, p’, q, r and s guarantee that all the exponents of 4(x) + E and ID+(x)1 in the square brackets are positive, and, thus, the functions in the square brackets containing the function 4 are uniformly bounded above for E E (0, &*I. Therefore, we get the following inequality Lli, + j-(x, ii,, DC,) I M(c#J(x) + &)‘-*(-co
+ C,M_’
+ c2M-(1-p’)
+ C3M-(I-@
+ C4M-(1-‘-S)),
in Sz, where all CjS are positive constants independent of M and E. From this inequality we see immediately that there exists a constant M, > 0, which is independent of E, such that the inequality Lii, + j-(x, ii,, Dt,)
I 0
(3.4)
holds in fi when M L M,,. Set M* = max(M,, (c*)‘-“). Then apart from (3.4) we have liejan = ME’ 1 E whenM L M* and 0 c E 5 E*, which implies that ii,(x) = M(c#J(x) + E)”is an upper solution of the problem (3.1) and (3.2). Now let us consider the following problem Lu + f,(x,
L u = E,
u, Du) = 0,
u > E,
in Q
on asz
(3.5) (3.6)
where the function f,(x, U, 4) is as in the condition (As). Obviously, the functions ii,(x) = c~(x) + E and G,(x) = M(d(x) + E)’ are also the lower and upper solutions of the problem (3.5) and (3.6), respectively, for any 0 < E I E*, 0 < c 5 c* and M 1 M*. Thus, by choosing the constant c E (0, c*] sufficiently small and the constant M E [M*, +a) sufficiently large so that the inequality c~(x) + E 5 M($(x)
+ E)‘,
VXESJ,
v & E (0, &*I
holds (which is possible because 0 < v < l), we conclude by applying the first theorem of Amann [2] that there exists a maximal solution of (3.5) and (3.6) which we denote by uE(x), such that c+(x) + E I u,(x) 5 M(+(x) + E)“,
VXEsz,
and u,(x) E C2+U(M) fI C(d). Notice that the first inequality above guarantees that f,(x, uE, Du,) = f(x, uE, Due), and, thus, u = U, is also a solution of the problem (3.1) and (3.2). Lemma 1 is thus proved.
188
GUI SHANOBIN
LEMMA2. For any C2+” -smooth domain a’ CC a, there exists a constant C independent of E, such that v &E (0, &*I.
lI%IICZ+~(W) 5 C,
(3.7)
Proof. Take three domains Qj, j = 1,2, 3, with C2+cY-smooth boundaries such that SY CC Q, CC Q2 cc Qr cc 0. Let m = c min 4(x), which is positive since r#~> 0 in a. Let f, be a function as in condition xe4?4 (A3) with E = m. Then for any E E (0, c*] we have J% + fmk %,DUE) = 0,
on Q3.
By the interior estimate theorem of Ladyzenskaya and Ural’treva p. 266]), we get a constant C1 independent of E, such that
(see [16, theorem
3.1,
From this we see that Du,(x) is uniformly bounded on Q2 for E E (0, a*] since ~7~ u,(x) I M(max 4(x) + e*)” and the right-hand side of this inequality is independent of a. Therefore, XEi=l the function _&(x) = f, (x, u,(x), Du, (x)) is uniformly bounded on Q2 for E E (0, &*I. Now let us consider the following linear elliptic boundary value problem Lv =A,
in
( v = 0,
Q2,
(3.8)
on aQa .
(3 -9)
Since x is continuous and bounded on Q2, we see x E LP(Q2) for any p E (1, +a~), and, thus, by theorem 9.15 of [17] we conclude that the problem (3.8) and (3.9) has a unique solution v = v, which belongs to HJ’~,~(Q~)for any p E (1, +m). Moreover, by lemma 9.17 of [17] we see that there exists a positive constant C2 independent of E such that iI veil Wz*J’(Qz)
5
c211J;E11rP(Q2)
v
3
& E
(0,
&*I.
Thus, since J’,(x) is uniformly bounded on Q2 which implies that is bounded by a constant independent of E, we conclude that is also bounded by a constant independent of E. Taking p = n/(1 - a) and applying Sobolev’s embedding inequality, we see is bounded by a constant independent z; v, belongs to Cl+“@,) and the norm (Iv,JI~I+~(Q~) 11 f,llrp(Q9
IIv,IIW~,P(Q2)
Next let us consider the boundary value problem Lw = 0,
1 w = UE)
in
Q2,
(3.10)
on aQ2.
(3.11)
Since u, E C2+u(Q2), we get the conclusion that this problem has a unique solution w = w, which belongs to C2+a (Q2) - by applying the theorem 6.14 of [17]. Furthermore, by Schauder’s interior estimate theorem (cf. corollary 6.3 in [17], for instance) we see that for some constant C, independent of E the following holds iI W~ib+U(QI)
-=
c311 weilC(&)
3
v
& E
(0,
&*I.
189
Semilinear elliptic equations
By the maximum principle we have Y
>
(
IIw&Q~) 5 max u,(x) 5 M max+(x) + E* xsfi
xeaQ2
,
VE E (0, &*I.
is also bounded by a constant independent of E. Hence, we see that ))~~JJo~+aco,, Now, since U, and u, + w, are both solutions of the problem Lu =J;,,
in Qz,
i u = UC,
on aQz,
we get the equality U, = u, + w, in Q2 by applying the uniqueness theorem. Thus, we have b&l+“(Q,)
s
b,IIC’+“(Q,)
+
b%~~C’+a(Q,)
s
v
c4,
E E
(0,
.?*I,
where C, is a constant independent of E. This enables us to get the following estimate iIJ;Eb(Q,)
=
IlfmcG W), ~%(-@)IIC”(Q,) 5 c59
where C, is constant independent
v &E (0,&*I,
of E. Now since u, is the solution of the linear equation
Lu = TE, by applying Schauder’s interior estimate theorem once more we then get
Il”,llc2+~@‘,5
db&(Q,)
+
tl.&acQ1,)
s
v
c7,
E E
(0,
.?*I,
where C, and C, are positive constants independent of E. This finishes the proof of lemma 2. Proof of theorem 1. Set uk = uEI- , where &k = (l/k)&*, k = 1, 2, . . . . Take a sequence [Sz,],“,1 of domains with C2+a-smooth boundaries, such that
Q2, cc n2 cc
***cc szj cc $+i
cc
*--
and lJT= I Qj = Cl. By lemma 2, we see that IIu~JI~z+~(~~) is bounded by a constant Nj independent of E for each j = 1,2, . . . . Thus, by the compactness of the embedding operator of C2’ol(&) into c2(ni), we get a subsequence (U{*‘)T= 1 Of the sequence (u&, 1 COIWfXghg in C2(n,). Similarly, by the compactness of the embedding operator of C2’ol(sZ2)into C2(n2), we get a subsequence 1~~~‘)~~ 1 of the sequence (uj”]T= 1 converging in C2(sZ2). By an induction procedure, we get for each j = 1,2, . . . a sequence (u!“)T= 1 which is convergent in C”(~i> and has the property that (u!j+i) lT=1 is a subsequence of (u,‘j’)T=1. We set uj = uy’ forj = 1,2, . . . . It is obvious that the sequence u/!is convergent in C2(n’) for any domain &2’CC i2. Especially, the sequence {u,!l~=1 converges pointwisely in Q. Denote by u the limit of this sequence. Then u E C2(SZ) and it satisfies the equation Lu = f(x, u, Du)
in n because lujlj”=1 converges to u in C2(a’) for any LY CC Sz. Furthermore, inequalities (3.3) we get c@(x) 5 u(x) 5 M(4+))‘,
from the
VXEQ
which implies that lim u(x) = 0. We, thus, get a solution of the problem (1.1) and (1.2) which x-an belongs to C2(Q) n C(G) if we set u = 0 on XL Applying Schauder’s interior regularity theorem we see u E C2+Ol(n) and, thus, theorem 1 is proved.
190
GUI SHANGBIN
Acknowledgement-The author wishes to express his thanks to Dr Zhang Zhijun for his discussions with him on the problem considered in this paper. He also wishes to thank Professor H. Amann for sending him the article [18] which greatly benefited him. REFERENCES 1. AMANNH., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. math. J. 21, 125-146 (1971). 2. AE~ANNH., Existence and multiplicity theorems for semilinear elliptic boundary value problems, Math. Z. 150, 567-597 (1976). 3. NOUSSAIR E. S., On semilinear elliptic boundary value problems in unbounded domains, J. diff. Eqns 41, 334-348 (1981). 4. STUARTC. A., Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities, Math. Z. 163, 239-249 (1978). 5. BAXLEYJ. V., Some singular nonlinear boundary value problems, SIAM J. mnth. Analysis 22, 463-479 (1991). 6. COCLITEM. M. & PALMIERIG., On a singular nonlinear Dirichlet problem, Commune partial diff. Eqns 14, 1315-1327 (1989). 7. CRANDALL M. G., RABINOWITZ P. H. &TARTARL., On a Dirichlet problem with a singular nonlinearity, Communs partial diff. Eqns 2, 193-222 (1977). 8. DALMASSO R., Solutions d’equations elliptiques semi-lineaires singulieres, Annuli. Mat. puru uppl. 153, 191-201 (1988). 9. DIAZJ. L., MORELJ. M. & OSWALDL., Elliptic equations with singular nonlinearity, Communspurtiul diff. Eqns 12, 1333-1344 (1987). 10. FINK A. M., GAT~CAJ. A., HERNANDEZG. E. & WALTMANP., Approximation of solutions of singular second order boundary value problems, SIAM J. math. Analysis 22, 440-462 (1991). 11. GOMESS. M., On a singular nonlinear elliptic problem, SIAM J. math. Analysis 17, 1359-1369 (1986). 12. LAZERA. C. & MCKENNAP. J., On a singular nonlinear elliptic boundary value problem, Proc. Am. math. Sot. 111, 721-730 (1991). 13. USAMIH., On a singular elliptic boundary value problem in a ball, Nonlinear Analysis 13, 1163-l 170 (1989). 14. Cur SHANGBIN, Comparison principles and uniqueness theorems for nonlinear elliptic boundary value problems, J. Lunzhou Univ. Science Series 22(3), 8-14 (1986). 15. CHENZ. C. & Luo X. B., Comparison and uniqueness of positive solutions for the mixed problem for semilinear parabolic equations, Communspurtiul diff. Eqns 11, 1285-1295 (1986). 16. LADYZENSKAYA 0. A. & URAL’TREVA N. N., Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968). (English translation.) 17. GILBARGD. & TRUDINGER N. S., Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer, Berlin (1983). 18. AMANNH., Periodic solutions of semilinear parabolic equations, A collection of papers in honor of Erich Rothe, Nonlinear Analysis 1, 29 (1978).
Methods & Applicalions,
Vol. 21, No. 3, pp. 181-190,
1993. 0
0362-546X/93 $6.00 + .OO 1993 Pergamon Press Ltd
POSITIVE SOLUTIONS FOR DIRICHLET PROBLEMS ASSOCIATED TO SEMILINEAR ELLIPTIC EQUATIONS WITH SINGULAR NONLINEARITY GUI SHANGBIN Department (Received
of Mathematics, 1 March
Lanzhou
1991; received
Key words and phrases: solution.
University,
Lanzhou,
in revised form
8 January
Semilinear
elliptic equation,
Gansu
730000,
People’s
Republic
1993; received for publication
singular
nonlinearity,
Dirichlet
of China
1 March 1993)
problem,
positive
1. INTRODUCTION
IN THIS PAPER
we are
going to study the existence of the solution of the following problem Lu
I
+ f(x,
u = 0,
24, Du)
= 0,
on X&
u > 0,
in 0,
(1.1) (1.2)
where Q is a bounded domain in R” with smooth boundary, L is a second order uniformly elliptic linear operator, and&x, u, c) is a C’ function defined on Sz x (0, +co) x R” which may be singular at u = 0 with respect to the second variable U. Existence of a positive solution for the problem (1.1) and (1.2) is an interesting subject of study. A great number of results have been obtained in the case wheref(x, U, <) is defined and of C’ class on Q x (-00, +a~) x R”, both for bounded domains Q and unbounded domains Q, all concerning either the uniqueness of solutions or the existence of multiple solutions (cf. [l-3] for instance). Some results in the case wheref(x, U, 0 is defined on Q x (-co, +m) x R” and is continuous but not of C’ class or even not continuous with respect to the second variable at u = 0, have also been obtained (cf. [4], for instance). Recently, some attention has been paid to the case wheref(x, u, <) is only defined on fl x (0, +CCJ)x R” and has no limit as u -+ O+, that is, the nonlinearity is singular (cf. [5-131, for instance). However, all the discussions concerning singular nonlinearity are greatly limited at the point that the problems considered have very special nonlinear termf(x, u, c) as well as a very special domain Q. In particular, only nonlinear terms of the form f(]xl , u), as well as domains which are balls or annuli, are considered if the domain fi in the problem (1.1) and (1.2) is required to be bounded. In the present paper, we are going to apply the upper and lower solutions method, combined with the perturbation method, to derive a general existence result for positive solutions of problem (1 .l) and (1.2) with singular nonlinearity. Our restriction on the nonlinear term f(x, U, <), as well as the domain Sz, is very weak and the result obtained here can, thus, be used to deal with a wide class of nonlinear elliptic boundary value problems with singular nonlinearity which may have convection terms. In the following section we state our main result and give some applications of it. The proof of the main result is left until Section 3. 181
182
CU!
2.
THE
MAIN
%lANGBlN
RESULT
AND
ITS APPLICATIONS
Our basic assumptions are as follows: (A,) n is a bounded domain in R" with smooth boundary; (AJ L = Cy,j= 1a,(X)(d'/dXi 8X,> + I;=1bi(X)(d/dXi), where (0 c a < l), and there is a positive constant L such that i i,j=
aij(xlyiYj
2
vy E
VXEsz,
LlYl2~
au, bi are
all
R";
in
P(a)
(2.1)
1
(A,) for any E > 0, the restriction of the function f(x, u, [) on 0 x [E, +oo) x R" has an extension&(x, u, 4) on a x (-00, +oo) x R" which satisfies the conditions in [2], i.e. for any fixed (u, r) E (-a~, +a) x R",.j,(x, u,t)belongs to C=(Q) as a function of x, andf,(x, u, r) is differentiable with respect to the variables u and r, and df,/au, af,/lgj (j = 1,2, . . . , n) are continuous in 0 x (-00, +w) x R"; (AJ apart from the condition (A&, the functionf(x, u, c) satisfies the following conditions: (i) there is a constant p < 1 such that for sufficiently small u > 0 and any r E R" sufficiently near to the origin the following holds f(x, u, 0 2 AUP, (ii) thereexistconstantsp’<
l,O
J-(x, u, 0 s B, + &UP + &l&r +
vxen;
l,Osr<2ands<
&uSlrl’,
VXEQ,
(2.2) 1 -rsuchthat
v u E (0,+a),
v[eR". (2.3)
Here A, Bjdenote different positive constants. The main result is the following theorem. THEOREM1. Under the above assumptions, the problem (1.1) and (1.2) has at least one solution which belongs to C2’,(Q) fl C(n). This theorem will be proved in the following section. We now give some applications of it. Let us first consider the following problem
Lu + u(x)uP - b(x)uP’(l + IDul2)q = 0, u = 0,
on
in Sz,
an.
(2.4)
(2.5)
Here, as well as below, n denotes the bounded domain with smooth boundary, elliptic operator as given in the condition (A,).
and L is an
THEOREM2. Suppose that a(x), b(x) E Ca(@ (0< (Y< 1) and 0 < Cl 5 u(x) I
c,,
0 s b(x) I
c,,
VXEC-L
If p < 1, p’ > p and q I 1, then the problem (2.4) and (2.5) has at least one positive solution U(X) E c2+yn)
n c(Q).
183
Semilinear elliptic equations
Proof. The conditions
(Ai)-
are obviously satisfied. We have
a(x)zP - b(x)u@(l + l
- 2c2uP’-p) 1 &.P, VI<1 5 1,
VUE(0.(4(1CtiC2))l’(il’-il)).
and a(x
- b(x)LP’(l + lr1*)4 I c*lP,
VXE!&
v U E (0, +oo),
v
Thus, by theorem 1 we see that the problem (2.4) and (2.5) has at least one positive solution U(X)E c*+U(Q) n C@). Remark.
If p c 0, then the nonlinearity in (2.4) is singular. If furthermore, p 5 -1, then the nonlinearity in (2.4) is singular and nonintegrable. The same remark can be made for the following results. Next let us consider the following problem Lu + a(x
u = 0, THEOREM 3.
+ b(x)(l + IDu12)q - g(x)up’ - h(x)IDulq’ = 0,
in L2,
on an
(2.6) (2.7)
Suppose that a(x), b(x), g(x), h(x) E C”(n) and
0 < C, 5 a(x) 5 C,,
c, I b(x) 5 c,,
0 I h(x) I c,,
Olg(x)lC*,
VXEsz.
If p < 1,O < q < 3, p’ > p and 2q < q’ I 2, then the problem (2.6) and (2.7) has at least one positive solution U(X)E C*+Ol(Q)fl C@). Proof. Obviously, the conditions (A,)-(A,) a(x
are satisfied by this problem. Moreover, we have
+ b(x)(l + l<12)q- g(x)up’ - /z(x)l~lq’
L cr2.P + c,l
- c*up’-P) + lrl*q(ci - c*l~l9’-*9)
and a(x
+ b(x)(l
+ /
v 24E (0, +oo),
VEER”,
which means that the condition (AJ is also satisfied. Hence, by applying theorem 1 we get our conclusion.
184
GUI
SHANGBW
Our next application shows that in some cases the solution guaranteed unique. Consider the following problem Lu f m(x)uP(l + ]Dz@)Q = 0, u = 0,
by theorem
in Sz,
1 is (2.8)
on XX
(2.9)
THEOREM4. Suppose that m(x) E C”(Q) and 0 < C, I m(x) I C,. If either (i) qlOandp
f(x, u90 5
vu>o,
VXEQ
f(x, u, 0 2 2qc1up, c2up,
v ItI 5 1,
VXEQ
vu>o,
v
vxen,
vu>o,
v
and under condition (ii) we have f(x, u, 0 2 CIUP, f(& u, 0 5
c2up
+
vu>o,
VXEQ,
c2~plt12q,
v
Thus, by a direct application of theorem 1, we get the conclusion. To end this section, we give an example that can be solved exactly. Consider the following singular boundary value problem (2.10)
-R < x < R,
u”(x) + u(x)-P = 0 , i u(+R) = 0,
(2.11)
where p > 1 and R > 0. According to theorem 4, we know that this problem has a unique positive solution. We now construct its solution. One may easily verify that for any m > 0, the integral y&P
_
&-P
) -l/2
*
s0 is finite, and the function g(m) defined by this integral is monotone Moreover, the following holds lim g(m) = 0, m+o+
increasing in (0, +oo).
lim g(m) = +=J. ?Pl++CG
Thus, for any A > 0 there exists exactly one positive number, which we denote as m(R), such that g(m(R)) = R. Set fR(u)
=
’ (s’-~ - r~~(R)‘-~)-“~ds, 0
v u E (0, m(R)].
Semilinear
185
elliptic equations
It is obvious that the function fR(u)is monotone increasing in (0, m(R)], and Jim+ fR(u)= 0,
fR(m(R))= R. We now define a function uR(x) on [-R, +R] as follows
uR(x)
=
fi’(R + 4,
if x E (-R, 01,
fi’(R - X),
if x E [0, +R), ifx=
1 0,
&R.
One may easily verify that this function belongs to C”(-R, +R) n C[-R, +R] and it is the unique solution of the problem (2.10) and (2.11). A similar method can be used to construct the unique solution of the following boundary value problems U”(X)+ CzP( 1 + &(x)2)4 = 0, u(kR)
-R
< x < +R,
= 0,
where C is a positive constant and p, q are as in theorem 4. We leave it to the interested reader. 3. THE
PROOF
OF THE
MAIN
RESULT
This section is arranged to prove theorem 1. The method of proof used here can be outlined as follows: first, we prove that for sufficiently small E > 0, the perturbed problem
u > E,
f(x, 2.4, Du) = 0,
Lu +
in Sz,
on afi
i U = E,
(3.1) (3.2)
has a solution u = u, by using the upper and lower solutions method; secondly, we prove that there is a sequence Ej > 0 (j = 1,2, . . .) converging to zero, such that the corresponding sequence of functions ucj (j = 1,2, . . .) converges to a solution of the problem (1.1) and (1.2). By the Krein-Rutman theorem, we know that under the conditions (A,) and (AZ), there exist a function 4(x) E C2+,(fi) tl Cl@) and a positive number p such that Uj
=
U(x)
= -M(x),
where n denotes the outward-pointing LEMMA
1. Under
the assumptions
in Sz,
4(x) > 0,
unit normal vector on XL
(A1)-(A4),
there
exist
constants
E* > 0, c > 0, A4 > 0 and
v E
(0, I), such that for any E E (0, e*], the problem (3.1) and (3.2) has a solution u = u, which belongs to C2+,(Q) fl C(a) and satisfies the following inequalities VXEQ.
c+(x) + E I U,(X) I M(4(x) + E)“,
(3.3)
Proof. From the assumption (AJ (i) we see that there exist constants a > 0 and b > 0 such that the inequality (2.2) holds when 0 < u 5 a and I<[ I b. Set E,, = $a, and take v E (0, 1) such that 2 1 -p”l
(
v 5 min -
2-r -r--s
>’
186
GUI SHANGB~
For 0 < E c .eO, let ii,(x) = &J(X) + E and h,(x) = M($(x) + a)“, where c and M are positive constants to be determined. We now prove that ii,(x) and i;,(x) are lower and upper solutions of the problem (3.1) and (3.2), respectively, if c and E are sufficiently small and A4 is sufficiently large. and let Mr = maxlD$(x)l. Let MO = yy$d~(~~l,
Then if 0 < c I min(a/2M,,
xek=l
have
we
VXEf-2,
l~~,(x)I5 b,
E < ii,(x) 5 a,
b/M,),
and, thus, by applying the inequality (2.2) we get Lii, + f-(x, ii,, Ix,)
2 -pop
+ A(&
r -p(&
+ &)p
+ E) + A(@
+ &)p
= (a$ + &)p(A - p(cc$ +
&y-p) El1-P 1,
Therefore,
in Q.
by setting
and c* = min &, (
$,
&(;~‘op)).
we see that the inequality Lii, + f(x, ii,, lx,)
2 0
holds in a if 0 c c I c* and 0 c E 5 E*. This, combined with the fact that @Jan = E, means that ii, is a lower solution of (3.1) and (3.2) when 0 < c d c* and 0 < E I a*. Next we notice that D&(x) = vM(+(x) + &)“-%#J(X), and L;,(X)
=
=
V(V
-V(l
l)M($(X)
-
-
+ &)‘-’
V)M(+(X)
i
i,j=
+ &)“-’
&j(X)
84(X) x
1
I
%4x)
aij(X) x 1
I
-Lv(l
-
v)M(9(x)
+ &)“-21Dr$(x)12
5
-Av(l
-
v)M(+(x)
+ &pg(x),
x I
-
+ vM($(x)
+
Ey+‘L~(X)
J
Mx)
i
i,j=
84(X) x
-
PVMHX)
J
/wA4(4(x)
where
g(x)= ImJ(x)12 + *(I r.- v)w2*
+ Ey-+qX)
+ dY-%(X)
187
Semilinear elliptic equations
Thus, by applying the inequality (2.3) we get LB, + f(x, ii,, DC,) I - Av(l - v)M(+(x) + @‘-*g(x) + B, + B,MP’(+(x)
+ E)“@
+ B, v*Mq(4(x) + &)(y-l)qlDqb(x)lq + B, ~‘M’+~(c$(x) + &)+1)r+vs/D4(x)lr = M(c$(x) + c)‘-*[+(l + B,M-“-P”($(X) + B, ,fM-(1-r-s)
- v)g(x) + B,M-‘(4(x)
+ e)*-’
+ E)2-d’-P’) + B2v~M-‘1-4)(~(x)
+ e)*-W’(‘-4)jD+(x)14
(4(x) + ~)(*-~)-“(‘-~-~)lD~(x)l~].
Notice that the properties of the eigenfunction
4 guarantee that ming(x) > 0. Moreover, the xsii assumptions on the constants v, p’, q, r and s guarantee that all the exponents of 4(x) + E and ID+(x)1 in the square brackets are positive, and, thus, the functions in the square brackets containing the function 4 are uniformly bounded above for E E (0, &*I. Therefore, we get the following inequality Lli, + j-(x, ii,, DC,) I M(c#J(x) + &)‘-*(-co
+ C,M_’
+ c2M-(1-p’)
+ C3M-(I-@
+ C4M-(1-‘-S)),
in Sz, where all CjS are positive constants independent of M and E. From this inequality we see immediately that there exists a constant M, > 0, which is independent of E, such that the inequality Lii, + j-(x, ii,, Dt,)
I 0
(3.4)
holds in fi when M L M,,. Set M* = max(M,, (c*)‘-“). Then apart from (3.4) we have liejan = ME’ 1 E whenM L M* and 0 c E 5 E*, which implies that ii,(x) = M(c#J(x) + E)”is an upper solution of the problem (3.1) and (3.2). Now let us consider the following problem Lu + f,(x,
L u = E,
u, Du) = 0,
u > E,
in Q
on asz
(3.5) (3.6)
where the function f,(x, U, 4) is as in the condition (As). Obviously, the functions ii,(x) = c~(x) + E and G,(x) = M(d(x) + E)’ are also the lower and upper solutions of the problem (3.5) and (3.6), respectively, for any 0 < E I E*, 0 < c 5 c* and M 1 M*. Thus, by choosing the constant c E (0, c*] sufficiently small and the constant M E [M*, +a) sufficiently large so that the inequality c~(x) + E 5 M($(x)
+ E)‘,
VXESJ,
v & E (0, &*I
holds (which is possible because 0 < v < l), we conclude by applying the first theorem of Amann [2] that there exists a maximal solution of (3.5) and (3.6) which we denote by uE(x), such that c+(x) + E I u,(x) 5 M(+(x) + E)“,
VXEsz,
and u,(x) E C2+U(M) fI C(d). Notice that the first inequality above guarantees that f,(x, uE, Du,) = f(x, uE, Due), and, thus, u = U, is also a solution of the problem (3.1) and (3.2). Lemma 1 is thus proved.
188
GUI SHANOBIN
LEMMA2. For any C2+” -smooth domain a’ CC a, there exists a constant C independent of E, such that v &E (0, &*I.
lI%IICZ+~(W) 5 C,
(3.7)
Proof. Take three domains Qj, j = 1,2, 3, with C2+cY-smooth boundaries such that SY CC Q, CC Q2 cc Qr cc 0. Let m = c min 4(x), which is positive since r#~> 0 in a. Let f, be a function as in condition xe4?4 (A3) with E = m. Then for any E E (0, c*] we have J% + fmk %,DUE) = 0,
on Q3.
By the interior estimate theorem of Ladyzenskaya and Ural’treva p. 266]), we get a constant C1 independent of E, such that
(see [16, theorem
3.1,
From this we see that Du,(x) is uniformly bounded on Q2 for E E (0, a*] since ~7~ u,(x) I M(max 4(x) + e*)” and the right-hand side of this inequality is independent of a. Therefore, XEi=l the function _&(x) = f, (x, u,(x), Du, (x)) is uniformly bounded on Q2 for E E (0, &*I. Now let us consider the following linear elliptic boundary value problem Lv =A,
in
( v = 0,
Q2,
(3.8)
on aQa .
(3 -9)
Since x is continuous and bounded on Q2, we see x E LP(Q2) for any p E (1, +a~), and, thus, by theorem 9.15 of [17] we conclude that the problem (3.8) and (3.9) has a unique solution v = v, which belongs to HJ’~,~(Q~)for any p E (1, +m). Moreover, by lemma 9.17 of [17] we see that there exists a positive constant C2 independent of E such that iI veil Wz*J’(Qz)
5
c211J;E11rP(Q2)
v
3
& E
(0,
&*I.
Thus, since J’,(x) is uniformly bounded on Q2 which implies that is bounded by a constant independent of E, we conclude that is also bounded by a constant independent of E. Taking p = n/(1 - a) and applying Sobolev’s embedding inequality, we see is bounded by a constant independent z; v, belongs to Cl+“@,) and the norm (Iv,JI~I+~(Q~) 11 f,llrp(Q9
IIv,IIW~,P(Q2)
Next let us consider the boundary value problem Lw = 0,
1 w = UE)
in
Q2,
(3.10)
on aQ2.
(3.11)
Since u, E C2+u(Q2), we get the conclusion that this problem has a unique solution w = w, which belongs to C2+a (Q2) - by applying the theorem 6.14 of [17]. Furthermore, by Schauder’s interior estimate theorem (cf. corollary 6.3 in [17], for instance) we see that for some constant C, independent of E the following holds iI W~ib+U(QI)
-=
c311 weilC(&)
3
v
& E
(0,
&*I.
189
Semilinear elliptic equations
By the maximum principle we have Y
>
(
IIw&Q~) 5 max u,(x) 5 M max+(x) + E* xsfi
xeaQ2
,
VE E (0, &*I.
is also bounded by a constant independent of E. Hence, we see that ))~~JJo~+aco,, Now, since U, and u, + w, are both solutions of the problem Lu =J;,,
in Qz,
i u = UC,
on aQz,
we get the equality U, = u, + w, in Q2 by applying the uniqueness theorem. Thus, we have b&l+“(Q,)
s
b,IIC’+“(Q,)
+
b%~~C’+a(Q,)
s
v
c4,
E E
(0,
.?*I,
where C, is a constant independent of E. This enables us to get the following estimate iIJ;Eb(Q,)
=
IlfmcG W), ~%(-@)IIC”(Q,) 5 c59
where C, is constant independent
v &E (0,&*I,
of E. Now since u, is the solution of the linear equation
Lu = TE, by applying Schauder’s interior estimate theorem once more we then get
Il”,llc2+~@‘,5
db&(Q,)
+
tl.&acQ1,)
s
v
c7,
E E
(0,
.?*I,
where C, and C, are positive constants independent of E. This finishes the proof of lemma 2. Proof of theorem 1. Set uk = uEI- , where &k = (l/k)&*, k = 1, 2, . . . . Take a sequence [Sz,],“,1 of domains with C2+a-smooth boundaries, such that
Q2, cc n2 cc
***cc szj cc $+i
cc
*--
and lJT= I Qj = Cl. By lemma 2, we see that IIu~JI~z+~(~~) is bounded by a constant Nj independent of E for each j = 1,2, . . . . Thus, by the compactness of the embedding operator of C2’ol(&) into c2(ni), we get a subsequence (U{*‘)T= 1 Of the sequence (u&, 1 COIWfXghg in C2(n,). Similarly, by the compactness of the embedding operator of C2’ol(sZ2)into C2(n2), we get a subsequence 1~~~‘)~~ 1 of the sequence (uj”]T= 1 converging in C2(sZ2). By an induction procedure, we get for each j = 1,2, . . . a sequence (u!“)T= 1 which is convergent in C”(~i> and has the property that (u!j+i) lT=1 is a subsequence of (u,‘j’)T=1. We set uj = uy’ forj = 1,2, . . . . It is obvious that the sequence u/!is convergent in C2(n’) for any domain &2’CC i2. Especially, the sequence {u,!l~=1 converges pointwisely in Q. Denote by u the limit of this sequence. Then u E C2(SZ) and it satisfies the equation Lu = f(x, u, Du)
in n because lujlj”=1 converges to u in C2(a’) for any LY CC Sz. Furthermore, inequalities (3.3) we get c@(x) 5 u(x) 5 M(4+))‘,
from the
VXEQ
which implies that lim u(x) = 0. We, thus, get a solution of the problem (1.1) and (1.2) which x-an belongs to C2(Q) n C(G) if we set u = 0 on XL Applying Schauder’s interior regularity theorem we see u E C2+Ol(n) and, thus, theorem 1 is proved.
190
GUI SHANGBIN
Acknowledgement-The author wishes to express his thanks to Dr Zhang Zhijun for his discussions with him on the problem considered in this paper. He also wishes to thank Professor H. Amann for sending him the article [18] which greatly benefited him. REFERENCES 1. AMANNH., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. math. J. 21, 125-146 (1971). 2. AE~ANNH., Existence and multiplicity theorems for semilinear elliptic boundary value problems, Math. Z. 150, 567-597 (1976). 3. NOUSSAIR E. S., On semilinear elliptic boundary value problems in unbounded domains, J. diff. Eqns 41, 334-348 (1981). 4. STUARTC. A., Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities, Math. Z. 163, 239-249 (1978). 5. BAXLEYJ. V., Some singular nonlinear boundary value problems, SIAM J. mnth. Analysis 22, 463-479 (1991). 6. COCLITEM. M. & PALMIERIG., On a singular nonlinear Dirichlet problem, Commune partial diff. Eqns 14, 1315-1327 (1989). 7. CRANDALL M. G., RABINOWITZ P. H. &TARTARL., On a Dirichlet problem with a singular nonlinearity, Communs partial diff. Eqns 2, 193-222 (1977). 8. DALMASSO R., Solutions d’equations elliptiques semi-lineaires singulieres, Annuli. Mat. puru uppl. 153, 191-201 (1988). 9. DIAZJ. L., MORELJ. M. & OSWALDL., Elliptic equations with singular nonlinearity, Communspurtiul diff. Eqns 12, 1333-1344 (1987). 10. FINK A. M., GAT~CAJ. A., HERNANDEZG. E. & WALTMANP., Approximation of solutions of singular second order boundary value problems, SIAM J. math. Analysis 22, 440-462 (1991). 11. GOMESS. M., On a singular nonlinear elliptic problem, SIAM J. math. Analysis 17, 1359-1369 (1986). 12. LAZERA. C. & MCKENNAP. J., On a singular nonlinear elliptic boundary value problem, Proc. Am. math. Sot. 111, 721-730 (1991). 13. USAMIH., On a singular elliptic boundary value problem in a ball, Nonlinear Analysis 13, 1163-l 170 (1989). 14. Cur SHANGBIN, Comparison principles and uniqueness theorems for nonlinear elliptic boundary value problems, J. Lunzhou Univ. Science Series 22(3), 8-14 (1986). 15. CHENZ. C. & Luo X. B., Comparison and uniqueness of positive solutions for the mixed problem for semilinear parabolic equations, Communspurtiul diff. Eqns 11, 1285-1295 (1986). 16. LADYZENSKAYA 0. A. & URAL’TREVA N. N., Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968). (English translation.) 17. GILBARGD. & TRUDINGER N. S., Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer, Berlin (1983). 18. AMANNH., Periodic solutions of semilinear parabolic equations, A collection of papers in honor of Erich Rothe, Nonlinear Analysis 1, 29 (1978).