ON THE NUMBER OF POSITIVE SOLUTIONS FOR NONLINEAR
ELLIPTIC EQUATIONS WHEN A PARAMETER IS LARGE
SONG-SUN LIN
Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu, Taiwan 30050, Republic of China (Received 21 August 1989; received for publication 30 July 1990)
Key words and phrases: Positive solution, semilinear eigenvalue problems, uniqueness theorem.
1. INTRODUCTION
THE MULTIPLICITY problems of positive solutions of a class of semilinear eigenvalue problems are studied
-Au(x) = Aj-(u(x)) for x E SJ
(1.1)
U(X) = 0 for x E an
wheref(u) > 0 for u > 0 and A > 0, and Q is a bounded smooth domain in R”.
Equation (1.1) arises in nonlinear heat generation, in models of combustion, etc. We refer to the survey paper by Lions [ 121.
In [lo], the author proved that the positive solution of (1.1) is unique for A is large when f is bounded and satisfies a “concavity” condition. More precisely, (1.1) has an unique positive solution for large A if f satisfies the following assumptions:
(i) f E C’W, CQ)),
(ii) f(u) 2 m > 0 for each u I 0 and some m > 0, (iii) lim f(u)/u = 0,
U’+m
(iv) lim inf f(u) > lim sup f’(u)u.
U’+m ll++CC
We proved that solutions z+, of (1.1) satisfies z+, L ,lCu, if A is large, where u0 is the solution of -Au(x) = 1 for x E Sz
(1.2) V(X) = 0 for x E aQ
and some constant C > 0. Then, condition (iv) implies that (1.1) has an unique positive solution when 1 is large. Condition (iv) permits f has a logarithmic growth when u tends to infinite. In the case of an ODE, a similar result has been obtained by Shivaji [18].
In this paper, we shall study the multiplicity problem of (1 .l) when f is sublinear. More precisely, f(u) - u ‘, 0 < /3 < 1, when u is large. By constructing some appropriate super- and subsolutions and applying Serrin’s sweeping principle, we can prove that for “large solution” ux of (l.l), ux 2 Au, for large A. And then improve this estimate to t+, - A1’o-B)~,r by a “singular perturbation method” used in [lo], where ug is the unique positive solution of
-AU(X) = up(x) for x E Sz
(1.3) U(X) = 0 for x E an.
The uniqueness of large solution can be obtained if f satisfies a concavity condition liEEp% < 1.
u
If f has power growth at u = 0 and + 00, then problem (1.1) can be classified into four types, (8,8), (01, P), (P, a) and (a, a), 0 < P < 1 and cy > 1, where type (a, PI is
“f(u) - MU when u - 0 and
f(u) - up when u - +co
etc. This classification makes it easier to identify large or small solution when it exists. In this paper the existence and uniqueness problems for large solutions of types (8, p) and (CY, /3) are studied. Problems of type (p, (Y) and (CY, CX) are more difficult and complicated, depending on geometrical and/or topological properties of a. For reference, see e.g. Brezis and Nirenberg [5] and Lions [ 121.
In Section 2, we give a special case of Serrin’s sweeping principle and recall some results in [lo] which are useful in this paper.
In Section 3, we prove that ux - A l’(l-B)~B and an uniqueness theorem for large A, in the case of type (P, P).
In Section 4, we prove that ux - I l’“-p)~p for large solution ux and an uniqueness theorem for large solution in the case of type (01, p).
2. PRELIMINARIES
In this section we shall recall some results obtained in [lo], which are useful in getting apriori estimates of ux when A is large. We also give a theorem (with a proof) which is a special case of Serrin’s sweeping principle. The principle can tell us where solutions do not exist, therefore gives us some a priori estimates of solutions. The principle is useful in various problems and is very useful in the case when f is sublinear.
In this paper it is always assumed that Q is a bounded smooth domain in R”, for example, a E C3. We denote by t(x) the distance from x E a to the boundary 80, and by s(x) the point of afi which is closest to x (which is uniquely defined if x is close enough to XJ). We choose sufficiently small 6, > 0 such that the boundary strip Q0 of width a,, i.e.
Qzb” = lx E a 10 < t(x) < 6,)
is covered (and only covered) by the straight lines in the inner normal direction -Q) and emanating from s(x).
Given p E Lp(sZ), p > 1, let u E IV2*p(a) be the solution of Poisson equation with Dirichlet boundary condition, i.e.
-Au(x) = p(x) for x E Q
(2.1) u(x) = 0 for x E aQ.
We prove the following proposition by using L!‘-estimates and applying maximum and strong maximum principle.
PROPOSITION 2.1. Assume that p L 0 and p f 0 on Sk
(i) Suppose that u E C2(Q) fl C’(b) is the solution of (2.1). Then there exist constants I 2 k > 0, such that
kt(x) I u(x) I b(x) on a. (2.2) (ii) Suppose that p E L”(Q) and supp(p) z Q2,, 0 < 6 < &,, then for p > n, solution u of (2.1) satisfies
u(x) 5 C,~l’PllPllco~(x) on si where constants C, depends only on p, n, Cl, but not p.
(iii) For 0 < 6 < 6,) define the function ps by P*(X) =
l
l
for x E !&0 forxEQ - fi2,. Then the solution Z.Q of (2.1) with p(x) = p*(x) satisfies
z+(x) I CP@V,(X) on Sz. Remark. u6 is given by u*(x) = I G(x, Y) dy on fi I ns
where G(x, y) is the Green’s function of -A with zero Dirichlet boundary condition.
(2.3)
(2.4)
(2.5)
(2.6)
Proof. (i) Since p 2 0 and p f 0. By maximum principle, U(X) > 0 in Q and by strong maximum principle
-g(s)
<
0 on XL sSince an is compact, there exists 6’, 0 < 6’ < a,, and 1’ 1 k’ > 0 such that -1’ 5 $5 (x) I -k’ < 0
S(X) (2.7)
for any x E OS,. (2.2) follows by u(x) = -t(x)
.i 1 au
- (x + W)n,,,) dt
o anscxj (2.8)
for any x E Sz,, , and by choosing I r k > 0 appropriately for x E Sz - sZs,. It is clear that k and I depend on u and then on p.
(ii) In the following, constants Ci, i = 1,2, 3, . . . , may vary but depend only on p, n and Q. By P-estimate, there exists a constant C, such that
IMl2,p 5 GIIPllp 5 c2ll PIL w~*Yp
5 C3 II pllm area(%2)“P LVp = Gil Pll.J1’p.
For p > n, by Sobolev imbedding theorem, u E C’+“(a), (Y = 1 - n/p. Therefore, for any
XELi
Since (2.8) is also valid for u, u(x) I c,G~‘~~~P]]~~(x) on Q, with C, depends on p, n and Q only.
(iii) This follows by (i) and (ii). The proof is complete.
We need the following comparison lemma in order to prove Serrin’s sweeping principle. LEMMA 2.2. Suppose that f E C’ and u and v E C’(Q) satisfying
AU + f(u) 2 Au + f(u) in s2 and
VkU in Sz.
Then, either u = u or u > u in a.
Proof. Let w(x) = V(X) - u(x). Then
w(x) L 0 in Sz and
Aw(x) + c(x)w(x) 5 0 in Q where
c(x) = f ‘(G(x)) with $(x) E [u(x), u(x)].
Let s/’ be any subdomain of Q with a’ C SJ and m’ = max(O, I’], where I’ = max(f’(w) 1 u(x) I w 5 u(x), x E a’). Then, for x E Q’, Aw(x) + (c(x) - m’)w(x) i -m’w(x) I 0. By maximum principle, either w = 0 in Sz’ or w(x) > 0 for all x E Sz’. Since 0’ c Sz is arbitrary, either w = 0 in Q or w(x) > 0 for x E a.
The proof is complete.
In the remainder of this paper, it is always assumed that f satisfies the following conditions: (I-I-O)
i
(i) f E C’((0, 00)) fl CVO, a)) for some y E (0, 1) (ii) f(u) > 0 for u > 0.
Then, it is easy to see that the following definition applies.
Definition 2.3. A function 4 E C’(Q) fl C(a) is called a supersolution of semilinear elliptic boundary value problem
Au(x) + f(u(x)) = 0 for x E Q
(2.9) U(X) = 0 for x E &2
if $ satisfies
A4 + f(d4x)) 5 0 for x E Q $6) 2 0 for x E %2.
Similarly, a subsolution r,u E Cz(&2) n C(a) of (2.9) satisfies At/&) + f(w(x)) 2 0 for x E Q
u/(x) 5 0 for x E an. We state the monotone iteration schemes as below.
(2.11)
PROPOSITION 2.4. Let $J be a supersolution and I,V a subsolution of (2.9) with w 5 4. Then there exist solution u and D with
where 41 is the minimum solution between v/ and I$, and TJ is the maximum solution between I+V and 4. u may be equal to ii. For the proof see, e.g. Sattinger [15, 161 or Amann [l].
We now give the following theorem which is a special case of Serrin’s sweeping principle, see Serrin [17] and Sattinger [16]. For completeness a short proof is given.
THEOREM 2.5. Let {&] be a family of supersolutions of (2.9) which is increasing in r, 0 I t 5 1, i.e.
4,(x) 5 4,(x) for x E Sz if fi 9 r2, and satisfy the following conditions:
(i) 4, E C’(Q) n C’(Q, for ‘5 E [0, 11, (ii) 4, = 0 on %2,
(iii) 4, is not a solution of (2.9) for r E (0, 11,
(iv) 4, is continuous in T, in the sense that for any E > 0, there exists 6 = a(e) > 0 if 1r’ - 71 < 6
b,(x) - M4l 5 W) for x E Sz.
Suppose that u E C2(sZ) n Cl@) is a solution of (2.9) with 2.4 5 41 in Q.
Then either u = I&, or U(X) < &(x) for x E Sz. A similar result holds for a family of subsolutions lw,l*
Proof. Let t0 = inf(r E [0, l] 1$,(x) > u(x) in 01. Then 4,(x) 1 u(x) in 0. By lemma 2.2, either I& E u or 4,(x) > U(X) for x E Q. We shall prove r0 = 0. If r,, > 0, then (D,,(X) > U(X) for x E Q by (iii).
By strong maximum principle
for s E XJ and some k > 0. A similar argument as in proving proposition 2.1(i) implies that 4,(x) - u(x) 1 M(x)
for x E Q and some q > 0. By (iv), there exists 0 < T’ < r0 such that $,4X) - U(X) 2 (4,(x) - N(x)) - 4X) z 0 in Sz a contradiction to the definition of r, . Hence r0 = 0. This completes the proof.
Remark. The conditions in theorem 2.5 are easily verified in our constructions of super- or subsolutions of (1.1). It is clear that some kind of continuity condition of ($,] with respect to 7 has to be satisfied in order that the conclusion of theorem 2.5 holds. (iv) suits our purpose and is easily obtained by applying proposition 2.1(i).
3. (b,b)-TYPE
In the remainder of the paper we shall also assume thatf has a power growth at u + co, i.e. f satisfies
(H-1) lim ‘(‘) 7=, 1 0</3<1. U++m U
In the case of (/3, P)-type, we shall prove that for large ;1 (1.1) has the large solution only. Consider the linear eigenvalue problem
-Au(x) = Au(x) for x E Q
(3.1) U(X) = 0 for x E an.
Let A, (20) be the least eigenvalue and q > 0 be the corresponding eigenfunction with normalization 1) u, jlrn = 1. We first prove the following lemma.
LEMMA 3.1. Suppose that f satisfies (H-O), (H-l) and (H-2) f(u) > CU.4 in (0, PI for some cr > 0 and p > 0. Then the solution ux of (1.1) satisfies
4, 2 PV, if A 1 Ii/a.
Furthermore, (1 .l) has the minimum positive solution u for ;1 2 A,/a.
(3.2)
Proof. For r E (0, ,u] and A 2 1,/a, we have
A(zv,(x)) + hf(rvr(x)) > -r&vi(x) + Aoru,(x) = 7q(x)(Ao - A,)
1 0.
Therefore, (sui], 7 E (0, ~1, is a family of subsolutions of (1.1) which is strictly increasing in 7 and is not a solution for 7 E (0, ,u]. By proposition 2.1(i), ux 2 tx vi in Q for some 7x E (0, P). For ,4 2 2,/a, by theorem 2.5, ux 2 ,uu, in 0. This proves (3.2).
To prove (1.1) has the minimum positive solution, it suffices to construct a supersolution & of (1.1) such that &, r pu, , in s2. This can be obtained if f satisfies (H-O) and
(H-l’) lim
f(u) =
0.u-+m 2.4
However, in the case off satisfying (H-O) and (H-l), & = 1 1’o-p)7u0 is a supersolution for large 7. For details, see the proof of theorem 3.3.
It is known that (1.3) has the unique positive solution in C2’p(fi) (unique in the class of C’(Q) fl C’(a)), for example, see Aronson and Peletier [3]. We shall give a short proof as below.
PROPOSITION 3.2. There exists a unique positive solution vg of -Au(x) = u”(x) for x E Q
u(x) = 0 for x E %2 where 0 < /3 < 1.
(1.3)
Proof. It suffices to prove that (1.3) has a minimum positive solution. In fact, if f satisfies for u E (0, 00) (3.3) and (2.9) has the minimum positive solution u. Then (2.9) has the unique positive solution. In fact, if u is a positive solution of (2.9), then
0 = (uAzj - uAu)d_x D
This implies that u = u.
To prove (1.3) has the minimum positive solution. We note that uB > I,u for all u E (091, “(‘-‘)), the existence of minimum solution follows by lemma 3.1. The proof is complete.
We can now prove the following asymptotic theorem for ux when 1 is large. THEOREM 3.3. Suppose that f satisfies (H-O), (H-l) and (H-2). Then
lim u,(x)/(A i’(l-@u&)) = I uniformly on fi A--+-
where ux is a solution of (1.1).
Proof. We shall divide the proof into three steps:
Step 1. There exists II, > 0, if I > &, then
ui 2 AU@.
Step 2. For any E > 0 and cl > 0 there exists AE = A(&, .sl) > 0, if 13 > &, then Ux 2 ~l/c’-a,(l _ s)‘N’-13)(1 _ E1)r/o-8)Ug.
Step 3. For any E > 0 and E, > 0 there exists A: = A’(&, q) > 0, if ;1 > Ai, then nx 5 ;l1’(1-@)(1 + e)l’(‘-P)(l + er)r’(r-@r,$.
Step 1. We shall first prove that there exists m > 0 such that ux 2 Amv, if A L 2,/a. By proposition 2.1 (i), there exists k, > 0 such that vr L k, v,, . Hence
% 2 Pl vo if A 1 A,/0
where p1 = pk, > 0. Let ,u’ E (0, r(lJ and m’ = min]f(u): u E [P’, 00)). Since f satisfies (H-O) and (H-l), m’ > 0. Denote by [$ 2 U] = (x E Q 1 v(x) 2 U], etc. If A 2 2,/o, then
ux(x) =
a
WA Y)~(uA(Y)) dy ndy
2 Am’(v,(x) - Cp(p’/,uI)l’pvo(x))
here (2.5) is used, m = m’/2 and p’ is chosen with C&‘/&)“” = 5.
Next, we prove that there exists A2 2 A,/a, such that A 1 A2 implies ux 2 Au,. By proposition 2.1(i), there exists A4 > 0 such that
A4
-?I() 2 vg. 2
For this M, there exists U = U(M) > 0, if u 2 U, thenf(u) 2 M. Therefore Ux(X) 1 AM
s [Amu t U1 ‘XG Y) dr I AM{1 - CP(U/Am)“P)vo(x)
1 Avg,
here A2 is chosen such that CP(U/l,m)l’P = t. Step 2. By proposition 2.1(i) and (ii), with p = v;,
u,(x) = G(x, Y)$(Y) dy 5
C~@‘~vp(x)
0.5
(3.4) for 0 < 6 < 6,) and constant C, depends on p, n, Q and vg. We note that (3.4) can be improved to
for 0 < 6 < &, and some constant CL > 0. By (H-l), for any E > 0, there exists U = U(E) > 0 such that u 2 U implies
f(u) 1 (1 - &)U? Therefore,
24x(x) 2 A(1 - E)
s ]&J/j 3 Lq G(x, _WqANB du 1 A’+@(1 - &)?I/311 - c,(u/n)r’q
2 A’+%g(l - &)(l - Er)
whenever ,I 1 max(l,, A&, here, A3 = (C,/&JPU, i.e. IZ,(U/A,)“~ = q. Moreover, after repeating this argument once, we obtain
#x(X) 2 A(1 - E)
s [A’+~(l-e)(l-,*)“pz w G(w)V’+~(~ - s)(l - O+(-Y)lBdy
111+fi+q1 - &)l+ql - &,)@ * 1 - c, l (
_$l _ E ) l/P AI+p(1
1 >I us (xl = A1+p+ql - &)‘+ql - &r)‘+@up(X)
whenever A 1 max(A,, A3, A,), here A., is given in
i.e.
(
u
>
l/P cl3 A;+@(1 - &)(l - El) = e’
i
u
1
1/o+m*4 = wql _ E)(l _ El) . Repeating this argument, we can prove that for any n 1 2
u,(x) 1 p+p+-+q(l - E)(l _ ~,))‘+B+“.+B”-‘u,(x)
whenever 1 z maxI&, &, . . . . An+*), with
i
U
1
1/(1+8+...+/3”-*)A II+2 = G’QP [(I _ E)(l _ E1)]l+B+...+o”-~
Since 0 < p < 1, A, -+ A, as n + 00, where
a, =
(C~/Elp+) ul-” (1 - &)(l - El) * Let A, = max&,A,, . ..) If A az AE, then= A(&, El) < +aJ.
ux(x) z A 1way1 _ e) W14)(1 _ &J”“-mva(x) on !A
(3.6)
Step 3. It is easy to see that (H-O) and (H-l) implies that
f(U) c/l + 2u” for u 2 0 (3.7)
where A 2 0 is a constant. We first prove that (A1’(‘-@) 5~~) is a family of supersolutions of (1.1) if T is large. In fact
A(2 i’(‘-fi)r&J + ~j-(~““-@rv,) 5 -A”(‘-+ + ;1(A + 2(~i’(‘-~)rQJ”) = _~l/‘l-P’lr _ 2*“v,p _ A-%-B)AJ 10
if
5 L 2(jU,ll&” + A-O’(l-B)A. Furthermore, it is easy to verify that (3.8) is true if
(3.8)
r 2 max(A-“(‘-“)A”“, (2(lU,)It + l)l’o-p)). (3.9) For A 2 1, we can choose
r 2 A4; 3 max(A1’P, (21lu,I(~ + 1)“‘‘-“‘l. (3.10) Therefore, A1”-@ tug, z 2 ML, is not a solution of (1.1) if T is large enough. Hence, by Serrin’s sweeping principle, we have
Uh 5 P(l-O)M;Uo
for A 2 A;, for some A; > 0. Next, by an argument similar to that in step 2, we can prove that for any E > 0, e1 > 0, there exists A: = A’(&, ei) < +co, such that I 2 A:, implies
U,(X) I Ai’(i-@(l + &)““+)(l + &i)““-“‘v&) on b. The proof is complete.
During the proof, we have obtained the following corollary.
COROLLARY 3.4. Suppose thatf satisfies (H-O) and (H-l). If (ux) is a family of solutions of (1 .l) with ux 2 pvo for some P > 0 and A 1 A r 0, then
lim u~/(A~‘~~-~)v~) = 1
X-+oS uniformly on a.
To prove an uniqueness result for positive solutions of (1. l), we need the following lemma. LEMMA 3.5. For any w E Wan’ (=H,‘(Q)
(3.11) and i n Ivwl2 2 c, i v,‘w2 (3.12) n where C, > 0 is a constant.
Proof. Consider the linear eigenvalue problem
-Aw(x) = ptv!-‘(x)w(x) in &2 W(X) = 0 on an.
Then ,u~ = 1 is an eigenvalue with positive eigenfunction uB. Hence 1 is the first eigenvalue. A variational principle which characterizes the first eigenvalue gives (3.11). For (3.12), see Brezis and Turner [6, lemma 2. l] and Lions and Magenes [ 11, p. 761.
We can now prove the following uniqueness theorem.
THEOREM 3.6. Suppose that f satisfies (H-O), (H-l) and (H-2)
(H-3) lim_;upI f ‘(u)lu2 < +m
and
(H-4) ]izrtpF < 1.
u
Then there exists I* L 0 such that (1.1) has an unique positive solution for ,I > A*.
Proof. Let ux be the minimum solution and z+, be any positive solution of (1 .l). Let wk = uh - ux. Then
-Aw, = lf’(8,)w, (3.13)
where 8,(x) E (ux(x), ux(x)).
For any E > 0, which is sufficiently small, let U = U(E) be chosen such that u 1 U imply (1 - &)zP I f(u) 5 (1 + &)UP
and
f ‘(UN f(u)
If’(u)1 5 AU-~ + B for u E (0, U]
where A = A(E) and B = B(E) 2 0. By theorem 3.3, there exists A, 2 0 such that I 1 A, implies
Al’(l_@(l - &)Ua 5 & 5 e, I Ui 5 ~l’(‘-@(l + &)Up. From (3.13), we have
For the first term of right-hand side of (3.14), if 13 L A,, we have
by (3.11). For the second term of right-hand side of (3.14), we have A f’Q3,)w,2 I A 1e,r ul
s
Ie_*(AK2 + 04
hC r/l i A{/l”“-~‘(l - &)up)-2w,2 [A "('-ql-e)"$.js U] + AB i _, _,,(l_p)4uS2wh2)
[lJo'U(l-&) A II A(-l-@‘+@(l _ c)-2(A + BU2)
i ui2wf n
I $-l-8)‘(1-fl)(l _ c)-2(A + BU2)C;’
Ivd2
R by (3.12). Hence, for I 2 AE
(1 - ~(1 + &)(I - E)@-’ - A(-p-1)‘(1-8)(1 - E)-~(A + BU2)C,-‘) !
IVwA12 I 0. (3.15) n
By choosing sufficiently small E > 0 and large 12: 2 A,, then ,J z Ad implies (. . .) > 0 in (3.15). Hence, for A 2 Al
.I
n (vwx12 = 0 which implies wA = 0. The proof is complete.4. &/I)-TYPE
In this section we shall study the multiplicity problem of (1.1) in the case of (a, /?)-type, i.e. we consider f(u) - uoL for u - O+, 1 < CY when n = 1,2 and 1 < CY < (n + 2)/(n - 2) when n 1 3, andf(u) - uB for u - + 03, 0 < j3 < 1. For (01, /I)-type, (1.1) has no positive solution if A is too small. More precisely, we have the following lemma.
LEMMA 4.1. Suppose that f satisfies
0 < f(u) 5 h4u for u > 0 (4.1)
for some constant M > 0. Then (1.1) has no positive solution if A < 2,/M. Proof. Multiplying (1.1) by ul and integrating over 0, gives
- Hence
ul Au, = A f(ux)ul.
cl 3 a
n n
In the remainder of this section, we always assume that f satisfies
(H-2’) lim
f(u) =
1u+o+ 24 01
where (Y > 1. (H-2’) implies that there exists p > 0 such that f(u) I 2u* for u E [O, ~1. Therefore for any solution u of (1.1)
2.4x 5 ~1’(1-Q4p ug where A1 K > -l/(1-13) M, = max z A 1’8, (2\(u,II: + l)l’(l+) 1
is given in (3.9) with A = 1,/M. We now give an existence theorem for type (a, p) problem. THEOREM 4.2. Suppose that f satisfies (H-O), (H-l) and (H-2’), with
(i)cr>lifn=1,2,
(ii) 1 < Q! < (n + 2)/(n - 2), if n L 3. Then there exists A2, 2 Al/M such that
(i) when A < I,, (1.1) has no positive solution,
(ii) when J > A*, (1.1) has at least one positive solution.
Proof. The existence of positive solution of (1.1) for large ,l can be proved by a variational method, see for example, Ambrosetti and Rabinowitz [2] and Nirenberg [13]. It will also be proved in theorem 4.4 by constructing an appropriate subsolution of (1.1). Let
A* = inf (A > 0 : (1.1) has positive solution at A].
Then A* 2 Al/M by lemma 4.1. Once (1 .l) has a positive solution ux at A, then ux is a sub- solution of (1.1) for A’ > 1. In fact
implies
Au, + Af(uJ = 0
AU, + A’f(uJ = (A’ - A)f(uJ > 0
if A’ > A. As in theorem 3.3, II 1’(1-8)tvo is a supersolution if t is large enough. Hence (1.1) has solution for A > A*. The proof is complete.
We need the following well-known results, see e.g. [2,6, 121. PROPOSITION 4.3. The equations
-Au = uLy in Q
(4.2)
u=o
on an.have a positive solution if (i) OL > 1 for n = 1,2, and
(ii) 1 < Q! < (n + 2)/(n - 2), for n 2 3.
Denote by U, (any) solution of (4.2). We can prove an uniqueness result for large solution of (1.1) of type (a, P).
THEOREM 4.4. Suppose thatf satisfies (H-O), (H-l), (H-2’), (H-3) and (H-4). Then there exists A* > 0 such that for Iz 1 I*, (1.1) has an unique positive solution ux with u,_ 2 uO. Moreover, for these uk
lim u,/(~““-~‘u~) = 1 uniformly on G. X++CC
Proof. By corollary 3.4 and theorem 3.6, the solution uh of (1.1) is unique if uk 2 u0 for A r A* and some A* > 0. It remains to prove that there exists positive solution ux of (1.1) satisfying uX 2 u0 if A 2 A for some A > 0. In the following, we shall show that ;I1’(l-~)C~, is a subsolution of (1.1) by an appropriately chosen C and large 1 (depends on C). (H-l) and (H-2’) imply that there exist U, and CJz, 0 < U, < U, < 00, such that
and Let
f(u) 2 +ua if u E (0, U,)
f(u) 2 $4” if u E (U,, ~0). m = min(f(u) 1 U, 5 u 5 U,] > 0.
(i) For x E Q with ~l’(l-B)Cu,(x) 2 U,
A(?“1-8’Cu,(x)) + Af(A1’(l-B)Cua(x)) 1 -A1’(l-‘%~(x) + $(A”‘‘-“‘Cu,(x))”
= pw)CPu~(x)~~ _ C’-Bu;-8) if + L C1-pllu,[l~-“. Let 10 c = ( 2l,&-fl > i/(1 -8) * (ii) For x E kX with U, I A1’(l-B)Cu,(x) 5 U,,
A(A”‘‘-“‘Cu,(x)) + Af(A1’(‘-s)Cu,(x)) 2 --;ll’(l-p)Cu~(x) + Am 1 -U, u,“-‘(x) + Am
if A 2 (l/m)U,I(u,I(~-‘.
(iii) For x E Q with A”‘‘-“‘Cu,(x) I U,
A@.“(-%u,(x)) + Af(A”(‘-‘ha(x)) 2 -~l’(l-p)Cu~(x) + $(A”‘‘-“‘Cu,(x))” = ~l/‘l-“‘Cu,“(x)(t~‘“-s)/(l-~)C”-l _ 1) 20
if A 2 (2C1-ol)(1-B)‘(ol-p). Therefore, 2 l’(l-‘)Cu cz is a subsolution when A 1 IIu,IJ:-i - max(2, U,/m).
This completes the proof.
Acknowledgemen&This work was partially supported by the National Science Council of the Republic of China.
REFERENCES
1. AMANN H., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. math. JI. 21, 125-146 (1971).
2. AMBROSETTI A. & RABINOWITZ P., Dual variational methods in critical point theory and applications, J. funct.
Analysis 14, 349-381 (1973).
3. ARONSON D. G. & PELETIER L. A., Large time behavior of solutions of the porous medium equation in bounded domains, J. diff. Eqns 39, 378-412 (1981).
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
BERESTYCKI H. & LIONS P. L., Some applications of the method of super and subsolutions, in Bifurcation and nonlinear eigenvalue problems, Lecfure Notes in Mathematics 782, 16-41. Springer, Berlin (1980).
BREZIS H. & NIRENBERG L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,
Communs pure appl. Math. 36, 437-447 (1983).
BREZIS H. & TURNER R. E. L., On a class of superlinear elliptic problems, Communspartial diff. Eqns 2, 601-604 (1977).
CRANDALL M. & RABINOWITZ P., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Archs ration Math. Analysis 58, 207-218 (1975).
DANCER E. N., Uniaueness for elliutic eauation when a parameter is large, Nonlinear Analysis 8, 835-836 (1984).
JOSEPH D. D. s( LU~DRCEN T. S.,-Quasilinear Dirichlet-problems driven by positive source, Archs ration Mech. Analysis 49, 241-269 (1973).
LIN S. S., Some uniqueness results for positone problems when a parameter is large, Chinese J. Math. 13, 67-81 (1985).
LIONS J. L. & MAGENES E., Problems aux Limites non Homo&es et Applications, Vol. I. Dunod, Paris (1968). L:ONS P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24,441-467 (1982).
NIRENBERG L., Variational and topological methods in nonlinear problems, Bull. Am. math. Sot. 4, 267-302 (1981).
PROTTER M. H. & WEINBERGER H. F., Maximum Principles in Differential equations. Prentice-Hall, NJ (1967). SATTINGER D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. math. JI. 21, 979-1000 (1972).
SATTINGER D. H., Topics in stability and bifurcation theory, Lecture Notes in Mathematics 309. Springer, Berlin (1973).
17. SERRIN J., Nonlinear elliptic equation of second order (unpublished).