# Existence of positive radial solutions for elliptic systems (1).

Abstract

In this paper, we study the existence of positive radial solutions for the elliptic system by fixed point index theory.

AMS subject classification: 34B15, 45G15.

Keywords: Positive radial solution, fixed point index, cone.

1. Introduction and Preliminaries

There are many results on the study of positive radial solutions in the annulus for elliptic equations, see [1, 7, 9, 10] and references. However we are interested in problems of superlinearity and sublinearity for the elliptic system. In this paper, we study existence of positive radial solutions of the elliptic system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

with one of the following sets of boundary conditions

u = v =0 on r = a, r = b, (1.2a)

u = v = 0 on r = a, [partial derivative]u/[partial derivative]r = [partial derivative]v/[partial derivative]r = 0 on r = b, (1.2b)

[partial derivative]u/[partial derivative]r = [partial derivative]v/[partial derivative]r = 0 on r = a, u = v = 0 on r = b, (1.2c)

where {x [member of] [R.sup.n] : a < [absolute value of x] < b} is an annulus, r = [absolute value of x] = [square root of [x.sup.2.sub.1] + [x.sup.2.sub.2] + ... + [x.sup.2.sub.n]] (n [greater than or equal to] 2), whereas f [member of] C([R.sup.+] x [R.sup.+], [R.sup.+]), g [member of] C([R.sup.+], [R.sup.+]), f (0, 0) = g(0) = 0, [h.sub.i] [member of] C((a, b), [R.sup.+]) (i = 1, 2), [R.sup.+] = [0,+[infinity]).

The existence of positive radial solutions for the elliptic system is studied in [5, 8]. The paper  only deals with the sublinear case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In , Jiang and Liu studied the case of sublinearity ([f.sub.0] = 0, [f.sub.[infinity]] =[infinity]) or superlinearity ([f.sub.0] =[infinity], [f.sub.[infinity]] = 0), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The purpose of this paper is to study the existence of positive radial solutions of the system (1.1)-(1.2a). Our particular interest is that f (u, v) and g(u) grow both superlinearly and sublinearly in u, v respectively. So our results are different from the ones of [1, 5, 7-11] and the conditions that we use are more general than the ones used in [1, 5, 7-11]. (1.1)-(1.2) is equivalent to the boundary value problems

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

u(a) = v(a) = u(b) = v(b) = 0, (1.4a)

u(a) = v(a) = u'(b) = v'(b) = 0, (1.4b)

u'(a) = v'(a) = u(b) = v(b) = 0. (1.4c)

Let s = - [[integral].sup.b.sub.r][t.sup.1-n] dt, m = - [[integral].sup.b.sub.a][t.sup.1-n] dt, w(s) = u(r(s)), z(s) = v(r(s)). Then (1.3)-(1.4) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, let t = m - s/m, [psi](t) = w(s) and [psi](t) = z(s). Then (1.1)-(1.2) can also be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

[psi](0) = [psi](0) = [psi](1) = [psi](1) = 0, (1.6a)

[psi](0) = [psi](0) = [psi]'(1) = [psi]'(1) = 0, (1.6b)

[psi]'(0) = [psi]'(0) = [psi](1) = [psi](1) = 0, (1.6c)

where [p.sub.i] (t) = [m.sup.2][r.sup.2(n-1)](m(1 - t))[h.sub.i](r(m(1 - t)))(i = 1, 2).

From now on, we concentrate on (1.5)-(1.6). Indeed, (1.1)-(1.2) has a positive radial solution for any annulus if we can prove that there exists a positive solution to BVP (1.5)-(1.6) for any m [not equal to] 0 (cf. ).

For convenience of notation, we list the following assumptions:

([H.sub.1]) f [member of] C([R.sup.+] x [R.sup.+],[R.sup.+]), g [member of] C([R.sup.+],[R.sup.+]), hi [member of] C((a, b),[R.sup.+]), hi (t) [??] 0 in any subinterval of (a, b), and

[[integral.sup.b.sub.a][h.sub.i] (t)dt < +[infinity] (i = 1, 2).

([H.sub.2]) There exists [alpha] [member of] (0, 1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly with respect to u [member of] [R.sup.+].

([H.sub.3]) There exists [beta] [member of] (0,+[infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly with respect to u [member of] [R.sup.+].

([H.sub.4]) There exists p [member of] (0,+[infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly with respect to u [member of] [R.sup.+].

([H.sub.5]) There exists q [member of] (0, 1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly with respect to u [member of] [R.sup.+].

([H.sub.6]) f (u, v) and g(u) are increasing in u and v and there existsR > 0 such that

[[gamma].sub.1]f (R,[[gamma].sub.2]g(R)) < R,

where [[gamma].sub.i] = [[integral].sup.1.sub.0][p.sub.i] (t)dt (i = 1, 2).

The following examples to illustrate ([H.sub.2])-([H.sub.5]) are in order.

Example 1.1. Let f (u, v) = (1+[e.sup.v-u])[v.sup.2], g(u) = [u.sup.3], [alpha] = 1/2, [beta] = 2. Then ([H.sub.2])-([H.sub.3]) hold. Here f (u, v) grows sublinearly in u and superlinearly in v respectively, whereas g(u) grows superlinearly in u.

Example 1.2. Let f (u, v) = (1 + [e.sup.-(u+v)])[v.sup.1/2], g(u) = [u.sup.3], [alpha] = [beta] = 1/2. Then ([H.sub.2])-([H.sub.3]) hold, in which f (u, v) grows sublinearly, whereas g(u) grows superlinearly in u.

Example 1.3. Let f (u, v) = (1 + [e.sup.-(u+v)])[v.sup.1/2], g(u) = [u.sup.3/2] or g(u) = [u.sup.1/2], p = q = 1/2. Then ([H.sub.4]) and ([H.sub.5]) hold, f (u, v) grows sublinearly, whereas g(u) grows superlinearly or sublinearly.

Example 1.4. Let f (u, v) = (1 + [e.sup.u+v])[v.sup.1/2] or f (u, v) = (1 + [e.sup.u])[v.sup.1/2], g(u) = [u.sup.1/2] + [u.sup.3], [alpha] = q = 1/2. Then ([H.sub.2]) and ([H.sub.5]) hold, f (u, v) and g(u) are increasing in u and v. At +[infinity], f (u, v) grows superlinearly or f (u, v) grows superlinearly and sublinearly in u and v respectively, whereas g(u) grows superlinearly at +[infinity].

By virtue of ([H.sub.1]), we can define the integral operator A : C[0, 1] [right arrow] C[0, 1] by

(A[psi])(t) = [[integral].sup.1.sub.0][G.sub.i] (t, s)[p.sub.1](s)f ([psi](s), (T [psi])(s))ds, (1.7)

where

(T [psi])(t) = [[integral].sup.1.sub.0][G.sub.i] (t, s)[p.sub.2](s)g([psi](s))ds (i = 1, 2, 3), (1.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9c)

Then the positive solutions of BVP (1.5)-(1.6) are equivalent to the positive fixed points of A.

Let J = [0, 1], 0 < c < d < 1, [J.sub.0] = [c, d], [[epsilon].sub.0] = c(1 - d),E = C[0, 1], [parallel]u[parallel] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for u [member of] E,

K = {u [member of] C[0, 1] : u(t) [greater than or equal to] 0, u(t) [greater than or equal to] t (1 - t)u, t [member of] J}.

It is easy to show that (E, [parallel]*[parallel]) is a real Banach pace and K is a cone in E. From (1.9) we get that

[G.sub.1](t, s) [greater than or equal to] [[epsilon].sub.0], [G.sub.2](t, s) [greater than or equal to] c, [G.sub.3](t, s) [greater than or equal to] 1 - d, (t, s) [member of] [J.sub.0] x [J.sub.0], (1.10)

t (1 - t)[G.sub.1](r, s) [less than or equal to] [G.sub.1](t, s) [less than or equal to] s(1 - s), t[G.sub.2](r, s) [less than or equal to] [G.sub.2](t, s) [less than or equal to] s, (1 - t)[G.sub.3](r, s) [less than or equal to] [G.sub.3](t, s) [less than or equal to] (1 - s), t, s, r [member of] J. (1.11)

Lemma 1.5. Let ([H.sub.1]) hold. Then A : K [right arrow] K is a completely continuous operator.

See [2, 6] for the proof of Lemma 1.5.

To prove our main results, we also need the following fixed point index theorems.

Let (E, [parallel]*[parallel]) be a real Banach space, P be a cone in E, and [B.sub.[rho]] = {u [member of] E : [parallel]u[parallel] < [rho]} ([rho] > 0) be the open ball of radius [rho]. Let A : [[bar.B].sub.[rho]] [intersection] P [right arrow] P be a completely continuous operator, i(A,[B.sub.[rho]][intersection] P,P) denote the fixed point index of A on [B.sub.[rho]] [intersection] P. For the details of the fixed point index, one can refer to .

Lemma 1.6.  Assume that A : [[bar.B].sub.[rho]][intersection] P [right arrow] P is a completely continuous operator. If there exists [u.sub.0] [member of] P \ {[theta]} such that

u - Au [not equal to] [lambda][u.sub.0] for all [lambda] [greater than or equal to] 0, u [member of] [partial derivative][B.sub.[rho]] [intersection] P,

then i(A,[B.sub.[rho]] [intersection] P,P) = 0.

Lemma 1.7.  Assume that A : [[bar.B].sub.[rho]] [intersection]P [right arrow] P is a completely continuous operator and has no fixed point on [partial derivative][B.sub.[rho]] [intersection] P.

(1) If [parallel]Au[parallel] [less than or equal to] [parallel]u[parallel] for any u [member of] [partial derivative][B.sub.[rho]] [intersection] P, then i(A,[B.sub.[rho]] [intersection] P,P) = 1.

(2) If [parallel]Au[parallel] [greater than or equal to] u for any u [member of] [partial derivative][B.sub.[rho]] [intersection] P, then i(A,[B.sub.[rho]] [intersection] P,P) = 0.

2. Main Results

Theorem 2.1. Let ([H.sub.1]), ([H.sub.2]) and ([H.sub.3]) hold. Then (1.1)-(1.2a) has a positive radial solution for any annulus a < r < b.

Proof. Firstly, we consider (1.5)-(1.6a). By ([H.sub.2]), there are v > 0 and sufficiently large M >0 such that

f ([psi],[psi]) [greater than or equal to] v[[psi].sup.[alpha]] for all [psi] [member of] [R.sup.+], [psi] > M, (2.1)

g([psi]) [greater than or equal to] [C.sub.0][[psi].sup.1/[alpha]] for all [psi] > M, (2.2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[gamma]'.sub.2] = [[integral].sup.d.sub.c][p.sub.2](r)dr. Let N = (M + 1)[[epsilon].sup.-1.sub.0], [[psi].sub.0] (t) = sin [pi]t [member of] K \ {[theta]}. We claim that

[psi] - A[psi] [not equal to] [lambda][[psi].sub.0] for all [lambda] [greater than or equal to] 0, [psi] [member of] [partial derivative][B.sub.N] [intersection] K.

In fact, if there are [lambda] [greater than or equal to] 0, [psi] [member of] [partial derivative][B.sub.N] [intersection] K such that [psi] - A[psi] = [lambda][[psi].sub.0], then

[psi](t) [greater than or equal to] (A[psi])(t) [greater than or equal to] [[integral].sup.d.sub.c] [G.sub.1](t, s)[p.sub.1](s)f [psi](s), [[integral].sup.1.sub.0][G.sub.1](s, r)[p.sub.2](r)g([psi](r))dr)ds, t [member of] J. (2.3)

Owing to [alpha] [member of] (0, 1] and [psi](t) [greater than or equal to] [[epsilon].sub.0][parallel][psi][parallel] = [[epsilon].sub.0]N = M + 1 > M, [psi](t) [member of] K, t [member of] [J.sub.0], (2.2) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

By using 0 [less than or equal to] [G.sub.1](t, s) [less than or equal to] 1, [alpha] [member of] (0, 1] and Jensen's inequality, it follows from (2.1)--

(2.4) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This is a contradiction. By Lemma 1.6 we get

i(A,[B.sub.N] [intersection] K,K) = 0. (2.5)

On the other hand, according to the second limit of ([H.sub.3]), there exists a sufficiently small [[rho].sub.1] [member of] (0, 1) such that

[C.sub.1] =: sup {f ([psi],[psi])/[psi][beta] : [for all] [psi] [member of] [R.sup.+], [psi] [member of] (0, [[rho].sub.1]]} < +[infinity]. (2.6)

Let [[epsilon].sub.1] = min {[[rho].sub.1][[gamma].sup.-1.sub.2], [[gamma].sup.-1.sub.2] [(1/2[C.sub.1][[gamma].sub.1]).sup.1/[beta]]} > 0. By the first limit of ([H.sub.3]), there exists a sufficiently small [[rho].sub.2] [member of] (0, 1) such that

g([psi]) [less than or equal to] [[epsilon].sub.1][[psi].sup.1/[beta]], [psi] [member of] [0, [[rho].sub.2]]. (2.7)

Let [rho] = min{[[rho].sub.1], [[rho].sub.2]}. (2.6) and (2.7) imply that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus [parallel]A[psi][parallel] [less than or equal to] 1/2[parallel][psi][parallel] < [parallel][psi][parallel] for any [psi] [member of] [partial derivative][B.sub.[rho]] [intersection] K. Lemma 1.7 yields

i(A,[B.sub.[rho]] [intersection] K,K) = 1. (2.8)

(2.5) together with (2.8) imply that

i(A, ([B.sub.N] \ [[bar.B].sub.[rho]]) [intersection] K,K) = i(A,[B.sub.N] [intersection] K,K) - i(A,[B.sub.[rho]] [intersection] K,K) = -1.

So A has a fixed point [psi] [member of] ([B.sub.N] [[bar.B].sub.[rho]]) [intersection] K and satisfies 0 < [rho] < [parallel][psi][parallel] [less than or equal to] N. We know that [psi](t) > 0, t [member of] (0, 1) by definition of K. This show that BVP (1.5)-(1.6a) has a positive solution [psi],[psi] [member of] [C.sup.2](0, 1) [intersection] C[0, 1], and satisfies [psi](t) > 0,[psi](t) > 0 for any t [member of] (0, 1). Similarly we can get the conclusions of (1.5)-(1.6b) and (1.5)-(1.6c). This completes the proof of Theorem 2.1.

Theorem 2.2. Let ([H.sub.1]), ([H.sub.4]) and ([H.sub.5]) hold. Then (1.1)-(1.2) has a positive radial solution for any annulus a < r < b.

Proof. First consider (1.5)-(1.6a). By ([H.sub.4]), there exist [delta] > 0,[C.sub.2] > 0 and [C.sub.3] > 0 such that

f ([psi],[psi]) [less than or equal to] [delta][[psi].sup.p] + [C.sub.2], g([psi]) [less than or equal to] [([psi]/2[delta][[gamma].sub.1][[gamma].sup.p.sub.2]).sup.1/p] + [C.sub.3], [psi],[psi] [member of] [R.sup.+]. (2.9)

(2.9) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

By means of simple calculation, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then there exists a sufficiently large G > 0 such that

[delta][[gamma].sub.1] [[[([parallel][psi][parallel]/2[delta][[gamma].sub.1]).sup.1/p] + [C.sub.3][[gamma].sub.2]].sup.p] + [[gamma].sub.1][C.sub.2] < 3/4[parallel][psi][parallel], [parallel][psi][parallel] > G.

From this and (2.10) we obtain that

[parallel]A[psi][parallel] < [parallel][psi][parallel], [psi][member of][partial derivative][B.sub.G] [intersection] K.

This, along with Lemma 1.7, yields that

i(A,[B.sub.G] [intersection] K,K) = 1. (2.11)

In addition, by ([H.sub.5]), there exist [eta] > 0 and sufficiently small [xi] > 0 such that

f ([psi],[psi]) [greater than or equal to] [eta][[psi].sup.q], [psi][member of] [R.sup.+], 0 [less than or equal to] [psi] [less than or equal to] [xi], (2.12)

[g.sup.q]([psi]) [greater than or equal to] [C.sub.4][psi], 0 [less than or equal to] [psi] [less than or equal to] [xi], (2.13)

where [C.sub.4] = 2 [([eta][[epsilon].sup.2.sub.0] [[integral].sup.d.sub.c][G.sub.1] (1/2, s)[p.sub.1](s)ds [[integral].sup.d.sub.c][p.sup.q.sub.2](r)dr).sup.-1]. Since g(0) = 0, g [member of] C([R.sup.+],[R.sup.+]), there exists [sigma] [member of] (0, min{[xi], [[gamma].sup.-1.sub.2] [xi]}) such that g([psi]) [less than or equal to] [[gamma].sup.-1.sub.2] [xi], for any [psi] [member of] [0, [sigma]]. This implies that

[[integral].sup.1.sub.0] [G.sub.1](s, r)[p.sub.2](r)g([psi](r))dr [less than or equal to] [xi], [psi] [member of] [[bar.B].sub.[sigma]] [intersection] K, s [member of] [0, 1]. (2.14)

By using Jensen's inequality and 0 < q [less than or equal to] 1, from (2.12)-(2.14) we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

thus

[parallel]A[psi][parallel] > [parallel][psi][parallel] for all [psi] [member of] [partial derivative][B.sub.[sigma]] [intersection] K. (2.15)

(2.15) together Lemma 1.7 yield

i(A,[B.sub.[sigma]] [intersection] K,K) = 0. (2.16)

(2.11) and (2.16) imply that

i(A, ([B.sub.G] \ [[bar.B].sub.[sigma]]) [intersection] K,K) = i(A,[B.sub.G] [intersection] K,K) - i(A,[B.sub.[sigma]] [intersection] K,K) = 1.

So A has a fixed point [psi] [member of] ([B.sub.G] [[bar.B].sub.[sigma]]) [intersection] K and satisfies 0 < [sigma] < [parallel][psi][parallel] < G. This show that BVP (1.5)-(1.6a) has a positive solution [psi], [psi] [member of] [C.sup.2](0, 1) [intersection] C[0, 1], and [psi](t) > 0,[psi](t) > 0 for any t [member of] (0, 1). Similarly we can get the conclusions of (1.5)-(1.6b) and (1.5)-(1.6c). This completes the proof of Theorem 2.2.

Theorem 2.3. Let ([H.sub.1]), ([H.sub.2]), ([H.sub.5]) and ([H.sub.6]) hold. Then (1.1)-(1.2) has two positive radial solutions for any annulus a < r < b.

Proof. We take N > R > [sigma] such that either (2.5) or (2.16) hold. (1.7), (1.8) and ([H.sub.6]) indicate that

(A[psi])(t) [less than or equal to] [[integral].sup.1.sub.0][p.sub.1](s)f ([psi](s), [[integral].sup.1.sub.0] [p.sub.2](r)g([psi](r)dr)ds [less than or equal to] [[gamma].sub.1]f (R,[[gamma].sub.2]g(R)) < R

for any [psi] [member of] [partial derivative][B.sub.R] [intersection] K, then [parallel]A[psi][parallel] < [parallel][psi][parallel] for any [psi] [member of] [partial derivative][B.sub.R] [intersection] K. Lemma 1.7 implies

i(A,[B.sub.R] [intersection] K,K) = 1.

Consequently,

i(A, ([B.sub.N] \ [[bar.B].sub.R]) [intersection] K,K) = i(A,[B.sub.N] [intersection] K,K) - i(A,[B.sub.R] [intersection] K,K) = -1, i(A, ([B.sub.R] \ [[bar.B].sub.[sigma]]) [intersection] K,K) = i(A,[B.sub.R] [intersection] K,K) - i(A,[B.sub.[sigma]] [intersection] K,K) = 1.

So A has two fixed points [[psi].sub.1] [member of] ([B.sub.R] [[bar.B].sub.[sigma]]) [intersection] K and [[psi].sub.2] [member of] ([B.sub.N] \ [[bar.B].sub.R]) [intersection] K respectively, and 0 < [sigma] < [parallel][[psi].sub.1][parallel] < R < [parallel][[psi].sub.2][parallel] [less than or equal to] N. Then BVP (1.5)-(1.6) has two positive solution ([[psi].sub.1],[[psi].sub.1]), ([[psi].sub.2],[[psi].sub.2]), and satisfy [[psi].sub.i] (t) > 0, [[psi].sub.i] (t) > 0(i = 1, 2) for any t [member of] (0, 1). This completes the proof of Theorem 2.3.

Remark 2.4. From Examples 1.1-1.4 we know that all conclusions in this paper are different from the ones in [1, 5, 7-11] and the conditions that we use are more general than the ones in papers [1, 5, 7-11].

Received April 28, 2006; Accepted January 28, 2007

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(1) This work is supported by Natural Science Foundation of the EDJP (05KGD110225), JSQLGC, National Natural Science Foundation 10671167, and EDAP2005KJ221, China.

(2) Corresponding author

Shengli Xie Department of Mathematics and Physics, Anhui Institute of Architecture and Industry, Hefei 230022, People's Republic of China E-mail: xieshengli200@sina.com

Jiang Zhu (2) Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, People's Republic of China E-mail: jzhuccy@yahoo.com.cn