0 Boundary Value Problems

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Positive solutions for a class of superlinear semipositone systems on exterior domains

Abraham Abebe1, Maya Chhetri1*, Lakshmi Sankar2 and R Shivaji1

Correspondence: maya@uncg.edu 1 Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, USA

Full list of author information is available at the end of the article

Abstract

We study the existence of a positive radial solution to the nonlinear eigenvalue problem-Au = XK1 (|x|)f(v) in i2e,-Av = XK2(|x|)g(u) in i2e, u(x) = v(x) = 0if |x| = r0 (> 0), u(x) — 0, v(x) — 0 as |x| — to, where X > 0 is a parameter, Au = div(Vu) is the Laplace operator, = {x e R" | |x| > r0,n > 2}, and Ki e C1 ([r0, to),(0, to)); i =1,2are such that K(|x|) — 0 as |x| — to. Here f,g : [0,to) — R are C1 functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for X small via degree theory and rescaling arguments. We also discuss a non-existence result for X > 1 for the single equations case.

MSC: 34B16; 34B18

Keywords: superlinear; semipositone; positive solutions; existence; non-existence; exterior domains

ft Spri

1 Introduction

We consider the nonlinear elliptic boundary value problem

-Au = XK1(\x\)f (v) in Qe,

-Av = XK2(\x\)g(u) in

u(x) = v(x) = 0 if \x\ = r0 (> 0), u(x) — 0, v(x) — 0 as \x\ —^ to,

where X > 0 is a parameter, Au = div(Vu) is the Laplace operator, and = {x e rn | |x| > r0, n >2} is an exterior domain. Here the nonlinearities f,g: [0, to) — r are C1 functions which satisfy:

(Hi) f (0) < 0 and g(0) < 0 (semipositone).

(H2) For i = 1,2 there exist bi >0 and qi >1 such that lims—TO = b1, and lims—TO gj2 = b2.

Further, for i = 1,2, the weight functions Ki e C1([r0, to), (0, to)) are such that Ki(|x|) — 0 as |x| —^ to. In particular, we are interested in the challenging case, where Ki do not decay too fast. Namely, we assume

(H3) There exist cL1 > 0, d2 > 0, p e (0, n - 2) such that for i = 1,2

d1 / \ d2

< Kt{ \x\)<—n- for \x\»1.

ringer

|x|n+^ — v - |x|n+p

© 2014 Abebe et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

We then establish the following.

Theorem 1.1 Let (Hi)-(H3) hold. Then (1.1) has a positive radial solution (u, v) (u > 0, v > 0 in Qe) when k is small, and || u II to —^ (TO, 11 v 11 to —^ (TO as k —^ 0.

We prove this result via the Leray-Schauder degree theory, by arguments similar to those used in [1] and [2]. The study of such eigenvalue problems with semipositone structure has been documented to be mathematically challenging (see [3,4]), yet a rich history is developing starting from the 1980s (see [5-7]) until recently (see [8-12]). In [1, 2] the authors studied such superlinear semipositone problems on bounded domains. In particular, in [12] the authors studied the system

-Au = kf (v) in -Av = kg(u) in u = v =0 on 3^,

where ^ is a bounded domain in Rn, n > 1, and establish an existence result when k is small. The main motivation of this paper is to extend this study in the case of exterior domains (see Theorem 1.1). We also discuss a non-existence result for the single equation model:

-Au = kK1(|x|)f (u) in

u(x) = 0 if |x| = r0 (> 0),

u(x) — 0 as |x| — to,

for large values of k, whenf, K1 satisfy the following hypotheses:

(H4) f e C1([0, to), r),f'(z) > 0 for all z > 0,f(0) < 0, and there exists m0 > 0 such that

limz—TO > m0.

(H5) The weight function K1 e C1([r0, to), (0, to)) is such that s K1(r0s2-n) is decreasing for s e (0,1].

We establish the following.

Theorem 1.2 Let (H3)-(H5) hold. Then (1.2) has no nonnegative radial solution for k » 1.

We establish Theorem 1.2 by recalling various useful properties of solutions established in [13], where the authors prove a uniqueness result for k »1 for such an equation in the case whenf is sublinear at to. However, the properties we recall from [13] are independent of the growth behavior off at to. Non-existence results for such superlinear semipositone problems on bounded domain also have a considerable history starting from the work in the 1980s in [14] leading to the recent work in [15]. Here we discuss such a result for the first time on exterior domains.

Finally, we note that the study of radial solutions (u(r), v(r)) (with r = |x|) of (1.1) corresponds to studying

-(rn-1u'(r))' = krn-1Ki(r)f (v(r)) for r > r0,

-(rn-1v'(r))' = Xrn-1K2(r)g(u(r)) for r > ro,

u(r) = v(r) = 0 if r = r0 (> 0),

u(r) — 0, v(r) — 0 as r — to,

which can be reduced to the study of solutions (u(s), v(s)); s e [0,1] to the singular system:

-u"(s) = Xh1(s)f (v(s)), 0< s < 1,

-v"(s)=Xh2(s)g(u(s)), 0< s <1,

u(0) = u(1) = 0, v(0) = v(1) = 0,

r2 -2(n-1) 1

via the Kelvin transformation s = (r)2-n, where hi(s) = (n-°)2s (n-2) Ki(r0s2-n), i = 1,2 (see [16]). r0 n

Remark 1.3 The assumption (H3) implies that lims—0+ hi(s) = to, for i = 1,2, h = infte(0,1){h1(t), h2(t)} > 0, and there exist d > 0, n e (0,1) such that hi(s) < sdn for s e (0,1], and for i = 1,2. When in addition (H5) is satisfied, h1 is decreasing in (0,1].

We will prove Theorem 1.1 in Section 2 by studying the singular system (1.3), and Theorem 1.2 in Section 3 by studying the corresponding single equation

-u"(s) = Xh\(s)f (u(s)), 0 < s < 1,1 u(0) = u(1) = 0.

2 Existence result

We first establish some useful results for solutions to the system

-u"(s) = ¿1h:(s)|v(s) + /|?1, 0 < s < 1, -v"(s) = b2h2(s)u(s) + W2, 0 < s < 1, u(0) = u(1) = 0, v(0) = v(1) = 0,

where l > 0 is a parameter. (Clearly, any solution (ui, vi) of (2.1) for l >0 must satisfy ui(s) > 0, vl(s) > 0 for s e (0,1). This is also true for any nontrivial solution when l = 0.) We prove the following.

Lemma 2.1

(i) There exists l0 > 0 such that 2.1 has no solution if l > l0.

(ii) For each l e [0, l0), there exists M >0 (independent of l) such that if (ul, vl) is a solution of (2.1), then max{|ul||TO, ||vl|TO} < M.

Proof of (i) Let k1 := n2, := sin(n s). Here k1 is the principal eigenvalue and a corresponding eigenfunction of (s) = k$(s) in (0,1) with ^(0) = 0 = ^(1). Let a > ,,k1, c >0

V0102 h

be such that (s + l)qi > as - c for all s > 0 and for i = 1,2. Now let (ul, vl) be a solution of

(2.1). Multiplying (2.1) by < and integrating, we obtain

A J ul<1 ds = b1 I h1(s)(vl + l)qi <1 ds > b1 I h1(s)(avl - c)<1 ds Jo Jo Jo

A1 I vl<1 ds = b2 I h2(s)(ul + l)q2< ds > b2 / h2(s)(aul - c)< ds.

By Remark 1.3, h = infte(0,1)|h1(i),h2(i)} > 0, and Hh;^ := /0h;(s)ds < to for j = 1,2. Then from the above inequalities we obtain

i vl<1 ds < 1 ^ \A1 i ui< ds + b1c|h11|: Jo abih\ Jo

i ui<1 ds < (k1 i vi<1 ds + b2c|h2|H. Jo ab2h\ Jo /

Hence we deduce that

Z*1 . . I ul<1 as < — := w2,

where w := (ab2h - -tt), and w1 := A1c||h1111 + b2c||h21|1. This implies

ab1h ah

f1 A1W2

(vl + l)q1 < ds < —— := w3.

In particular, this implies A4 lq1ds < inf W3 < . Since w3 is independent of l, clearly this is

1 inf 13 i<1

4 [ 4,4 ]

a contradiction for l » 1, and hence there must exists an lo > o such that for l > lo, (2.1) has no solution.

Proof of (ii) Assume the contrary. Then without loss of generality we can assume there exists {ln} c (o,lo) such that ||uln ||to ^to as n ^ to. Clearly u![n(s) < o, and V((s) < o for all s e (o, 1). Let s1(ln) e (o, 1), s2(ln) e (o, 1) be the points at which uln and vln attain their maximums. Now since u'[ (s) < o for all s e (o, 1), we have

uln (s) >

sUiiSnfi2 for s e ^ sK^

for s e (s1(ln),1).

Hence uin(s) > min{s|||TO, ^S-gp}, and in particular, for s e [4, f ],

Uln(s) > minj 4 ||Uln ||to,4 ||Uln htoJ = 4 ||Uln ||to.

Let sjn,K e [4, |1 be such that min[i,|] ui„(s) = utn(si), and min^^ vtn(s) = vtn(sin). Now for s e [±,f], "4

vin(s) > b2hm \uin(t) +1\02 dt,

where m := min[i,3]x[i,3] G(s, t) (> 0), and Gis the Green's function of-Z" withZ(0) = 0 = Z(1). In particular, vjsi) > b2h^(uin(si))q2. Similarly uin(si) > bihff(vin(sin))qi. Hence, there exists a constant A >0 such that

uin (sin) > A(un (sin))qiq2

This is a contradiction since qlq2 > l and uln (sin) > 4 lluin ||to — to as n — to. Thus (ii) holds. □

Proof of Theorem l.l We first extend f and g as even functions on r by setting f (-s) =f (s) and g(-s) = g(s). Then we use the rescaling, X = ys, wl = yu, and w2 = y6v with y > 0, 6 = q+l,and 5 = s^2-1. With this rescaling, (l.3) reduces to

-wi'(s)=F(s, y , w2), 0< s < 1,

-w2'(s) = G(s, y , wi), 0< s < 1,

Wi(0) = Wi(1) = 0, W2(0) = W2(1) = 0,

F(s, y, W2):=y 1+5hi(s) f -2 - bi

+ bi|w2 |qi hi(s), and

G(s, y, Wi):=y6+5h2(s)(g(W^J- b2 y ^ + b2|wi|q2h2(s).

Note that by our hypothesis (H2), F(s, y, w2) — bi|w2 |qihi(s) and G(s, y, wi) — b2|wi|q2 x h2(s) as y — 0.HencewecancontinuouslyextendF(s, y, w2) and G(s, y, wi) toF(s, 0, w2) = bi|w2|qihi(s) and G(s, 0, wi) = b2|wi|q2h2(s), respectively. Note that proving (1.3) hasapos-itive solution for X small is equivalent to proving (2.2) has a solution (wi, w2) with wi > 0, w2 > 0 in (0,1) for small y >0. We will achieve this by establishing that the limiting equation (when y =0)

-wi'(s) = F(s,0,w2) = b^^s) | w2|qi, 0 < s <1, -w2'(s) = G(s, 0, wi) = ^^(s^wi02, 0 < s <1, Wl(0) = Wi(1) = 0, W2(0) = W2(1) = 0

(which is the same as (2.1) with i = 0) has a positive solution wi > 0, w2 > 0 in (0,1) that persists for small y >0.

Let X = Co [0,1] x Co [0,1] be the Banach space equipped with ||w||X = ||(wi, w2)||X = max{||wi|| 11 w2 N^o}, where || • ^oo denotes the usual supremum norm in C0([0,1]). Then for fixed y > 0, we define the map S(y , •): X — X by

S(y, w) :=w - {K{F(s, y, w2)),K{G(s, y, w^)),

whereK(H(s, y,Z(s))) = G(t,s)H(t, y,Z(t)) dt. NotethatF(s, y, •), G(s, y, •): Co([o,1]) ^ L1 (o, 1) are continuous and K: L1 (o, 1) ^ Cj ([o, 1]) is compact. Hence S(y, Oisa compact perturbation of the identity. Clearly for y > o, if S(y, w) = o, then w = (w1, w2) is a solution of (2.2), and if S(o, w) = o, then w = (w1, w2) is a solution of (2.3).

We first establish the following.

Lemma 2.2 There exists R > o such thatS(o, w) = oforallw = (w1, w2) e X with || w||X = R and deg(S(o, •),Br(o), o) = o.

Proof Define Sl (o, w):X ^ X by

Sl(o, w) := w - (K(bih1(s)iw2 + l|q1),K(b2h2(s)|w1 + l|q2))

for l > o. (Note So(o, w) = S(o, w).) By Lemma 2.1, if l > lo then Sl(o, w) = o and if Sl(o, w) = o for l e [o,lo), then ||w||X < M. This implies that there exists R » 1 such that Sl(o, w) = o for w e dBR(o) for any l > o. Also, since (2.1) has no solution for l > lo, deg(Slo(o, •), Br(o),o) = o. Hence, using the homotopy invariance of degree with the parameter l e [o, lo] we get

deg(S(o, •), Br(o),o) = deg(Slo(o, •), Br(o),o) = o. □

Next we establish the following.

Lemma 2.3 There exists r e (o, R) small enough such thatS(o, w) ¥Qfor allw = (w1, w2) e X with ||w|x = r and deg(S(o, •),Br(o),o) = 1.

Proof Define TT (o, w): X ^ X by

TT(o,w):=w - (K(tb1h1(s)|w2|q1),K(tb2h2(s)|w1|q2))

for t e [o, 1]. Clearly T 1(o, w) = S(o, w), and To(o, w) = / is the identity operator. Note that Tt(o, w) = o if w = (w1, w2) is a solution of

-w1'(s) = tb1h1(s)iw2iql, o < s < 1,

-w2'(s) = tb2h2(s)iWliq2, o < s <1,

w1(o) = w1(1) = o, w2(o) = w2(1) = o,

and for t = 1, (2.4) coincides with (2.3). Assume to the contrary that (2.4) has a solution w = (w1, w2) with ||w||X = r > o. Without loss of generality assume || w1 ||to = r. Now,

w1(s) = t i G(s,t)b1h1(s)iw2iq1 ds.

Then ||w11to < CC|w2|toto for some constant C > o independent of t e [o,1]. Similarly ||w2||to < CHw1 ||TOTO for some constant C > o. This implies that

r = ||w1||to< CHw1Hq!1q2 = Crq1q2

for some constant C > 0. But q1q2 > 1, and hence this is a contradiction if r > 0 is small. Thus there exists small r >0 such that (2.4) has no solution w with \\w\\X = r for all t e [0,1]. Now using the homotopy invariance of degree with the parameter t e [0,1], in particular using the values t = 1 and t = 0, we obtain

deg(S(0, •), Br (0), 0) = deg(T 1(0, •), Br (0),0) = deg(T0(0, •), Br (0), 0) =1. □

By Lemma 2.2 and Lemma 2.3, with 0 < r < R,we conclude that

deg(S(0, •),Br(0) \ B(0),0) = -1,

and hence (2.3) has a solution w = (wi, w2) with w1 > 0, w2 > 0 in (0,1), and r < \\w\\X < R. Now we show that the solution obtained above (when y =0) persists for small y >0 and remains positive componentwise.

Lemma 2.4 Let R, r be as in Lemmas 2.2, 2.3, respectively. Then there exists yo>0 such that:

(i) deg(S(y, •),Br(0) \ B(0), 0) = -1 for all y e [0, y„].

(ii) If S(y, w) = 0for y e [0, y0] with r < \\w\\X < R, then w1 > 0, w2 > 0 in (0,1).

Proof of (i) We first show that there exists y0 > 0 such that S(y, w) = 0 for all w = (w1, w2) e X with \\w\\X e {R,r}, for all y e [0, y0]. Suppose to the contrary that there exists {yn} with yn — 0, S(Yn, wn) = 0 and \\wj\x e {r,R}. Since K = (K,K) : L1(0,1) x L1(0,1) — CJ([0,1]) x C1([0,1])is compact, and {F(s, yn, w2n), G(s, yn, w1n)} are bounded in L1(0,1) x L1(0,1), wn — Z = (Z1,Z2) e C1([0,1]) x C1([0,1]) (up to a subsequence) with \\Z\\x = R or r and S(0, Z) = 0. This is a contradiction to Lemma 2.2 or 2.3 and hence there exists a small y0 > 0 satisfying the assertions. Now, by the homotopy invariance of degree with respect to y e [0, y0],

deg(S(Y, •), Br(0)\B;(0),0) = deg(S(0, •), Br(0)\B;(0),0) = -1

for all y e [0, y0].

Proof of (ii) Assume to the contrary that there exists yn — 0 and a corresponding solution wn = (w1n, w2n) such that r < \\wn \\X < R and

Q.n := {x e (0,1) | w1n(x) < 0 or w2n(x) < ^ = 0.

Arguing as before, — Z e C1([0,1]) x ^,([0,1]) with S(0,Z) = 0 (up to a subsequence). Note that Z ^ 0 since \\Z\X > r > 0. By the strong maximum principle Z1 > 0, Z2 > 0, Z1 (0) > 0, Z2(0) > 0, Z1 (1) < 0 and Z2(1) < 0. Now suppose there exists {xn} e (0,1) with {xn} e Qn and w1n(xn) < 0. Then {xn} must have a subsequence (renamed as {xn} itself) such that xn — x e [0,1]. But Z1 > 0 in (0,1) implies that x e {0,1}. Suppose x = 0. Since w1n(xn) < 0 and w1n(0) = 0, there existsyn e (0,xn) suchthat w'1n(yn) < 0, and hence taking the limit as n — to we will have Z1 (0) < 0, which is a contradiction since Z1 (0) > 0. A similar contradiction follows if x = 1, using the fact that Z1 (1) < 0. Further, contradictions can

be achieved if there exists [x„]e Q with {xn} e Qn and w2n(xn) < 0 using the facts that Z2 (0) > 0 and Z2 (1) < 0. This completes the proof of the lemma. □

We now easily conclude the proof of Theorem 1.1. From Lemma 2.4, since w = (w1, w2) is a positive solution of (2.2) for y small, (u, v) = (y-1w1, y-0 w2) with 0 = is a positive solution of (1.3) for X = yS where S = Further, since w1 > 0 and w2 > 0 in (0,1) for y e [0, y0], ||u||TO ^ to and ||v||TO ^to as X(= yS) ^ 0. This completes the proof of Theorem 1.1. □

3 Non-existence result

We first recall from [13] that, when (H5) is satisfied, one can prove via an energy analysis that a nonnegative solution u of (1.4) must be positive in (0,1) and have a unique interior maximum with maximum value greater than 0, where 0 is the unique positive zero of F(s) = JSf(y) dy. Further, for X » 1 and s1, s1 e (0,1) such that si > s1, u(s1) = u(sl) = ¡3 (see Figure 1), where ¡3 > 0 is the unique zero off, there exists a constant C such that s1 < CX-2 and (1 - si) < CX-2. Hence we can assume (si - s1)> 2 for X » 1. Now we provide the proof of Theorem 1.2.

Proof of Theorem 1.2 Let v := u - 3. Then v > 0 in (s1, si) and satisfies

vfo) = v(sl) = 0.

-v" = Xh^s)^ v, s1< s < si, I

Note that $(s) = -(sin(#-4)) > 0 in fo, si), $(si) = $(s1) = 0, and it satisfies -<j>" = $

- (Sl-S1) )) > 0 in (l1, l1), = YCD = alld iL satisfies - = (L s1)2 1

in (s1,si). Hence using the fact that fLL1 (-$v" + v$") ds = 0, we obtain

£( x ur-l ^(STLF1 * = 0

In particular,

f(u(sx

*u(sx)-P'X (si- S1)2

f(u(s\)) n2 X———-h1(sx) = -p-rr, for some sx e (s1,s^. (3.1)

Figure 1 Graph of u.

But h = inf(0,i) hi(s) > 0, and (si - si) > | for X ^ 1. Thus clearly (3.1) can hold when X ^ro, only if Z = u(sX) ^ro with U((^j-ß ^ 0. But by (H4), this is not possible since limZ^ro — m0 > 0. Hence the nonnegative solution cannot exist for X ^ 1. □

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details

1 Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, USA. 2Department of Mathematics & NTIS, University of West Bohemia, Univerzitni 22, Plzen, 30614, Czech Republic.

Acknowledgements

The third author is funded by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.

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doi:10.1186/s13661-014-0198-z

Cite this article as: Abebe et al.: Positive solutions for a class of superlinear semipositone systems on exterior domains. Boundary Value Problems 2014 2014:198.