MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation64084110.1155/2010/640841640841Research ArticlePositive Solution for the Elliptic Problems with Sublinear and Superlinear NonlinearitiesYuanChunmeiGuoShujuanTongKaiyuChouJyh HorngCollege of Physics and MathematicsChangzhou UniversityChangzhouJiangsu 213164Chinacczu.edu.cn20102212201020100810201013122010131220102010Copyright © 2010 Chunmei Yuan et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities -Δu=λa(x)up+b(x)uq in Ω, u>0 in Ω, u=0 on Ω, where λ>0 is a real parameter, 0<p<1<q. Ω is a bounded domain in N   (N 3), and a(x) and b(x) are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.

1. Introduction

In this paper, we consider the elliptic problems with sublinear and superlinear nonlinearities -Δu=λa(x)up+b(x)uqin  Ω,u>0in  Ω,u=0on  Ω, where λ>0  is a real parameter,  0<p<1<q.  Ω is a bounded domain in N(N3), and a(x) and b(x) are some given functions which satisfies the following assumptions:

a(x),b(x)L(Ω),  a(x)c0,  b(x)-c1,  where c0,  c1 are positive constants,

or

a(x), b(x)L(Ω),  a(x),b(x)c0,  where c0 is a positive constant.

For convenience, we denote (1)λ with hypothesis (H1) or (H2) by (1)λ- and (1)λ+, respectively.

Such problems occur in various branches of mathematical physics and population dynamics, and sublinear analogues or superlinear analogues of (1)λ have been considered by many authors in recent years (see  and their references). But most of such studies have been concerned with equations of the type involving sublinear nonlinearity (see [36, 8, 9]), with only few references dealing with the elliptic problems with sublinear and superlinear nonlinearities. In , Ambrosetti et al. deal with the analogue of (1)λ with a(x)=b(x)1. It is known from  that there exist λ*(0,), such that problem (1)λ has a solution if λλ*  and has no solution if λ>λ*,  provided b(x)1 on Ω.

Our goal in this paper is to show how variational method and super-subsolution method can be used to establish some existence results of problem (1)λ. We work on the Sobolev space H01(Ω)  equipped with the norm  x=(Ω|u|2dx)1/2.   For uH01(Ω)  we define Iλ:H01(Ω)  by Iλ(u)=12Ω|u|2dx-λp+1Ωa(x)|u|p+1dx-1q+1Ωb(x)|u|q+1dx.

Let λ1 be the first eigenvalue of -Δu=λu,xΩ,u=0,xΩ.

φ1  denotes the corresponding eigenfunction satisfying  0φ1(x)1.  Lp(Ω),  (1p), denotes Lebesgue spaces, and the norm in Lp is denoted by ·p.

2. The Existence of Positive Solution of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M60"><mml:mrow><mml:msubsup><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

It is well known that φ1(x)0,xΩ. Define  a=minΩ|φ1|2;   from (2.1) we know  a>0,  so we can split the domain Ω into two parts: Ωɛ and ΩΩɛ, where Ωɛ={xΩ:|φ1|2a/2}{xΩ:φ1(x)ɛ,ɛis small enough}. Let  b=infΩΩɛφ1(x);  we obtain that  bɛ  by the positivity of φ1  in  Ω,  and  ΩΩɛ  is nonempty when  ɛ  is small enough.

Theorem 2.1.

Let  a(x),  b(x) satisfy assumption (H1),  and  0<p<1<q<2*-1, where  2*=2N/(N-2) is the limiting exponent in the Sobolev embedding. Then there exists a constant λ̃>0  such that (1)λ- possesses at least a weak positive solution u*(x)H01(Ω)  for  λλ̃.

Proof.

Let  e(x)  denote the positive solution of the following equation: -Δe=1,xΩ,e=0,xΩ. Here and hereafter we use the following notations: A=a,   B=b,   E=e. Since 0<p<1, for all λ+,  there exists  T=T(λ)>0  satisfying TλATpEp. Observing that  b(x)-c1<0,  as a consequence, the function  Te  verifies T=-Δ(Te)λA(Te)pλa(x)(Te)p+b(x)(Te)q, and hence it is a supersolution of (1)λ+.  Let  v(x)=φ1l,  xΩ,  l>1. For xΩ, we have xΩɛ or xΩΩɛ. We will discuss it from two conditions.

(I)  For all xΩɛ, observing that l>1 and when ɛ is small enough, we have al(l-1)s-22-Bsl(q-1)>λ1l,s(0,ɛ). Since xΩɛ, then it follows that φ1(x)ɛ,  |φ1|2a/2.  From (2.5) we infer λ1ll(l-1)φ1-2|φ1|2-Bφ1(x)l(q-1),xΩɛ. Multiplying (2.6) with φ1l,  we get l(1-l)φ1l-2|φ1|2+λ1lφ1l-Bφ1lq. It follows that -Δ(φ1l)λa(x)(φ1l)p-b(x)(φ1l)q.

(II)  For all  xΩΩɛ, there exists  λ̃>0,  such that  for all  λλ̃, and we have λc0spl-Bsqlλ1lsl,sR,bs1. Since xΩΩɛ,  then we have φ1(x)b  (and  φ1(x)1).  From (2.9), it follows that -Δ(φ1l)λ1lφ1lλc0φ1lp-Bsqlλa(x)(φ1l)p+b(x)(φ1l)q. From (2.8) and (2.10), we derive that there exists  λ̃>0  such that   for all  xΩ,   for all  λλ̃, -Δ(φ1l)λa(x)(φ1l)p+b(x)(φ1l)q, that is,  v(x)=φ1l(x) is a subsolution of (1)λ-. Taking T as sufficiently large, we also have Te>φ1l  by minimal principle. Define  w(x)=Te(x),  and let K={uH01(Ω):  v(x)u(x)w(x),for all  xΩ},   then K is closed and convex (and weakly closed). Let f(s)=λa(x)sp+b(x)sq,  for all  s,s>0.  We consider the function Iλ(u)=12Ω|u|2dx-Ω0uf(s)dsdx. Observe that  b(x)<0,  0<p<1<q<2*-1;  we infer that  Iλ is coercive, bounded, since it is blow and weakly lower semicontinuous. Using this fact, we conclude that there exists u*K, such that Iλ(u*)=infKIλ (see ). In the following, we will prove that u* is a solution of problem (1)λ-.

For ϕK, define h:[0,1], such that h(t)=I(tϕ+(1-t)u*). Clearly, h(t) achieves its minimum at  t=0, and   h(t)|t=0=Ω[u*(ϕ-u*)]dx-Ωf(u*)(ϕ-u*)dx0. For all  φH01(Ω),  η>0,  define Ψ(x)={v,  whenu*+ηφ<v,  u*+ηφ,  whenvu*+ηφw,  w,  whenu*+ηφ>w.   Obviously,  ΨK,  and inserting (2.15) into (2.14),  we find 0vu*+ηφw[u*(ηϕ)-f(u*)(ηφ)]dx+u*+ηφ>w[u*(w-u*)-f(u*)(w-u*)]dx+u*+ηφ<v[u*(v-u*)-f(u*)(v-u*)]dx=ηvu*+ηφw[u*φ-f(u*)φ]dx+u*+ηφ>w[w(w-u*)-f(w)(w-u*)]dx+u*+ηφ<v[v(v-u*)-f(v)(v-u*)]dx-u*+ηφ>w|w-u*|2dx-u*+ηφ<v|v-u*|2dx+u*+ηφ>w[f(w)-f(u*)](w-u*)dx+u*+ηφ<v[f(v)-f(u*)](v-u*)dx. Since w(x) and v(x) are supersolution and subsolution, respectively, then u*+ηφ>w[w(w-u*)-f(w)(w-u*)]dxηu*+ηφ>w[wφ-f(w)φ]dx,u*+ηφ<v[v(v-u*)-f(v)(v-u*)]dxηu*+ηφ<v[vφ-f(v)φ]dx. Observe that meas[u*+ηφ>w]0, meas[u*+ηφ<v]0, as η0, u*+ηφ>w[wφ-f(w)φ]dx0,u*+ηφ<v[vφ-f(v)φ]dx0. Since  u*K,b(x)<0, it follows that u*+ηφ>w[f(w)-f(u*)](w-u*)dx=u*+ηφ>wλa(x)(wp-u*p)(w-u*)dx+u*+ηφ>wb(x)(wq-u*q)(w-u*)dxu*+ηφ>wλa(x)(wp-u*p)(w-u*)dx. Similar to (2.19), we have u*+ηφ<v[f(v)-f(u*)](v-u*)dx=u*+ηφ<vλa(x)(vp-u*p)(v-u*)dx+u*+ηφ<vb(x)(vq-u*q)(v-u*)dxu*+ηφ<vλa(x)(vp-u*p)(v-u*)dx. Similar to (2.18), as  η0, it follows that u*+ηφ>wλa(x)(wp-u*p)(w-u*)dx0,u*+ηφ<vλa(x)(vp-u*p)(v-u*)dx0. As η0, we also have vu*+ηφw[u*φ-f(u*)φ]dxΩ[u*φ-f(u*)φ]dx. Inserting (2.17), (2.19), and (2.20) into (2.16), we find 0η{vu*+ηφw[u*φ-f(u*)φ]dx+u*+ηφ>w[wφ-f(w)φ]dx+u*+ηφ<v[vφ-f(v)φ]dx}+u*+ηφ>wλa(x)(wp-u*p)(w-u*)dx+u*+ηφ<vλa(x)(vp-u*p)(v-u*)dx. Dividing by η and letting  η0, using (2.18), (2.21), and (2.22), we derive Ω[u*φ-f(u*)φ]dx0. Noting that φ is arbitrary, this holds equally for  -φ, and it follows that u* is indeed a weak solution of (1)λ-,  and the strong maximum principle yields  u*>φ1l,  in Ω. Therefore it is a weak positive solution of (1)λ-.

3. The Existence of Positive Solution of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M178"><mml:mrow><mml:msubsup><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>Theorem 3.1.

Let a(x),  b(x) satisfy assumption (H2),  and  0<p<1<q<+. Then there exists Λ,  Λ>0,  such that

for all λ(0,Λ) problem (1)λ+ has a minimal solution uλ such that Iλ(uλ)<0. Moreover uλ is increasing with respect to λ;

for  λ=Λ problem (1)λ+ has at least one weak solution uHLp+1;

for all  λ>Λ  problem  (1)λ+ has no solution.

To prove Theorem 3.1, let us define Λ=sup{λ>0:(1)λ+  has a solution}.First of all we prove a useful lemma.

Lemma 3.2.

One has 0<Λ<+.

Proof.

Let e(x) denote the solution of the following equation: -Δe=1,xΩ,e=0,xΩ. Since  0<p<1<q,  we can find  λ0>0  such that for all  0<λλ0  there exists  T=T(λ)>0  satisfying TλATpEp+BTqEq. As a consequence,  the function Te verifies T=-Δ(Te)λA(Te)p+B(Te)qλa(x)(Te)p+b(x)(Te)q, and hence it is a supersolution of (1)λ+.  Moreover, let u0 denote the solution of the following problem: -Δu=λa(x)u0p,xΩ,u0=0,xΩ. (From  we know that u0 exists.)  Then ɛu0 is a subsolution of (1)λ+, provided -Δ(ɛu0)=λɛa(x)u0pλa(x)(ɛu0)p+b(x)(ɛu0)q, which is satisfied for all  ɛ>0  small enough and all  λ. Taking  ɛ  as possibly smaller,  we also have ɛu0<Te. It follows that  (1)λ+ has a solution u,  ɛu0uTe whenever λλ0, and thus Λλ0.

Next, let λ* be such that c0(λ*tp+tq)>λ1t,t>0. If λ is such that  (1)λ+  has a solution u, multiplying (1)λ+ by φ1 and integrating over Ω we find λ1Ωuφ1dx=λΩa(x)upφ1dx+Ωb(x)uqφ1dxc0[Ω(λupφ1+uqφ1)dx]. This and (3.5)  immediately imply that  λ<λ*  and show that  Λλ*,  hence  0<Λ<+.

We are now ready to give the proof of Theorem 3.1.

Proof.

(i) From the proof of lemma, it follows that, for all  λ(0,Λ),   problem  (1)λ+  has a solution uλ.  Let u0 satisfy (3.5); the iteration -Δun+1=λa(x)unp+b(x)unq satisfies  unuλ  by making use of Lemma  3.3 of  and maximum principle. It is easy to check that uλ is a minimal solution of  (1)λ+. Indeed, if u is any solution of (1)λ+,  then  uu0  and u is a supersolution of  (1)λ+. Thus  unu,  for all  n, by induction, and uλu. Next, we will prove that Iλ(uλ)<0. Indeed, Iλ(u)=12Ω|u|2dx-λp+1Ωa(x)|u|p+1dx-1q+1Ωb(x)|u|q+1dx. Since uλ is a solution of  (1)λ+ we have Ω|uλ|2dx=Ωλa(x)uλp+1dx+Ωb(x)uλq+1dx. From Lemma  3.5 of , we know Ω[|φ|2-(λpa(x)uλp-1+qb(x)uλq-1)φ2]dx0,φH01. In particular with φ=uλ, we infer Ω|uλ|2dx-λpΩa(x)uλp+1dx-qΩb(x)uλq+1dx0. Combining (3.12) and (3.14), we obtain Iλ(uλ)=λ(12-1p+1)Ωa(x)uλp+1dx+(12-1q+1)Ωb(x)uλq+1dx1-p2(-1p+1+1q+1)Ωa(x)uλp+1dx<0. To complete the proof of (i), it remains to show that uλ<uλ1wheneverλ<λ1.Indeed, if  λ<λ1 then uλ1 is a supersolution of  (1)λ+. Since, for  ɛ>0  small,  ɛu0  is a subsolution of (1)λ+ and  ɛu0<uλ1,  then (1)λ+ possesses a solution v, with (ɛu0)vuλ1. Since uλ is the minimal solution of (1)λ+, we infer that  uλvuλ1.   Moreover -Δ(uλ1-uλ)=λ1a(x)uλ1p+b(x)uλ1q-(λa(x)uλp+b(x)uλq)λa(x)uλ1p+b(x)uλ1q-a(x)uλp-b(x)uλq0. Since  uλ1uλ  (because λ<λ1), then the Hopf Maximum principle yields  uλ<uλ1.

(ii) Let λn be a sequence such that  λnΛ; then from  Iλn(uλn)<0  we deduce that there exists C>0 such that un2C,unp+1p+1C. Then there exists u*H01 such that unu*>0 a.e. in Ω, strongly in Lp+1 and weakly in H01. Such a u* is thus a weak solution of (1)λ+  for λ=Λ.

(iii) This follows from the definition of Λ.

Acknowledgment

This work supported by the Physics and Mathematics Foundation of Changzhou University (ZMF10020065).

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