# Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

Authors: Chunmei Yuan; Shujuan Guo; Kaiyu Tong
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

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 ℝN(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.


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## Body

## 1. Introduction

In this paper, we consider the elliptic problems with sublinear and superlinear nonlinearities(1)λ-Δu=λa(x)up+b(x)uqinΩ,u>0inΩ,u=0on∂Ω,
where λ>0  is a real parameter,  0<p<1<q.  Ω is a bounded domain in ℝN(N≥3), and a(x) and b(x) are some given functions which satisfies the following assumptions:(H1)
a(x),b(x)∈L∞(Ω),  a(x)≥c0,  b(x)≤-c1,  where c0,  c1 are positive constants,or(H2)
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 [1–9] and their references). But most of such studies have been concerned with equations of the type involving sublinear nonlinearity (see [3–6, 8, 9]), with only few references dealing with the elliptic problems with sublinear and superlinear nonlinearities. In [1], Ambrosetti et al. deal with the analogue of (1)λ with a(x)=b(x)≡1. It is known from [2] that there exist λ*∈(0,∞), such that problem (1)λ has a solution if λ≤λ*  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 u∈H01(Ω)  we define Iλ:H01(Ω)→ℝ  by (1.1)Iλ(u)=12∫Ω|∇u|2dx-λp+1∫Ωa(x)|u|p+1dx-1q+1∫Ωb(x)|u|q+1dx.Letλ1 be the first eigenvalue of (1.2)-Δu=λu,x∈Ω,u=0,x∈∂Ω.φ1  denotes the corresponding eigenfunction satisfying  0≤φ1(x)≤1.  Lp(Ω),  (1≤p≤∞), denotes Lebesgue spaces, and the norm in Lp is denoted by ∥·∥p.


## 2. The Existence of Positive Solution of(1)λ-

It is well known that(2.1)∇φ1(x)≠0,∀x∈∂Ω.
Define  a=min∂Ω|∇φ1|2;   from (2.1) we know  a>0,  so we can split the domain Ω into two parts: Ωɛ and Ω∖Ωɛ, where Ωɛ={x∈Ω:|∇φ1|2≥a/2}⋂{x∈Ω:φ1(x)≤ɛ,ɛis small enough}. Let  b=infΩ∖Ωɛφ1(x);  we obtain that  b≥ɛ  by the positivity of φ1  in  Ω,  and  Ω∖Ωɛ  is nonempty when  ɛ  is small enough.Theorem 2.1.
Leta(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  λ≥λ̃.Proof.
Lete(x)  denote the positive solution of the following equation:
(2.2)-Δe=1,x∈Ω,e=0,x∈∂Ω.
Here and hereafter we use the following notations: A=∥a∥∞,   B=∥b∥∞,   E=∥e∥∞. Since 0<p<1, for all λ∈ℝ+,  there exists  T=T(λ)>0  satisfying
(2.3)T≥λATpEp.
Observing that  b(x)≤-c1<0,  as a consequence, the function  Te  verifies
(2.4)T=-Δ(Te)≥λA(Te)p≥λa(x)(Te)p+b(x)(Te)q,
and hence it is a supersolution of (1)λ+.  Let  v(x)=φ1l,  x∈Ω,  l>1. For x∈Ω, we have x∈Ωɛ or x∈Ω∖Ωɛ. We will discuss it from two conditions.
(I)  For allx∈Ωɛ, observing that l>1 and when ɛ is small enough, we have
(2.5)al(l-1)s-22-Bsl(q-1)>λ1l,∀s∈(0,ɛ).
Since x∈Ωɛ, then it follows that φ1(x)≤ɛ,  |∇φ1|2≥a/2.  From (2.5) we infer
(2.6)λ1l≤l(l-1)φ1-2|∇φ1|2-Bφ1(x)l(q-1),∀x∈Ωɛ.
Multiplying (2.6) with φ1l,  we get
(2.7)l(1-l)φ1l-2|∇φ1|2+λ1lφ1l≤-Bφ1lq.
It follows that
(2.8)-Δ(φ1l)≤λa(x)(φ1l)p-b(x)(φ1l)q.
(II)  For allx∈Ω∖Ωɛ, there exists  λ̃>0,  such that  for all  λ≥λ̃, and we have
(2.9)λc0spl-Bsql≥λ1lsl,∀s∈R,b≤s≤1.
Since x∈Ω∖Ωɛ,  then we have φ1(x)≥b  (and  φ1(x)≤1).  From (2.9), it follows that
(2.10)-Δ(φ1l)≤λ1lφ1l≤λc0φ1lp-Bsql≤λa(x)(φ1l)p+b(x)(φ1l)q.
From (2.8) and (2.10), we derive that there exists  λ̃>0  such that   for all  x∈Ω,   for all  λ≥λ̃,
(2.11)-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={u∈H01(Ω):v(x)≤u(x)≤w(x),for allx∈Ω},   then K is closed and convex (and weakly closed). Let f(s)=λa(x)sp+b(x)sq,  for all  s∈ℝ,s>0.  We consider the function
(2.12)Iλ(u)=12∫Ω|∇u|2dx-∫Ω∫0uf(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 [10]). In the following, we will prove that u* is a solution of problem (1)λ-.
Forϕ∈K, define h:[0,1]→ℝ, such that
(2.13)h(t)=I(tϕ+(1-t)u*).
Clearly, h(t) achieves its minimum at  t=0, and
(2.14)h′(t)|t=0=∫Ω[∇u*∇(ϕ-u*)]dx-∫Ωf(u*)(ϕ-u*)dx≥0.
For all  φ∈H01(Ω),  η>0,  define
(2.15)Ψ(x)={v,whenu*+ηφ<v,u*+ηφ,whenv≤u*+ηφ≤w,w,whenu*+ηφ>w.
Obviously,  Ψ∈K,  and inserting (2.15) into (2.14),  we find
(2.16)0≤∫v≤u*+ηφ≤w[∇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=η∫v≤u*+ηφ≤w[∇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
(2.17)∫u*+ηφ>w[∇w⋅∇(w-u*)-f(w)(w-u*)]dx≤η∫u*+ηφ>w[∇w⋅∇φ-f(w)φ]dx,∫u*+ηφ<v[∇v⋅∇(v-u*)-f(v)(v-u*)]dx≤η∫u*+ηφ<v[∇v⋅∇φ-f(v)φ]dx.
Observe that meas[u*+ηφ>w]→0, meas[u*+ηφ<v]→0, as η→0,
(2.18)∫u*+ηφ>w[∇w⋅∇φ-f(w)φ]dx⟶0,∫u*+ηφ<v[∇v⋅∇φ-f(v)φ]dx⟶0.
Since  u*∈K,b(x)<0, it follows that
(2.19)∫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*)dx≤∫u*+ηφ>wλa(x)(wp-u*p)(w-u*)dx.
Similar to (2.19), we have
(2.20)∫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*)dx≤∫u*+ηφ<vλa(x)(vp-u*p)(v-u*)dx.
Similar to (2.18), as  η→0, it follows that
(2.21)∫u*+ηφ>wλa(x)(wp-u*p)(w-u*)dx⟶0,∫u*+ηφ<vλa(x)(vp-u*p)(v-u*)dx⟶0.
As η→0, we also have
(2.22)∫v≤u*+ηφ≤w[∇u*⋅∇φ-f(u*)φ]dx⟶∫Ω[∇u*⋅∇φ-f(u*)φ]dx.
Inserting (2.17), (2.19), and (2.20) into (2.16), we find
(2.23)0≤η{∫v≤u*+ηφ≤w[∇u*⋅∇φ-f(u*)φ]dx+∫u*+ηφ>w[∇w⋅∇φ-f(w)φ]dx+∫u*+ηφ<v[∇v⋅∇φ-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  η→0, using (2.18), (2.21), and (2.22), we derive
(2.24)∫Ω[∇u*⋅∇φ-f(u*)φ]dx≥0.
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(1)λ+

Theorem 3.1.
Leta(x),  b(x) satisfy assumption (H2),  and  0<p<1<q<+∞. Then there exists Λ∈ℝ,  Λ>0,  such that(i)
for allλ∈(0,Λ) problem (1)λ+ has a minimal solution uλ such that Iλ(uλ)<0. Moreover uλ is increasing with respect to λ;(ii)
forλ=Λ problem (1)λ+ has at least one weak solution u∈H∩Lp+1;(iii)
for allλ>Λ  problem  (1)λ+ has no solution.To prove Theorem3.1, let us define
(3.1)Λ=sup⁡{λ>0:(1)λ+has a solution}.First of all we prove a useful lemma.Lemma 3.2.
One has0<Λ<+∞.Proof.
Lete(x) denote the solution of the following equation:
(3.2)-Δe=1,x∈Ω,e=0,x∈∂Ω.
Since  0<p<1<q,  we can find  λ0>0  such that for all  0<λ≤λ0  there exists  T=T(λ)>0  satisfying
(3.3)T≥λATpEp+BTqEq.
As a consequence,  the function Te verifies
(3.4)T=-Δ(Te)≥λA(Te)p+B(Te)q≥λa(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:
(3.5)-Δu=λa(x)u0p,x∈Ω,u0=0,x∈∂Ω.
(From [3] we know that u0 exists.)  Then ɛu0 is a subsolution of (1)λ+, provided
(3.6)-Δ(ɛu0)=λɛa(x)u0p≤λa(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
(3.7)ɛu0<Te.
It follows that  (1)λ+ has a solution u,  ɛu0≤u≤Te whenever λ≤λ0, and thus Λ≥λ0.
Next, letλ* be such that
(3.8)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
(3.9)λ1∫Ωuφ1dx=λ∫Ωa(x)upφ1dx+∫Ωb(x)uqφ1dx≥c0[∫Ω(λupφ1+uqφ1)dx].
This and (3.5)  immediately imply that  λ<λ*  and show that  Λ≤λ*,  hence  0<Λ<+∞.We are now ready to give the proof of Theorem3.1.Proof.
(i) From the proof of lemma, it follows that, for allλ∈(0,Λ),   problem  (1)λ+  has a solution uλ.  Let u0 satisfy (3.5); the iteration
(3.10)-Δun+1=λa(x)unp+b(x)unq
satisfies  un↑uλ  by making use of Lemma  3.3 of [1] 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  u≥u0  and u is a supersolution of  (1)λ+. Thus  un≤u,  for all  n, by induction, and uλ≤u. Next, we will prove that Iλ(uλ)<0. Indeed,
(3.11)Iλ(u)=12∫Ω|∇u|2dx-λp+1∫Ωa(x)|u|p+1dx-1q+1∫Ωb(x)|u|q+1dx.
Since uλ is a solution of  (1)λ+ we have
(3.12)∫Ω|∇uλ|2dx=∫Ωλa(x)uλp+1dx+∫Ωb(x)uλq+1dx.
From Lemma  3.5 of [1], we know
(3.13)∫Ω[|∇φ|2-(λpa(x)uλp-1+qb(x)uλq-1)φ2]dx≥0,∀φ∈H01.
In particular with φ=uλ, we infer
(3.14)∫Ω|∇uλ|2dx-λp∫Ωa(x)uλp+1dx-q∫Ωb(x)uλq+1dx≥0.
Combining (3.12) and (3.14), we obtain
(3.15)Iλ(uλ)=λ(12-1p+1)∫Ωa(x)uλp+1dx+(12-1q+1)∫Ωb(x)uλq+1dx≤1-p2(-1p+1+1q+1)∫Ωa(x)uλp+1dx<0.
To complete the proof of (i), it remains to show that
(3.16)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
(3.17)(ɛu0≤)v≤uλ1.
Since uλ is the minimal solution of (1)λ+, we infer that  uλ≤v≤uλ1.   Moreover
(3.18)-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)uλq≥0.
Since  uλ1≠uλ  (because λ<λ1), then the Hopf Maximum principle yields  uλ<uλ1.
(ii) Letλn be a sequence such that  λn↑Λ; then from  Iλn(uλn)<0  we deduce that there exists C>0 such that
(3.19)‖∇un‖2≤C,‖un‖p+1p+1≤C.
Then there exists u*∈H01 such that un→u*>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 ofΛ.



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