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.

1. Introduction

In this paper, we consider the elliptic problems with sublinear and superlinear nonlinearities -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:

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 [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 Iλ(u)=12∫Ω|∇u|2dx-λp+1∫Ωa(x)|u|p+1dx-1q+1∫Ωb(x)|u|q+1dx.

Let λ1 be the first eigenvalue of -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 <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,∀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.

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 λ≥λ̃.

Proof.

Let e(x) denote the positive solution of the following equation:
-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
T≥λATpEp.
Observing that b(x)≤-c1<0, as a consequence, the function Te verifies
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 all x∈Ωɛ, 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 x∈Ωɛ, then it follows that φ1(x)≤ɛ, |∇φ1|2≥a/2. From (2.5) we infer
λ1l≤l(l-1)φ1-2|∇φ1|2-Bφ1(x)l(q-1),∀x∈Ωɛ.
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 x∈Ω∖Ωɛ, there exists λ̃>0, such that for all λ≥λ̃, and we have
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
-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,
-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
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-.

For ϕ∈K, define h:[0,1]→ℝ, 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*)dx≥0.
For all φ∈H01(Ω), η>0, define
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
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
∫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,
∫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
∫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
∫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
∫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
∫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
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
∫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 <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∈ℝ, 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 u∈H∩Lp+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,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
T≥λATpEp+BTqEq.
As a consequence, the function Te verifies
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:
-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
-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
ɛu0<Te.
It follows that (1)λ+ has a solution u, ɛu0≤u≤Te whenever λ≤λ0, and thus Λ≥λ0.

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
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 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 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,
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
∫Ω|∇uλ|2dx=∫Ωλa(x)uλp+1dx+∫Ωb(x)uλq+1dx.
From Lemma 3.5 of [1], we know
∫Ω[|∇φ|2-(λpa(x)uλp-1+qb(x)uλq-1)φ2]dx≥0,∀φ∈H01.
In particular with φ=uλ, we infer
∫Ω|∇uλ|2dx-λp∫Ωa(x)uλp+1dx-q∫Ωb(x)uλq+1dx≥0.
Combining (3.12) and (3.14), we obtain
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
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≤)v≤uλ1.
Since uλ is the minimal solution of (1)λ+, we infer that uλ≤v≤uλ1. 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)uλq≥0.
Since uλ1≠uλ (because λ<λ1), then the Hopf Maximum principle yields uλ<uλ1.

(ii) Let λn be a sequence such that λn↑Λ; then from Iλn(uλn)<0 we deduce that there exists C>0 such that
‖∇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 of Λ.

Acknowledgment

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

AmbrosettiA.BrezisH.CeramiG.Combined effects of concave and convex nonlinearities in some elliptic problemsTanZ.YaoZ.Existence of multiple solutions for semilinear elliptic equationBrezisH.KaminS.Sublinear elliptic equations in ℝNBrezisH.OswaldL.Remarks on sublinear elliptic equationsTairaK.UmezuK.Positive solutions of sublinear elliptic boundary value problemsGherguM.RădulescuV.Nonradial blow-up solutions of sublinear elliptic equations with gradient termGherguM.RădulescuV. D.KajikiyaR.Comparison theorem and uniqueness of positive solutions for sublinear elliptic equationsKajikiyaR.A priori estimates of positive solutions for sublinear elliptic equationsStruweM.