# On Existence of Solution for a Class of  Semilinear Elliptic Equations with Nonlinearities  That Lies between Different Powers

Authors: Claudianor  O. Alves; Marco  A. S. Souto
Journal: Abstract and Applied Analysis
 (2008)

---

## Abstract

We prove that the semilinear elliptic equation−Δu=f(u), in Ω, u=0, on ∂Ω has a positive solution when the nonlinearity f belongs to a class which
satisfies μtq≤f(t)≤Ctp at infinity and behaves like tq near the origin, where 1<q<(N+2)/(N−2) if N≥3 and 1<q<+∞ if N=1,2. In our approach,
we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity
does not satisfy any hypotheses such those required by the blowup method.
Furthermore, we do not impose any restriction on the growth of p.


---

## Body

## 1. Introduction

Since the 70s, several authors have been studying existence of solutions for the semilinear elliptic Dirichlet problem(P)−Δu=f(u)inΩ,u=0on∂Ω,where Ω is a bounded domain in ℝN, N≥2.
Most part of these results were done under some hypotheses on the nonlinearity f in order to make variational methods work. The
most used hypothesis is the Ambrosetti-Rabinowitz condition [1] which makes the
Euler-Lagrange functional associated to subcritical problem (P) satisfies the Palais-Smale condition. Also,
the blowup method, due to Gidas and Spruck [2], is used to get existence of
solutions and, to work, it needs an asymptotical behaviour like tp for the subcritical nonlinearity f (this is, for the case 1<p<(N+2)/(N−2),
when N≥3 or 1<p<∞, when N=2). Most recently, two existence results
related to the subject have been published: de Figueiredo and Yang [3] considered problem (P), where variational techniques were applied
together with Morse's index, in Azizieh and Clément [4] using also variational
techniques and the blowup method; problem (P) was studied for the m-Laplacean operator, with m>1.
In both papers [3, 4], the existence of
solutions depends on the following condition:(1.1)μtp≤f(t)≤Ctpforlarget.Our results complete those above in the sense that this
paper treats this subject for a class of nonlinearities that lies between two
different powerstq and tp,1<q<p,
with no restrictions on p.
A multiplicity of solutions result was done by Li and Liu [5] with critical growth
restriction on p.
Here, we will suppose that f:ℝ+→ℝ+ is a locally Hölder continuous function with
the following decomposition f(s)=z(s)+λg(s).
The hypotheses on the functions z and g are the following.(H1)
lims→0(z(s)/s)=0.(H2)
There existsθ>2 such that(1.2)0<θZ(s)≤z(s)sfors≥0,where Z(s)=∫0sz(t)dt.(H3)
IfN≥3, there exists q∈(1,(N+2)/(N−2)), and q∈(1,+∞) if N=1,2 such that(1.3)|z(s)|≤|s|q∀s∈ℝ.(H4)
There is a positive real sequence{Mn} satisfying limn→+∞Mn=+∞,g(Mn)>0,
and(1.4)g(Mn)Mnq≥g(s)sq∀s∈[0,Mn],∀n∈ℕ.For convenience, we rewrite problem (P) in the following way:(P)−Δu=z(u)+λg(u)inΩ,u=0on∂Ω.Observe that conditions(H1),(H3), and (H4) imply(1.5)lim sups→0+f(s)s<λ1,where λ1 is the first eigenvalue of (−Δ;Ho1(Ω)).
Moreover, since f(s)≥z(s), for all s>0,
a necessary condition for the existence of solution for problem (P) is that the nonlinearity f crosses the line λ1s.As an example, we may consider(1.6)z(s)=sq,g(s)=sp(1+coss),θ=q+1,Mn=2nπ. The function f(s)=λsp(1+coss)+sq satisfies the above conditions but does not
satisfy Ambrosetti-Rabinowitz conditions. Moreover, f also does not belong to any class of solutions
contained in the references [1–3] or [4].Our main result is the following.Theorem 1.1.
There existsλ*>0 such that problem (P) has a positive solution for all λ∈(0,λ*].In our
proof, we adapt an idea explored by Chabrowski and Yang in [6]. Comparing our approach to
others cited in the literature, it is important to stress that ours does not
use Morse Index method and consequently we do not need derivatives of the
nonlinearity as in [3], and we do not have any restriction 1<p≤N/(N−2) as required in [4], for the case m=2.


## 2. Proof of Theorem1.1

Hereafter, let us denote by∥v∥ the usual norm of Ho1(Ω),
that is,(2.1)∥v∥=(∫Ω|∇v|2dx)1/2,and by cq the minimax level obtained by the
mountain-pass theorem of Ambrosetti and Rabinowitz applied to the
functional(2.2)Jq(v)=12∫Ω|∇v|2dx−∫ΩZ(v+)dx,where v+(x)=max{0,v(x)}.The proposition below establishes an estimate
involving theL∞-norm of a solution related to a subcritical
problem. This estimate is an important point in our approach, and its proof is
an immediate consequence of bootstrap arguments. The constant λ* that appears in Theorem 1.1 depends only on the
constant k below and on the sequence (Mn) given by (H4),
as we can see in this section.Proposition 2.1.
Letv be a solution of the problem(2.3)−Δv=h(v)inΩ,v=0on∂Ω, where h:ℝ→ℝ is a continuous functions verifying |h(s)|≤2|s|q,
for all s>0.
Then, for all C>0,
there exists a constant k=k(Ω,q,C)>0 such that if(2.4)∥v∥2≤C,∥v∥∞≤k.Applying Proposition2.1, for the constant C=2cq(1/2−1/(q+1))−1 there exists a constant k=k(Ω,q) such that (2.4) holds.Let us fixn∈ℕ such that Mn>k=k(Ω,q), and let λ*>0 satisfy(2.5)λ*g(Mn)Mnq≤1,λ*(1θ+1q+1)g(Mn)Mn|Ω|≤cq,where |Ω| denotes the Lebesgue's measure of the set Ω.For thatn∈ℕ previously fixed , let us consider the
function hn:ℝ→ℝ given by (2.6)hn(s)={0,ifs≤0,g(s),if0≤s≤Mn,g(Mn)Mnqsq,ifs≥Mn.From condition (H4),(2.7)|hn(s)|≤g(Mn)Mnq|s|q,∀s,and then fn(s)=z(s)+λhn(s) satisfies the inequality fn(s)≤2|s|q,
since λ≤λ*.It is easy to check that the truncated functionfn satisfies Ambrosetti-Rabinowitz condition in
the interval [Mn,+∞),
that is,(2.8)θFn(s)−sfn(s)≤0,∀s≥Mn,where Fn(s)=∫osfn(t)dt.
In fact, denoting Hn(s)=∫oshn(t)dt we have(2.9)Hn(s)=∫oMng(t)dt+∫Mnsg(Mn)Mnqtqdt,for s>Mn,
which implies(2.10)Hn(s)=∫oMng(t)dt+1q+1g(Mn)Mnq[sq+1−Mnq+1].Integrating inequality (H4),
from 0 to Mn,
we have(2.11)∫oMng(t)dt≤1q+1g(Mn)MnqMnq+1.Consequently, from (2.10) and
(2.11), the inequality(2.12)Hn(s)≤1q+1g(Mn)Mnqsq+1holds. Thus,(2.13)(q+1)Hn(s)−shn(s)≤0∀s>Mn,which, together
with (H2), implies(2.14)θFn(s)−sfn(s)≤0∀s≥Mn,as we wanted to check (observe
that (H2) and (H3) imply that θ≤q+1).Now, consider the following truncated problem:(Pn)−Δu=z(u)+λhn(u)inΩ,u>0inΩ,u=0on∂Ω.Let us recall thatλ≤λ* and Mn>k.
It is well known that problem (Pn) has a positive solutions u∈Wo1,p(Ω).
This occurs since the energy functional related to (Pn) given by(2.15)Jn(v)=12∫Ω|∇v|2dx−∫ΩZ(v+)dx−λ∫ΩHn(v)dxor(2.16)Jn(v)=Jq(v)−λ∫ΩHn(v)dxsatisfies the hypotheses of the
mountain pass theorem [1]. Moreover, from the inequality Jn(v)≤Jq(v),
for all v∈Ho1(Ω),
the minimax level bn of functional Jn satisfies bn≤cq.Thus, from
equalitybn=Jn(u)=Jn(u)−(1/θ)Jn′(u)u,  we have(2.17)(12−1θ)∥u∥2+∫Ω(1θufn(u)−Fn(u))dx=bnand from (2.8) we have(2.18)(12−1θ)∥u∥2≤λ∫{u≤Mn}|1θuhn(u)−Hn(u)|dx+bn.From inequality (2.7), we have |(1/θ)uhn(u)−Hn(u)|≤(1/θ+1/(q+1))g(Mn)Mn in {x∈Ω:u(x)≤Mn},
hence(2.19)(12−1θ)∥u∥2≤λ(1θ+1q+1)g(Mn)Mn|Ω|+cq.From the choice of λ*,
we get (2.20)∥u∥2≤2cq(12−1q+1)−1.Using the constant C obtained in Proposition 2.1 and from the last
inequality, we have that u(x)≤k, for all x∈Ω.
But Mn>k implies that u(x)≤Mn and thus u is a solution for (P).
The proof is done.Remark 2.2.
Hypothesis(H4) can be replaced by
(H4)'
There is a positive realM>k=k(Ω,q) satisfying(2.21)g(M)Mq≥g(s)sq∀s∈[0,M].


## 3. Final Comments

The methods applied in this paper with few
modifications can be used to establish existence of positive solutions for the
problem(3.1)−Δpu=z(u)+λg(u)inΩ,u=0on∂Ω,with N>p>1 and z,g verifying(H5)
lims→0(z(s)/|s|p−2s)=0.(H6)
There existsθ>p such that(3.2)0<θZ(s)≤z(s)s,fors≥0,where Z(s)=∫0sz(t)dt.(H7)
There existsq∈(p−1,(N(p−1)+p)/(N−p)) such that(3.3)|z(s)|≤|s|q∀s∈ℝ.(H8)
There is a positive real sequence{Mn} satisfying limn→+∞Mn=+∞,g(Mn)>0,
and(3.4)g(Mn)Mnq≥g(s)sq∀s∈[0,Mn],∀n∈ℕ.The ideas developed in this paper may be used when we
deal with thep-Laplacian operator. The main difference is concerned with the
proof of Proposition 2.1, where we should replace bootstrap arguments by Moser's
iteration methods, and then repeat the same approach explored by Chabrowski
and Yang [6].



---

*Source: 578417-2008-06-11.xml*