A nonsmooth variational approach to semipositone quasilinear problems in RN

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arXiv:2210.14887v1 [math.AP] 26 Oct 2022

A nonsmooth variational approach to semipositone

quasilinear problems in RN

Jeﬀerson Abrantes Santosa,1, Claudianor O. Alvesa,2, Eugenio Massab,3

aUniversidade Federal de Campina Grande, Unidade Acadˆemica de Matem´atica,

CEP: 58429-900, Campina Grande - PB, Brazil.

bDepartamento de Matem´atica, Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao,

Universidade de S˜ao Paulo, Campus de S˜ao Carlos, 13560-970, S˜ao Carlos SP, Brazil.

Abstract

This paper concerns the existence of a solution for the following class of

semipositone quasilinear problems

−∆pu=h(x)(f(u)−a) in RN,

u > 0 in RN,

where 1 < p < N,a > 0, f: [0,+∞)→[0,+∞) is a function with subcritical

growth and f(0) = 0, while h:RN→(0,+∞) is a continuous function that

satisﬁes some technical conditions. We prove via nonsmooth critical points

theory and comparison principle, that a solution exists for asmall enough. We

also provide a version of Hopf’s Lemma and a Liouville-type result for the p-

Laplacian in the whole RN.

Mathematical Subject Classiﬁcation MSC2010: 35J20, 35J62 (49J52).

Key words and phrases: semipositone problems; quasilinear elliptic

equations; nonsmooth nariational methods; Lipschitz functional; positive

solutions.

1. Introduction

In this paper we study the existence of positive weak solutions for the p-

Laplacian semipositone problem in the whole space

−∆pu=h(x)(f(u)−a) in RN,

u > 0 in RN,(Pa)

Email addresses: jefferson@mat.ufcg.edu.br (Jeﬀerson Abrantes Santos),

coalves@mat.ufcg.edu.br (Claudianor O. Alves), eug.massa@gmail.com (Eugenio Massa)

1J. Abrantes Santos was partially supported by CNPq/Brazil 303479/2019-1

2C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7

3E. Massa was partially supported by grant #303447/2017-6, CNPq/Brazil.

Preprint submitted to ...

Abrantes Santos, Alves, Massa - arXiv 2

where 1 < p < N ,a > 0, f: [0,+∞)→[0,+∞) is a continuous function with

subcritical growth and f(0) = 0. Moreover, the function h:RN→(0,+∞) is a

continuous function satisfying

(P1)h∈L1(RN)∩L∞(RN),

(P2)h(x)< B|x|−ϑfor x6= 0, with ϑ > N and B > 0.

An example of a function hthat satisﬁes the hypotheses (P1)−(P2) is given

below:

h(x) = B

1 + |x|ϑ,∀x∈RN.

In the whole of this paper, we say that a function u∈D1,p(RN) is a weak

solution for (Pa) if uis a continuous positive function that veriﬁes

ZRN

|∇u|p−2∇u∇v dx =ZRN

h(x)(f(u)−a)v dx, ∀v∈D1,p(RN).

1.1. State of art.

The problem (Pa) for a= 0 is very simple to be solved, either employing the

well known mountain pass theorem due to Ambrosetti and Rabinowitz [AR73],

or via minimization. However, for the case where (Pa) is semipositone, that

is, when a > 0, the existence of a positive solution is not so simple, because

the standard arguments via the mountain pass theorem combined with the

maximum principle do not directly give a positive solution for the problem,

and in this case, a very careful analysis must be done.

The literature associated with semipositone problems in bounded domains

is very rich since the appearance of the paper by Castro and Shivaji [CS88]

who were the ﬁrst to consider this class of problems. We have observed that

there are diﬀerent methods to prove the existence and nonexistence of solutions,

such as subsupersolutions, degree theory arguments, ﬁxed point theory and

bifurcation; see for example [ACS93], [AAB94], [ANZ92], [AHS96] and their

references. In addition to these methods, also variational methods were used in

a few papers as can be seen in [AdHS19], [CCSU07], [CdL16], [CDS15], [CRT17],

[CTY06], [DC12], [FMS21], [GPPS03] and [Jea11]. We would like to point out

that in [CdL16], Castro, de Figueiredo and Lopera studied the existence of

solutions for the following class of semipositone quasilinear problems







−∆pu=λf(u) in Ω,

u(x)>0 in Ω,

u= 0 on ∂Ω,

(1.1)

where Ω ⊂RN,N > p > 2, is a smooth bounded domain, λ > 0 and f:R→R

is a diﬀerentiable function with f(0) <0. In that paper, the authors assumed

that there exist q∈(p−1,Np

N−p−1), A, B > 0 such that

A(tq−1) ≤f(t)≤B(tq−1),for t > 0

f(t) = 0,for t≤ −1.

Abrantes Santos, Alves, Massa - arXiv 3

The existence of a solution was proved by combining the mountain pass theorem

with the regularity theory. Motivated by the results proved in [CdL16], Alves,

de Holanda and dos Santos [AdHS19] studied the existence of solutions for a

large class of semipositone quasilinear problems of the type







−∆Φu=f(u)−ain Ω,

u(x)>0 in Ω,

u= 0 on ∂Ω,

(1.2)

where ∆Φstands for the Φ-Laplacian operator. The proof of the main result

is also done via variational methods, however in their approach the regularity

results found in Lieberman [Lie88,Lie91] play an important role. By using the

mountain pass theorem, the authors found a solution uafor all a > 0. After

that, by taking the limit when agoes to 0 and using the regularity results

in [Lie88,Lie91], they proved that uais positive for asmall enough.

Related to semipositone problems in unbounded domains, we only found the

paper due to Alves, de Holanda, and dos Santos [AdHdS20] that studied the

existence of solutions for the following class of problems

−∆u=h(x)(f(u)−a) in RN,

u > 0 in RN,(1.3)

where a > 0, f: [0,+∞)→[0,+∞) and h:RN→(0,+∞) are continuous

functions with fhaving a subcritical growth and hsatisfying some technical

conditions. The main tools used were variational methods combined with the

Riesz potential theory.

1.2. Statement of the main results.

Motivated by the results found in [CdL16], [AdHS19] and [AdHdS20], we

intend to study the existence of solutions for (Pa) with two diﬀerent types of

nonlinearities. In order to state our ﬁrst main result, we assume the following

conditions on f:

(f0)

lim

t→0+

F(t)

tp= 0 ;

(fsc) there exists q∈(1, p∗) such that lim sup

t→+∞

f(t)

tq−1<∞,

where p∗=pN

N−pis the critical Sobolev exponent;

(f∞)q > p in (fsc) and there exist θ > p and t0>0 such that

0< θF (t)≤f(t)t, ∀t > t0,

where F(t) = Rt

0f(τ)dτ.

Our ﬁrst main result has the following statement

Abrantes Santos, Alves, Massa - arXiv 4

Theorem 1.1. Assume the conditions (P1)−(P2),(fsc),(f0)and (f∞). Then

there exists a∗>0such that, if a∈[0, a∗), problem (Pa)has a positive weak

solution ua∈C(RN)∩D1,p(RN).

As mentioned above, a version of Theorem 1.1 was proved in [AdHdS20] in

the semilinear case p= 2. Their proof exploited variational methods for C1

functionals and Riesz potential theory in order to prove the positivity of the

solutions of a smooth approximated problem, which then resulted to be actual

solutions of problem (Pa). In our setting, since we are working with the p-

Laplacian, that is a nonlinear operator, we do not have a Riesz potential theory

analogue that works well for this class of operator. Hence, a diﬀerent approach

was developed in order to treat the problem (Pa) for p6= 2. Here, we make

a diﬀerent approximation for problem (Pa), which results in working with a

nonsmooth approximating functional.

As a result, the Theorem 1.1 is also new when p= 2, since the set of

hypotheses we assume here is diﬀerent. In fact, avoiding the use of the Riesz

theory, we do not need to assume that fis Lipschitz (which would not even be

possible in the case of condition ( e

f0) below), and a diﬀerent condition on the

decaying of the function his required.

The use of the nonsmooth approach turns out to simplify several

technicalities usually involved in the treatment of semipositone problems.

Actually, working with the C1functional naturally associated to (Pa), one

obtains critical points uathat may be negative somewhere. When working in

bounded sets, the positivity of uais obtained, in the limit as a→0, by proving

convergence in C1sense to the positive solution u0of the case a= 0, which is

enough since u0has also normal derivative at the boundary which is bounded

away from zero in view of the Hopf’s Lemma. This approach can be seen for

instance in [Jea11,CdL16,AdHS19,FMS21]. In Rn, a diﬀerent argument must

be used: actually one can obtain convergence on compact sets, but the limiting

solution u0goes to zero at inﬁnity as |x|(p−N)/(p−1) (see Remark 2), which

means that one needs to be able to do some ﬁner estimates on the convergence.

In [AdHdS20], with p= 2, the use of the Riesz potential, allowed to prove that

|x|N−2|ua−u0| → 0 uniformly, which then led to the positivity of uain the

limit.

In the lack of this tool, we had to ﬁnd a diﬀerent way to prove the positivity

of ua. The great advantage of our approach via nonsmooth analysis, is that

our critical points uawill always be nonnegative functions (see Lemma 2.3).

In spite of not necessary being week solutions of the equation in problem (Pa),

they turn out to be supersolutions and also subsolutions of the limit equation

with a= 0. These properties will allow us to use comparison principle in order

to prove the strict positivity of uawith the help of a suitable barrier function

(see the Lemmas 4.1 and 4.2). From the positivity it will immediately follow

that uais indeed a weak solutions of (Pa).

The reader is invited to see that by (f∞), there exist A1, B1>0 such that

F(t)≥A1|t|θ−B1, f or t ≥0.(1.4)

Abrantes Santos, Alves, Massa - arXiv 5

This inequality yields that the functional we will be working with is not bounded

from below. On the other hand, the condition (f0) will produce a “range of

mountains” geometry around the origin for the functional, which completes the

mountain pass structure. Finally, conditions (fsc) and (f∞) impose a subcritical

growth to f, which are used to obtain the required compactness condition.

Next, we are going to state our second result. For this result, we still assume

(fsc) together with the following conditions:

f0)

lim

t→0+

F(t)

tp=∞;

f∞)q < p in (fsc).

Our second main result is the following:

Theorem 1.2. Assume the conditions (P1)−(P2),(fsc),(e

f0)and (e

f∞). Then

there exists a∗>0such that, if a∈[0, a∗), problem (Pa)has a positive weak

solution ua∈C(RN)∩D1,p(RN).

In the proof of Theorem 1.2, the condition ( e

f0) will produce a situation

where the origin is not a local minimum for the energy functional, while ( e

f∞)

will make the functional coercive, in view of (fsc). It will be then possible to

obtain solutions via minimization. As in the proof of Theorem 1.1, we will work

with a nonsmooth approximating functional that will give us an approximate

solution. After some computation, we prove that this approximate solution is

in fact a solution for the original problem when ais small enough.

Remark 1. Observe that if f, h satisfy the set of conditions of Theorem 1.1

or those of Theorem 1.2 and uis a solution of Problem (Pa), then the rescaled

function v=a

−1

q−1uis a solution of the problem:

(−∆pv=a(q−p)

q−1h(x)( e

fa(v)−1) in RN,

v > 0in RN,(1.5)

which then takes the form of Problem (1.1), with λ:= a(q−p)

q−1and a new

nonlinearity e

fa(t) = a−1f(a1

q−1t), which satisﬁes the same hipotheses of f. In

particular, if f(t) = tq−1then e

fa≡f.

In the conditions of Theorem 1.1, where q > p, we obtain a solution of

Problem (1.5)for suitably small values of λ, while in the conditions of Theorem

1.2, where q < p, solutions are obtained for suitably large values of λ.

It is worth noting that, as a→0, the solutions of Problem (Pa)that we

obtain are bounded and converge, up to subsequences, to a solution of Problem

(Pa)with a= 0 (see Lemma 3.2). As a consequence, the corresponding solutions

of Problem (1.5)satisfy v(x)→ ∞ for every x∈RN.

Semipositone problems formulated as in (1.5)were considered recently in

[CRQT17,PSS20].

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arXiv:2210.14887v1[math.AP]26Oct2022AnonsmoothvariationalapproachtosemipositonequasilinearproblemsinRNJeﬀersonAbrantesSantosa,1,ClaudianorO.Alvesa,2,EugenioMassab,3aUniversidadeFederaldeCampinaGrande,UnidadeAcadˆemicadeMatem´atica,CEP:58429-900,CampinaGrande-PB,Brazil.bDepartamentodeMatem´atica,Inst...

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A nonsmooth variational approach to semipositone quasilinear problems in RN

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