Nikolaos S. Papageorgiou

Existence of positive solutions for nonlinear Dirichlet problems with gradient dependence

and arbitrary growth

Nikolaos S. Papageorgiou

, Calogero Vetro

and Francesca Vetro

1Department of Mathematics, National Technical University, Zografou campus, 15780, Athens, Greece

2Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy

3Department of Energy, Information Engineering and Mathematical Models (DEIM), University of Palermo, Viale delle Scienze ed. 8, 90128, Palermo, Italy

Received 6 October 2017, appeared 23 April 2018 Communicated by Dimitri Mugnai

Abstract. We consider a nonlinear elliptic problem driven by the Dirichlet p-Lapla- cian and a reaction term which depends also on the gradient (convection). No growth condition is imposed on the reaction term f(z,·,y). Using topological tools and the asymptotic analysis of a family of perturbed problems, we prove the existence of a positive smooth solution.

Keywords: convection reaction term, nonlinear regularity, nonlinear maximum princi- ple, pseudomonotone map, Picone identity, Hartman condition.

2010 Mathematics Subject Classification: 35J60, 35J92.

1 Introduction

LetΩ⊆_R^N be a bounded domain with aC²-boundary∂Ω. We study the following nonlinear Dirichlet problem with gradient dependence

−_∆pu(z) = f(z,u(z),∇u(z)) inΩ, u

∂Ω =0, u>0. (1.1) In this problem∆p denotes thep-Laplace differential operator defined by

∆pu=div(|∇u|^p−2∇u) for allu∈W₀^1,p(_Ω), 1< p<_∞.

The dependence of the reaction term f(z,x,y)on the gradient ∇u of the unknown func- tion u, precludes the use of variational methods in the study of problem (1.1). Instead our approach is topological and uses the theory of nonlinear operators of monotone type and the asymptotic analysis of a perturbation of the original problem. We prove the existence of a

BCorresponding author. Email: calogero.vetro@unipa.it

positive smooth solution, without imposing any growth condition on f(z,·,y). Instead, we employ a Hartman-type condition on f(z,·,y)which leads to an a priori bound for the posi- tive solutions. This absence of any growth condition on f(z,·,y)distinguishes our work from previous ones on elliptic equations with convection. We refer to the papers of de Figueiredo–

Girardi–Matzeu [3], Girardi–Matzeu [6] (semilinear problems driven by the Laplacian) and Faraci–Motreanu–Puglisi [2], Ruiz [9] (nonlinear problems driven by the p-Laplacian). We mention also the recent work of Gasi ´nski–Papageorgiou [5] on Neumann problems driven by a differential operator of the form div(a(u)∇u). In all the aforementioned works f(z,·,y) exhibits the usual subcritical polynomial growth.

2 Mathematical background – hypotheses

LetX be a reflexive Banach space andX^∗ its topological dual. Byh·,·iwe denote the duality brackets for the pair(X^∗,X). A map V : X → X^∗ is said to be “pseudomonotone”, if it has the following property:

“u_n−→^w uin X , V(u_n)−→^w u^∗ in X^∗ and lim sup

n→+_∞

hV(u_n),u_n−ui ≤0 imply that u^∗ =V(u) and hV(u_n),u_ni → hV(u),ui”.

A maximal monotone, everywhere defined operator is pseudomonotone. Moreover, ifV= A+K with A maximal monotone and everywhere defined and K is completely continuous (that is, un w

−→ u in X ⇒ K(un) → K(u) in X^∗), then V is pseudomonotone (see Gasi ´nski–

Papageorgiou [4], p. 336).

Pseudomonotone operators exhibit remarkable surjectivity properties. More precisely we have (see Gasi ´nski–Papageorgiou [4], p. 336).

Proposition 2.1. If V :X→X^∗ is pseudomonotone and strongly coercive, that is,

kuk→+lim_∞

hV(u),ui

kuk = +_∞, then V is surjective (that is, R(V) =X^∗).

The following two spaces will be used in the analysis of problem (1.1):

•the Sobolev spaceW₀^1,p(_Ω);

•the Banach spaceC₀¹(_Ω) ={u∈C¹(_Ω)_: u

∂Ω=₀}_.

By k · k we denote the norm of W₀^1,p(_Ω). On account of the Poincaré inequality, we can take

kuk=k∇uk_p for all u∈W₀^1,p(_Ω).

The Banach spaceC₀¹(_Ω)is an ordered Banach space with positive (order) cone C+ =ⁿu∈C₀¹(_Ω) _: u(z)≥_{0 for all}z∈_Ω^o_.

This cone has a nonempty interior given by intC+=

u∈C+ : u(z)>0 for allz∈ _Ω, ^∂u

∂n ∂Ω<0

,

where ^∂u_∂n = (∇u,n)_RN withn(·)being the outward unit normal on∂Ω.

Given x ∈ _{R, we set} x^± = max{±x, 0}. Then for u ∈ W₀^1,p(_Ω), we define u^±(·) = u(·)^±. We know that

u^±∈W₀^1,p(_Ω)_{, u}=_u⁺−u⁻, |u|=_u⁺+_u⁻_. Consider the following nonlinear eigenvalue problem:

−_∆pu(z) =bλ|u(z)|^p−2u(z) _inΩ, u

∂Ω=_{0. (2.1)}

We know that (2.1) has a smallest eigenvaluebλ₁ which has the following properties:

• bλ₁>_{0 and}bλ₁is isolated (that is, ifbσ(p)denotes the spectrum of (2.1), then we can find ε >0 such that(bλ₁,bλ₁+ε)∩_bσ(p) =_∅);

• bλ₁is simple (that is, ifu,b ueare eigenfunctions corresponding tobλ₁, thenub=ξuefor some ξ ∈ _R\ {0});

•

bλ₁=inf

"

k∇uk^p_p

kuk^p_p ^:u∈W₀^1,p(_Ω),u6=0

#

. (2.2)

The infimum in (2.2) is realized on the one-dimensional eigenspace corresponding tobλ₁. The above properties imply that the elements of this eigenspace, have fixed sign. By ub₁ we de- note the L^p-normalized (that is, ku_b1k_p = 1) positive eigenfunction corresponding to bλ₁. The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasi ´nski–

Papageorgiou [4], pp. 737–738) imply thatub₁ ∈intC+.

A function f :Ω×_R×_R^N →_Ris said to be “Carathéodory”, if

• _{for all}(x,y)∈_R×_R^N_,z → f(z,x,y)is measurable;

• for almost allz∈_Ω,(x,y)→ f(z,x,y)is continuous.

Such a function is necessarily jointly measurable (see Hu–Papageorgiou [8], p. 142).

The hypotheses on the reaction term f(z,x,y)are the following:

H(f) f :Ω×_R×_R^N →_Ris a Carathéodory function such that f(z, 0,y) =0 for a.a.z∈ _Ω, ally∈_R^N and

(i) there exist M>0 andδ>0 such that

f(z,x,y)≤0 for a.a.z∈ _{Ω, all}|x−M| ≤δ, all|y| ≤δ;

(ii) there existsa_M ∈ L^∞(_Ω)such that

|f(z,x,y)| ≤a_M(z)[1+|y|^p−1] for a.a.z ∈_Ω, all 0≤x ≤ M, ally∈_R^N; (iii) for every c>0, there existsη_c∈ L^∞(_Ω)such that

η_c(z)≥bλ₁ for a.a.z ∈_Ω,η_c6≡bλ₁, lim inf

x→0⁺

f(z,x,y)

x^p−1 ≥η_c(z) uniformly for a.a.z∈_{Ω, all}|y| ≤c.

Remark 2.2. Since we are looking for positive solutions and all the above hypotheses concern the positive semiaxisR+ = [0,+_∞), without any loss of generality we assume that

f(z,x,y) =0 for a.a. z∈ _{Ω, all}x ≤0, ally ∈_R^N.

Hypothesis H(f) (i)is essentially a condition due to Hartman [7] (p. 433). It was used by Hartman [7] for ordinary Dirichlet differential systems.

Example 2.3. The following function satisfies hypothesesH(f). For the sake of simplicity we drop thez-dependence.

f(x,y) =

(ηx^p−1−2ηx^r−1+x^τ−1|y|^p−1 if 0≤x ≤1, g₀(x,y) _{if 1}<x,

with g₀(·,·)any continuous function,η >bλ₁ > 0, p< r,τ. This particular f(·,·)satisfies the Hartman conditionH(f) (i)with M=1 andδ>0 small such thatδ≤ η

1 p−1.

Let M > 0 be as in hypothesis H(f) (i)and consider the nonexpansive function (that is, Lipschitz continuous with Lipschitz constant 1)p_M :R→_Rdefined by

p_M(x) =

(x⁺ if x≤ M, M if M< x.

Clearly|pM(x)| ≤ |x|for allx ∈R. We introduce the following function

bf(z,x,y) = f(z,p_M(x),y) +p_M(x)^p−1. (2.3) This is a Carathéodory function. Lete∈intC+andε >0. We consider the following auxiliary Dirichlet problem:

−_∆pu(z) +|u(z)|^p−2u(z) = fb(z,u(z),∇u(z)) +εe(z) inΩ, u

∂Ω =0. (2.4) Proposition 2.4. If hypotheses H(f) hold and ε > 0 is small, then problem (2.4) admits a solution uε ∈ intC+and0≤uε(z)≤M for all z ∈_Ω.

Proof. Let A : W₀^1,p(_Ω)→ W^−1,p0(_Ω) =W₀^1,p(_Ω)^∗ (¹_p+ _p¹0 = 1)be the nonlinear map defined by

hA(u)_,hi=

Z

Ω|∇u|^p−2(∇u,∇h)_RNdz for allu,h ∈W₀^1,p(_Ω)_.

It is well-known (see, for example Gasi ´nski–Papageorgiou [4]), that the map A(·) is bounded (that is, maps bounded sets to bounded ones), continuous, strictly monotone, hence maximal monotone too. Also letψ_p :W₀^1,p(_Ω)→ L^p0(_Ω)be

ψ_p(u) =|u|^p−2u for allu∈W₀^1,p(_Ω)_.

This map too is bounded, continuous and maximal monotone (recall that L^p0(_Ω) ,→ W^−1,p0(_Ω)).

Let N

bf denote the Nemitsky map corresponding to the Carathéodory function bf, that is, Nfb(u)(·) = ^bf(·,u(·),∇u(·)) for allu ∈W₀^1,p(_Ω)_.

Hypothesis H(f) (ii) and the Krasnoselskii theorem (see Gasi ´nski–Papageorgiou [4], p. 407) imply that

Nbf(·)is bounded, continuous.

The compact embeddings of

W^1,p(_Ω)intoL^p(_Ω) and of L^p0(_Ω)intoW^−1,p0(_Ω) =W₀^1,p(_Ω)^∗

(use the Sobolev embedding theorem and Lemma 2.2.27, p. 141 of Gasi ´nski–Papageorgiou [4]), imply thatψpandN_bf are both completely continuous maps.

LetV:W₀^1,p(_Ω)→W^−1,p0(_Ω)be defined by V(u) =A(u) +ψ_p(u)−N

bf(u)−εe for all u∈W₀^1,p(_Ω).

EvidentlyV(·) is bounded, continuous and pseudomonotone. Also, for allu ∈ W₀^1,p(_Ω) we have

hV(u)_,ui=k∇uk^p_p+kuk^p_p−

Z

Ω bf(z,u,∇u)u dz−ε Z

Ωeu dz

≥ kuk^p−c₁kuk^p−1−c₁ for somec₁>₀ (see hypothesisH(f) (ii)_),

⇒ V(·)is strongly coercive (recallp>1).

Then we can use Proposition2.1and finduε ∈W₀^1,p(_Ω),uε 6=0 such that V(u_ε) =0

⇒ hA(uε),hi+

Z

Ω|uε|^p−2uεh dz=

Z

Ω bf(z,uε,∇uε)h dz+ε Z

Ωeh dz (2.5) for allh ∈W₀^1,p(_Ω).

In (2.5) we choose h=−u⁻_ε ∈W₀^1,p(_Ω). Then

k∇u⁻_ε k^p_p ≤0 (see (2.3) and recall thate∈ intC+),

⇒ uε ≥0, uε 6=0.

From (2.5) we have











−_∆pu_ε(z) +u_ε(z)^p−1= f(z,p_M(u_ε(z)),∇u_ε(z)) +p_M(u_ε(z))^p−1+εe(z) for a.a. z∈ _Ω, u_ε

∂Ω=0.

(2.6)

From (2.6) and the nonlinear regularity theory (see Gasi ´nski–Papageorgiou [4], pp. 737–738), we have

u_ε ∈C+\ {0}. Letc= kuεk_C1

0(_Ω). Hypotheses H(f) (ii),(iii)imply that there existsξbc >0 such that f(z,x,y) +ξb_cx^p−1≥0 for a.a.z∈_Ω, all 0≤ x≤ M, all|y| ≤c.

From (2.6) we have

∆pu_ε(z)≤[1+ξb_c]u_ε(z)^p−1 for a.a.z∈_Ω,

⇒ u_ε ∈intC+,

by the nonlinear strong maximum principle (see Gasi ´nski–Papageorgiou [4], p. 738).

Finally we show that

0≤uε(z)≤ M for allz ∈_Ω. (2.7)

To show (2.7) we argue by contradiction. So, suppose that u_ε(z₀) =max

Ω u_ε > M.

Evidentlyz₀∈_Ω. Then forϑ>0 small we have

∂uε

∂n

∂Bϑ(z0)≤0 and uε(z)> M for allz∈ B_ϑ(z₀)⊆_Ω.

HereB_ϑ(z₀) ={z∈_Ω : |z−z₀| ≤ϑ}. We have

p_M(u_ε(z)) =M for all z∈ B_ϑ(z₀)and∇u_ε(z₀) =0. (2.8) We chooseϑ>0 small so that

|uε(z)−uε(z0)| ≤δ and |∇uε(z)| ≤δ for allz ∈B_ϑ(z0) (recallu_ε ∈_intC+),

⇒ |p_M(u_ε(z))−p_M(u_ε(z₀))| ≤δ and |∇u_ε(z)| ≤δ for allz ∈B_ϑ(z₀) (2.9) (recallp_M(·)is nonexpansive).

From (2.9) and hypothesis H(f) (i), we have

f(z,pM(uε(z)),∇uε(z))pM(uε(z))≤0 for allz∈ Bϑ(z0). (2.10) We multiply (2.6) with u_ε(z) and then integrate over B_ϑ(z₀). Using the nonlinear Green’s identity (see Gasi ´nski–Papageorgiou [4], p. 211), we have

Z

B_ϑ(z0)

|∇u_ε|^pdz−

Z

∂B_ϑ(z0)

|∇u_ε|^p−2∂uε

∂nu_εdσ+

Z

B_ϑ(z0)u^p_εdz

=

Z

Bϑ(z0)

[f(z,pM(uε),∇uε) +pM(uε)^p−1]uεdz+ε Z

Bϑ(z0)euεdz

=

Z

B_ϑ(z0)

[f(z,p_M(uε),∇uε) +p_M(uε)^p−1]p_M(uε)^uε Mdz+ε

Z

B_ϑ(z0)euεdz (see (2.8))

≤

Z

B_ϑ(z₀)M^p−1u_εdz+ε Z

B_ϑ(z₀)eu_εdz (see (2.10) and (2.8)),

⇒ −

Z

∂B_ϑ(z₀)

|∇u_ε|^p−2∂uε

∂nu_εdσ+

Z

B_ϑ(z₀)

[u^p_ε⁻¹−M^p−1−εe]u_εdz≤0.

But from (2.8) we see that for allε > 0 small (say forε∈ (0,ε₀)) the left hand side of the last inequality is strictly bigger than zero, a contradiction. Therefore (2.7) is true.

Proposition 2.5. If hypotheses H(f)hold, then there existγ∈(0, 1)and c₂>0such that uε ∈C^1,γ₀ (_Ω)andkuεk

C₀^1,γ(_Ω) ≤c₂ for allε∈ (0,ε₀).

Proof. Let u_ε ∈ intC+ (ε ∈ (0,ε₀)) be the solution of (2.4) produced in Proposition 2.4. We know that

0≤uε(z)≤ M for all z∈_Ω. (2.11)

Also from (2.3) and (2.5), we have

−_∆pu_ε(z) = f(z,u_ε(z),∇u_ε(z)) +εu_ε(z) for a.a.z∈ _Ω, u_ε

∂Ω =0. (2.12) From (2.11) and (2.12) and the nonlinear regularity theory (see Gasi ´nski–Papageorgiou [4], Theorem 6.2.7, p. 738), we infer that there exist γ∈(0, 1)andc₂ >0 such that

uε ∈ C₀^1,γ(_Ω)andkuεk_C1,γ

0 (_Ω) ≤c2 for allε∈(0,ε₀). Now we are ready for the existence result.

Theorem 2.6. If hypotheses H(f)hold, then problem(1.1)admits a positive solutionub∈intC+. Proof. Letεn∈ (0,ε₀), n ∈N, and assume that εn →0⁺. Let un =uεn ∈ intC+ for alln ∈ _N (see Proposition2.4). On account of Proposition2.5and sinceC₀^1,γ(_Ω)is embedded compactly intoC₀¹(_Ω), we may assume that

u_n →u_b inC¹₀(_Ω). We claim thatub6=0.

Arguing by contradiction, assume that ub = 0. If c = sup_n∈Nku_nk_C1

0(_Ω), then hypothesis H(f) (iii)implies that givenε>0, we can findδb=δb(ε)>0 such that

f(z,x,y)≥[η_c(z)−ε]x^p−1 for a.a. z∈_{Ω, all 0}≤ x≤δ, allb |y| ≤c. (2.13) Sinceun →0 inC¹(_Ω)(recall that we have assumed thatub=0), we can findn₀ ∈_Nsuch that 0≤u_n(z)≤δb for allz∈_Ω, all n≥n₀. (2.14) For n≥n₀, we consider the function

R(u_b1,un)(z) =|∇u_b1(z)|^p− |∇un(z)|^p−2 ∇un(z),∇ ^ub

p 1

u^p_n⁻¹

! (z)

!

R^N

. From the nonlinear Picone’s identity of Allegretto–Huang [1], for n≥n₀ we have

0≤

Z

ΩR(u_b1,u_n)dz

= k∇u_b1k^p_p−

Z

Ω|∇un(z)|^p−2 ∇un,∇ ^ub

p 1

u_n^p−1

!!

R^N

dz

= bλ₁−

Z

Ω f(z,u_n,∇u_n) ^ub

p 1

u^p_n⁻¹dz−ε_n Z

Ωe ub₁^p u^p_n⁻¹dz

≤ bλ₁−

Z

Ω(η_c(z)−ε)u_b^p₁dz (see (2.13), (2.14) and recall thate∈ intC+)

=

Z

Ω(bλ₁−η_c(_z))_ub1^p_dz+ε (recall thatku_b1k_p =₁)_{. (2.15)}

Sinceub₁∈intC+and hypothesis H(f) (iii)says that

η_c(z)≥bλ₁ for a.a.z∈ _Ω, η_c 6=bλ₁, we infer that

ξ^∗ =

Z

Ω(η_c(z)−bλ₁)u_b1^pdz>0.

So, choosingε∈ (0,ξ^∗), from (2.15) we have 0≤

Z

ΩR(u_b1,u_n)dz<0 for alln ≥n₀,

a contradiction. Thereforeub6= 0. Moreover, passing to the limits asn →+_∞(see (2.12) with ε=ε_n,n∈N), we obtain

−_∆pu_b(z) = f(z,ub(z),∇u_b(z)) for a.a.z∈ _Ω, ub

∂Ω =0.

As before the nonlinear regularity theory and the nonlinear maximum principle, imply that ub∈intC+.

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