an Orlicz–Sobolev space approach

The asymptotic behavior of solutions to a class of inhomogeneous problems:

an Orlicz–Sobolev space approach

Andrei Grecu

and Denisa Stancu-Dumitru

1Department of Mathematics, University of Craiova, 200585 Craiova, Romania

2Research group of the project PN-III-P1-1.1-TE-2019-0456, University Politehnica of Bucharest, 060042 Bucharest, Romania

3Department of Mathematics and Computer Sciences, Politehnica University of Bucharest, 060042 Bucharest, Romania

Received 10 January 2021, appeared 19 April 2021 Communicated by Roberto Livrea

Abstract. The asymptotic behavior of the sequence{vn}of nonnegative solutions for a class of inhomogeneous problems settled in Orlicz–Sobolev spaces with prescribed Dirichlet data on the boundary of domainΩis analysed. We show that{vn}converges uniformly inΩasn→∞, to the distance function to the boundary of the domain.

Keywords: weak solution, viscosity solution, nonlinear elliptic equations, asymptotic behavior, Orlicz–Sobolev spaces.

2020 Mathematics Subject Classification: 35D30, 35D40, 35J60, 35J70, 46E30, 46E35.

1 Introduction

Let Ω ⊂ _R^N (N ≥ 2) be a bounded domain with smooth boundary ∂Ω. We consider the family of problems







−div

ϕn(|∇v|)

|∇v| ∇v

=λe^v inΩ,

v=0 on∂Ω, (1.1)

where for each positive integer n, the mappings ϕ_n: R →_R are odd, increasing homeomor- phisms of class C¹satisfyingLieberman-type condition

N−1< ϕ⁻_n −1≤ ^tϕ

0n(t)

ϕ_n(t) ≤ ϕ⁺_n −1< _∞, ∀ t≥0 (1.2) for some constants ϕ⁻_n andϕ⁺_n with 1< ϕ⁻_n ≤ ϕ⁺_n < _∞,

ϕ⁻_n →_∞ asn→_∞, (1.3)

BCorresponding author.

E-mails: andreigrecu.cv@gmail.com (A. Grecu), denisa.stancu@yahoo.com (D. Stancu-Dumitru).

and such that

there exists a real constantβ>1 with the property that ϕ⁺_n ≤ βϕ⁻_n, ∀n≥_{1 (1.4)} and

nlim→_∞ϕ_n(1)^1/ϕ−n =1. (1.5) For some examples of functions satisfying conditions (1.2)–(1.5) the reader is referred to [5, p. 4398]. Here we just point out the fact that in the particular case whenϕ_n(t) =|t|ⁿ⁻²t,n≥2, the differential operator involved in problem (1.1) is the n-Laplacian, which for sufficiently smooth functionsvis defined as∆nv:=div(|∇v|ⁿ⁻²∇v). In this particular case problem (1.1) becomes

(−_∆nv=λe^v inΩ,

v=0 on∂Ω, (1.6)

which has been extensively studied in the literature (see, e.g. [3,7,12,14,15,18,19,32]). An existence result concerning problem (1.6) for each given n > N and λ > 0 sufficiently small was proved by Aguilar Crespo & Peral Alonso in [3] by using a fixed-point argument while Mih˘ailescu et al. [32] showed a similar result by using variational techniques. Moreover, in [32] was studied the asymptotic behavoir of solutions asn→∞. More precisely, it was proved that there exists λ^? > 0 (which does not depend on n) such that for each n > N and each λ∈(0,λ^?)problem (1.6) possesses a nonnegative solutionu_n∈W₀^1,n(_Ω)and the sequence of solutions{un} converges uniformly inΩ, as n → ∞, to the unique viscosity solution of the problem

(min{|∇u| −1,−_∆∞u}=0 in Ω,

u=0 on∂Ω, (1.7)

which is precisely the distance function to the boundary of the domain dist(·,∂Ω) (see [26, Lemma 6.10]). The result from [32] was extended to the case of equations involving variable exponent growth conditions by Mih˘ailescu & F˘arc˘as,eanu in [14]. Motivated by these results the goal of this paper is to investigate the asymptotic behaviour of the solutions of the family of problems (1.1), as n →_∞, forλ > 0 sufficiently small. We will show that the results from [32] and [14] continue to hold true in the case of the family of problems (1.1). In particular, our results generalise the results from [32] and complement the results from [14].

The paper is organized as follows. In Section 2we give the definitions of the Orlicz and Orlicz–Sobolev spaces which represent the natural functional framework where the problems of type (1.1) should be investigated. Section3is devoted to the proof of the existence of weak solutions for problem (1.1) whenλ is sufficiently small. Finally, in Section 4we analyse the asymptotic behavior of the sequence of solutions found in the previous section, as n → _∞, and we prove its uniform convergence to the distance function to the boundary of the domain.

2 Orlicz and Orlicz–Sobolev spaces

In this section we provide a brief overview on the Orlicz and Orlicz–Sobolev spaces and we recall the definitions and some of their main properties. For more details about these spaces the reader can consult the books [2,22,33,34] and papers [4,9,10,20,21].

First, we will introduce the Orlicz spaces. We assume that the function ϕ is an odd, increasing homeomorphism fromRontoRof classC¹. We defineΦ:[0,∞)→_Rby

Φ(t) =

Z _t

0 ϕ(s)ds.

Note that Φ is aYoung function, that isΦ vanishes whent = 0, Φis continuous, Φis convex and lim_t→_∞Φ(t) = ∞. Moreover, since Φ(0) = 0 if and only if t = 0, lim_t→0 Φ(t)

t = 0 and lim_t→_∞ ^Φ(t)

t = _{∞, then Φ} is called a N-function (see [1,2]). Next, we define the function Φ^? :[0,∞)→_Rgiven by

Φ^?(t) =

Z _t

0 ϕ⁻¹(s)ds.

Φ^? is called thecomplementary function ofΦ. The functionsΦandΦ^? satisfy Φ^?(t) =sup

s≥0

(st−_Φ(s)) for anyt ≥0.

We note thatΦ^? is also aN-function, too.

Throughout this paper, we will assume that 0< ϕ⁻−1≤ ^tϕ

0(t)

ϕ(t) ≤ ϕ⁺−1<_∞, for all t>0 (2.1) for some positive constants ϕ⁻ and ϕ⁺. By [28, Lemma 1.1] (see also [31, Lemma 2.1]) we deduce that

1< ϕ⁻≤ ^tϕ(t)

Φ(t) ≤ ϕ⁺< _∞, for allt >0. (2.2) By relation (2.2) it follows that for eacht>0 ands∈ (0, 1]we have

−_lns^ϕ− =

Z _t

st

ϕ⁻ τ dτ≤

Z _t

st

ϕ(τ)

Φ(τ) ^dτ=_lnΦ(t)−_lnΦ(st)≤

Z _t

st

ϕ⁺

τ dτ= −_lns^ϕ+ or

s^ϕ+Φ(t)≤_Φ(st)≤s^ϕ−Φ(t), ∀t >0, s ∈(0, 1]. (2.3) Similarly, for each t>_{0 and}s>_{1 we have}

lns^ϕ− =

Z _st

t

ϕ⁻ τ dτ≤

Z _st

t

ϕ(τ)

Φ(τ) ^dτ=lnΦ(st)−lnΦ(t)≤

Z _st

t

ϕ⁺

τ dτ=lns^ϕ+ or

s^ϕ−Φ(t)≤_Φ(st)≤s^ϕ+Φ(t), ∀ t>0, s>1 . (2.4) Inequalities (2.3) and (2.4) can be reformulated as follows

min{s^ϕ−,s^ϕ+}_Φ(t)≤_Φ(st)≤max{s^ϕ−,s^ϕ+}_Φ(t) for any s,t >0 . (2.5) Similarly, by [31, Lemma 2.1] we deduce that

min{s^ϕ−−1,s^ϕ+−1}ϕ(t)≤ ϕ(st)≤max{s^ϕ−−1,s^ϕ+−1}ϕ(t), ∀s,t>0. (2.6) Next, if we lets= ϕ⁻¹(t)then we have

t(ϕ⁻¹)⁰(_t)

ϕ⁻¹(t) = ^ϕ(_s) ϕ⁰(s)s.

By (2.1) we deduce that 1

ϕ⁺−1 ≤ ^t(ϕ⁻¹)⁰(t)

ϕ⁻¹(t) ≤ ¹

ϕ⁻−1, ∀ t>0 . The above relation implies that

1< ^ϕ

+

ϕ⁺−1 ≤ ^tϕ

−1(_t)

Φ^?(t) ≤ ^ϕ

−

ϕ⁻−1 < _∞ for allt >0. (2.7) Examples. We point out some example of functions ϕwhich are odd, increasing homeomor- phism from R ontoR, and ϕ and the corresponding primitive Φsatisfy condition (2.2) (see [10, Examples 1–3, p. 243]):

1. ϕ(t) =|t|^p−2t,Φ(t) = ^|t_p^|p with p>1 andϕ⁻ = ϕ⁺= p.

2. ϕ(t) = log(1+|t|^r)|t|^p−2t, Φ(t) = log(1+|t|^r)^|t_p^|p − _p^rR|t| 0 s^p+r−1

1+s^r ds with p,r > 1 and ϕ⁻= p, ϕ⁺= p+r.

3. ϕ(t) = _log^|t₍^|₁^p−_+|^2t_t|) for t 6= 0, ϕ(0) = 0, Φ(t) = _plog^|₍^t|₁^p_+|t|)+ ¹_pR|t| 0

s^p

(1+s)(log(1+s))² ds with p>2 andϕ⁻ = p−1, ϕ⁺= p=lim inft→_∞ ^logΦ(t)

logt .

For each bounded domain Ω⊂ _R^N_{, the} Orlicz space L^Φ(_Ω) defined by the N-function Φ (see [1,2,9]) is the set of real-valued measurable functionsu:Ω→_Rsuch that

kuk_LΦ(_Ω):=_sup Z

Ωu(x)v(x)dx;

Z

ΩΦ^?(|v(x)|)dx≤1

<_∞.

Then, the Orlicz spaceL^Φ(_Ω)endowed with theOrlicz normk · k_LΦ(_Ω) is a Banach space and its Orlicz normk · k_LΦ(_Ω) is equivalent to the so-calledLuxemburg normdefined by

kuk_Φ :=_inf

µ>_{0 ;}

Z

ΩΦ u(x)

µ

dx≤1

. (2.8)

In the case of Orlicz spaces, the following relations hold true (see, e.g. [17, Lemma 2.1]):

kuk_Φ^ϕ+ ≤

Z

ΩΦ(|u(x)|)dx≤ kuk^ϕ_Φ⁻ ∀ u∈ L^Φ(_Ω)withkuk_Φ <1, (2.9) kuk_Φ^ϕ− ≤

Z

ΩΦ(|u(x)|)dx≤ kuk^ϕ_Φ⁺ ∀u∈ L^Φ(_Ω)withkuk_Φ > 1 (2.10)

and _Z

ΩΦ(|u(x)|)dx=₁⇐⇒ kuk_Φ =_1, ∀u∈ L^Φ(_Ω)_{. (2.11)} Next, we recall that for each bounded domainΩ ⊂_R^N, theOrlicz–Sobolev space W^1,Φ(_Ω) defined by the N-function Φ is the set of all functions u such that u and its distributional derivatives of order 1 lie in Orlicz spaceL^Φ(_Ω). More exactly,W^1,Φ(_Ω)is the space given by

W^1,Φ(_Ω) =

u∈ L^Φ(_Ω); ∂u

∂x_j ∈L^Φ(_Ω), j∈ {1, . . . ,N}

.

It is a Banach space with respect to the following norm kuk_1,Φ :=kuk_Φ+k |∇u| k_Φ.

ByW₀^1,Φ(_Ω)we denoted the closure of all functions of class C^∞ with compact support overΩ with respect to norm ofW^1,Φ(_Ω), i.e.

W₀^1,Φ(_Ω):=C^∞₀ (_Ω)^k·k1,Φ. Note that the norms k · k_1,Φ and k · k_W1,Φ

0 := k |∇ · | k_Φ are equivalent on the Orlicz–Sobolev spaceW₀^1,Φ(_Ω)(see [21, Lemma 5.7]).

Under conditions (2.2) and (2.7),ΦandΦ^? satisfy the∆2-condition, i.e.

Φ(2t)≤CΦ(t)_, ∀ t≥_{0, (2.12)}

for some constant C > 0 (see [2, p. 232]). Therefore, L^Φ(_Ω), W^1,Φ(_Ω) and W₀^1,Φ(_Ω) are reflexive Banach spaces (see [2, Theorem 8.19] and [2, p. 232]).

Remark 2.1. For each real number p > 1 let ϕ(t) = |t|^p−2t, t ∈ R. It can be shown that ϕ⁻ = ϕ⁺ = p as mentioned above in Example 1 and the corresponding Orlicz space L^Φ(_Ω) reduces to the classical Lebesgue space L^p(_Ω) while the Orlicz–Sobolev spaces W^1,Φ(_Ω) and W₀^1,Φ(_Ω)become theclassical Sobolev spaces W^1,p(_Ω)andW₀^1,p(_Ω), respectively. Note also that by [2, Theorem 8.12] the Orlicz spaceL^Φ(_Ω)is continuously embedded in the Lebesgue spaces L^q(_Ω)for eachq∈(1,ϕ⁻].

3 Variational solutions for problem (1.1)

In this section we will show that there exists a certain constantλ^? >0 (independent ofn) such that for eachλ∈ (0,λ^?)problem (1.1) possesses a nonnegative weak solution for each integer n≥1.

We start by introducing the following notations: for each positive integernwe denote by Φna primitive of the function ϕ_n. More precisely, we defineΦn:[0,∞)→_Rby

Φn(t):=

Z _t

0 ϕ_n(s)ds.

Definition 3.1. We say that v_n is a weak solution of problem (1.1) if v_n ∈ W₀^1,Φn(_Ω) and the following relation holds true

Z

Ω

ϕ_n(|∇v_n|)

|∇vn| ∇v_n∇w dx=λ Z

Ωe^vnw dx, ∀w∈W₀^1,Φn(_Ω). (3.1) Note that the integral from the right-hand side of relation (3.1) is well-defined since the Orlicz–Sobolev spaceW₀^1,Φn(_Ω) is continuously embedded in the classical Sobolev space W₀^1,ϕ−n(_Ω)(see, e.g. [2, Theorem 8.12]) and forϕ⁻_n > Nwe haveW₀^1,ϕ−n(_Ω)⊂L^∞(_Ω). Moreover, we recall that Morrey’s inequality holds true, i.e. there exists a positive constantC_nsuch that

kvk_L∞(_Ω)≤Cnk |∇v| k

L^ϕ−n(_Ω), ∀ v∈W₀^1,ϕ−n(_Ω). (3.2)

By [8, Proposition 3.1] we know that we can chooseC_n as follows C_n:= ϕ⁻_n|B(0, 1)|⁻

1 ϕ−

n N⁻

N(_ϕ− n+₁) (ϕ−

n)² (ϕ⁻_n −1)

N(_ϕ− n−1) (ϕ−

n)² (ϕ⁻_n −N)

N−(ϕ− n)² (ϕ−

n)² [λ₁(ϕ⁻_n)]

N−ϕ− n (ϕ−

n)², (3.3) where|B(0, 1)|is the volume of the unit ball inR^Nand for each real numberp∈ (1,∞),λ₁(p) denotes the first eigenvalue for thep-Laplace operator with homogeneous Dirichlet boundary conditions, i.e.

λ₁(p):= inf

u∈C^∞₀(_Ω)\{0}

R

Ω|∇u|^pdx R

Ω|u|^p dx , ∀ p ∈(1,∞).

By [8, Proposition 3.1] (see also [13, Theorem 3.2] for a similar result) it is well known that

nlim→_∞C_n=kdist(·,∂Ω)k_L∞(_Ω), (3.4) where dist(x,∂Ω)_:=_{infy∈∂Ω}|x−y|_,∀ x∈ Ω, stands for the distance function to the boundary ofΩ.

For each positive integer n and each positive real number λ we introduce the Euler–

Lagrange functional associated to problem (1.1) as J_n,λ :W₀^1,Φn(_Ω)→_Rdefined by J_n,λ(v)_:=

Z

ΩΦn(|∇v|)dx−λ Z

Ωe^v dx, ∀v ∈W₀^1,Φn(_Ω)_. Standard arguments can be used in order to show thatJ_n,λ ∈C¹(W₀^1,Φn(_Ω),R)and

hJ_n,λ⁰ (v),wi=

Z

Ω

ϕ_n(|∇v|)

|∇v| ∇v∇w dx−λ Z

Ωe^vw dx, ∀ v, w∈W₀^1,Φn(_Ω).

Thus, it is clear that vn is a weak solution of (1.1) if and only if vn is a critical point of functionalJ_n,λ.

We point out that we cannot find critical points of J_n,λ by using the Direct Method in the Calculus of Variations since in the case of our problem J_n,λ is not coercive. For that reason we propose an analysis of problem (1.1) based on Ekeland’s Variational Principle in order to find critical points of J_n,λ.

For each positive integernwe denote λ^?_n:= ¹

2|_Ω|^e

−Cn

h|_Ω|+_Φn¹₍₁₎^i1/ϕ

−n

, (3.5)

whereCnis the constant given by relation (3.3) and|_Ω|stands for theN-dimensional Lebesgue measure ofΩ. The starting point of our approach is the following lemma.

Lemma 3.2. For each positive integer n letλ^?_n be given by relation (3.5). Then for each λ ∈ (0,λ^?_n) we have

J_n,λ(v)≥ ¹

2, ∀ v∈W₀^1,Φn(_Ω) with kvk_W1,Φn

0 =1 .

Proof. Let n be a positive integer arbitrary fixed. By relation (2.5) we get that Φn(s) ≥ Φn(1)s^ϕ−n, for all s>1 and thus,

s^ϕ−n ≤1+ ^Φn(s)

Φn(₁)^, ∀ s≥0.

Using this fact we deduce that Z

Ω|∇v|^ϕ−n dx ≤ |_Ω|+ ¹ Φn(1)

Z

ΩΦn(|∇v|)dx, ∀ v∈W₀^1,Φn(_Ω). (3.6) By the above inequality, and since for each v∈W₀^1,Φn(_Ω)_withkvk

W₀^1,Φn := k |∇v| k_Φn =_{1 we} haveR

ΩΦn(|∇v|)dx=1 (via relation (2.11)), it results k |∇v| k

L^ϕ−n(_Ω) ≤

|_Ω|+ ¹ Φn(1)

1/ϕ⁻_n

, ∀v ∈W₀^1,Φn(_Ω) withkvk_W1,Φn 0

=1. (3.7) Next, taking into account that W₀^1,Φn(_Ω) is continuously embedded in W₀^1,ϕ−n(_Ω) and using the fact that ϕ⁻_n > Nand Morrey’s inequality (3.2) we obtain

J_n,λ(v) =

Z

ΩΦn(|∇v|)dx−λ Z

Ωe^v dx

≥1−λ|_Ω|e^kvkL∞(Ω)

≥1−λ|_Ω|e^{Cnk |∇v| kLϕ−n(Ω)}, ∀v∈W₀^1,Φn(_Ω)withkvk_W1,Φn

0 =1.

Then for eachλ∈ (0,λ^?_n), combining the above estimates with relation (3.7) we get J_n,λ(v)≥1−λ|_Ω|e^Cn

h|_Ω|+_Φn¹₍₁₎^i1/ϕ

−n

≥1−λ^?_n|_Ω|e^Cn

h|_Ω|+_Φn¹₍₁₎^i1/ϕ

−n

= ¹ 2, for all v∈W₀^1,Φn(_Ω)with kvk_W1,Φn

0 =1. The proof of the lemma is complete.

Lemma 3.3. For each positive integer n letλ^?_nbe given by relation(3.5). Define λ^? := inf

n∈_N^∗λ^?_n. (3.8)

Thenλ^? >0.

Proof. First, we show that there exists a positive constantK>0 such that

|_Ω|+ ¹ Φn(1)

1/ϕ⁻_n

<K, ∀n≥1 . (3.9)

Indeed, since by (1.5) we have

nlim→_∞ϕn(1)^1/ϕ−n =1 , it yields that for each positive integernlarge enough we get

1

2 ≤ ϕ_n(1)^1/ϕ−n , which implies that

1

ϕ_n(₁) ≤2^ϕ−n .

By (1.2) (via (2.1) and (2.2)) we find that for each positive integernlarge enough the following inequalities hold true

1

Φn(1) ≤ ^ϕ

+n

ϕ_n(1) ≤ ϕ⁺_n2^ϕ−n ≤ βϕ⁻_n2^ϕ−n .

Using the above relations we deduce that for each positive integernlarge enough we obtain

|_Ω|+ ¹ Φn(1)

1/ϕ⁻_n

≤^h|_Ω|+βϕ⁻_n2^ϕ−ni1/ϕ⁻_n

≤βϕ⁻_n2^ϕ−n+11/ϕ⁻_n

. Now, taking into account the fact that lim_n→_∞ βϕ⁻_n2^ϕ−n+11/ϕ⁻_n

=2, the above approximations imply that relation (3.9) holds true.

Next, using (3.9) and the expression ofλ^?_nwe infer that λ^?_n = ¹

2|_Ω|^e

−Cn

h

|_Ω|+_Φn¹₍₁₎^i1/ϕ

−n

> ¹

2|_Ω|^e

−KCn, ∀ n≥1 .

Recalling that lim_n→_∞C_n=kdist(·,∂Ω)k_L∞(_Ω)(by (3.4)) and taking into account that function (1,∞)3 p −→λ₁(p)is continuous (see, Lindqvist [29] or Huang [23]) we conclude from the above estimates thatλ^? =inf_n∈_N^∗λ^?_n>0. The proof of Lemma3.3is complete.

The main goal of this section is to prove the existence of weak solutions of problem (1.1) for each positive integern. This result is the core of the following theorem.

Theorem 3.4. Let λ^? > 0be given by (3.8). Then for eachλ ∈ (0,λ^?)and each n ∈ _N^?, problem (1.1)has a nonnegative solution v_n ∈B₁(₀)⊂W₀^1,Φn(_Ω)identified by J_n,λ(v_n) =_inf

B1(0)J_n,λ, where B₁(0)is the unit ball centered at the origin in the Orlicz–Sobolev space W₀^1,Φn(_Ω).

Proof. We consider λ ∈ (0,λ^?) and n ∈ _N^? arbitary fixed. For each v ∈ W₀^1,Φn(_Ω) with kvk_W1,Φn

0 ≤1, in view of relations (2.9) and (2.11), we have kvk^ϕ−n

W₀^1,Φn ≥

Z

ΩΦn(|∇v|)dx≥ kvk^ϕ+n

W₀^1,Φn. (3.10)

Thus, taking into account (3.10), Morrey’s inequality (3.2) and relation (3.6), for each v ∈ B₁(0)⊂W₀^1,Φn(_Ω)we obtain

J_n,λ(v) =

Z

ΩΦn(|∇v|)dx−λ Z

Ωe^v dx

≥ kvk^ϕ+n

W₀^1,Φn −λ|_Ω|e^kvkL∞(Ω)

≥ −λ|_Ω|e^{Cnk |∇v| kLϕ−n(Ω)}

≥ −λ|_Ω|e^Cn

h

|_Ω|+_Φn¹₍₁₎^i1/ϕ

−n

. Computing J_n,λ(0) =−λ|_Ω|we deduce that

J_n,λ(0)<0 while by Lemma3.2we get

∂Binf1(0)J_n,λ ≥ ¹ 2 >0, which imply that

γ_n:= inf

B1(0)

J_n,λ ∈(−_∞, 0).

We considere>0 such that

e< inf

∂B1(0)J_n,λ− inf

B1(0)J_n,λ. (3.11)

Ekeland’s variational principle applied to J_n,λ restricted to B₁(0)provides the existence of v_e∈ B₁(₀)having the properties

i) J_n,λ(ve)< inf

B1(0)

J_n,λ+e,

ii) J_n,λ(v_e)< J_n,λ(v) +ekv−v_ek

W₀^1,Φn for all v6=v_e. Since inf_B

1(0)J_n,λ ≤inf_B1(0)J_n,λ andeis chosen small such that (3.11) holds true, using relation i)above we arrive at

J_n,λ(ve)< inf

B1(0)

J_n,λ+e≤ inf

B₁(0)J_n,λ+e< inf

∂B₁(0)J_n,λ,

from which we deduce that ve is not an element on the boundary of the unit ball of space W₀^1,Φn(_Ω), v_e ∈/ ∂B₁(0), and consequently, v_e is an element in the interior of this ball, that meansv_e∈ B₁(₀)_.

Next, we focus on the functional F_n,λ : B₁(0) → _R defined by F_n,λ(v) = J_n,λ(v) + ekv−v_ek_W1,Φn

0 . Obviously,v_eis a minimum point ofF_n,λ (viaii)) that infers F_n,λ(ve+tw)−F_n,λ(ve)

t ≥0

for smallt>0 and any w∈ B₁(0). Computing the above relation we find J_n,λ(ve+tw)−J_n,λ(ve)

t +ekwk_W1,Φn

0 ≥0

and then passing to the limit ast→0⁺it yields thathJ_n,λ⁰ (v_e)_,wi+ekwk_W1,Φn

0 ≥0 that implies kJ_n,λ⁰ (v_e)k_(W1,Φn

0 (_Ω))^? ≤e, where(W₀^1,Φn(_Ω))^? is the dual space ofW₀^1,Φn(_Ω).

In consideration of that, we draw to the conclusion that there exists a sequence {vm}_m ⊂ B₁(0)such that

mlim→_∞J_n,λ(v_m) =γ_n and lim

m→_∞J_n,λ⁰ (v_m) =0 . (3.12) The sequence{v_m}_mis certainly bounded inW₀^1,Φn(_Ω)sincev_m ∈ B₁(0)for allm∈_N^?and this fact induces the existence of vn ∈ W₀^1,Φn(_Ω)such that, up to a subsequence, {vm}_m con- verges weakly tov_ninW₀^1,Φn(_Ω)and uniformly inΩ, sinceϕ⁻_n >N, asm→∞. Furthermore, we infer that

mlim→_∞ Z

Ωe^vm(vm−vn)dx=0 and

mlim→_∞hJ_n,λ⁰ (v_m)_,v_m−v_ni=_{0 ,} which imply that

mlim→_∞ Z

Ω

ϕ_n(|∇v_m|)

|∇vm| ∇v_m∇(v_m−v_n)dx=_{0. (3.13)} Owing to the weak convergence of sequence {v_m}_m to v_n in W₀^1,Φn(_Ω), as m → _∞, we have that

mlim→_∞hJ_n,λ⁰ (v_n),v_m−v_ni=0

and it follows that

mlim→_∞ Z

Ω

ϕ_n(|∇v_n|)

|∇v_n| ∇v_n∇(v_m−v_n)dx=0. (3.14) Assembling relations (3.13) and (3.14), we conclude that

mlim→_∞ Z

Ω

ϕ_n(|∇v_m|)

|∇v_m| ∇v_m− ^ϕn(|∇v_n|)

|∇v_n| ∇v_n

∇(v_m−v_n)dx =0. (3.15) By [16, Lemma 3.2] we know that there exists a positive constantk_nsuch that

ϕ_n(|ξ|)

|ξ| ^ξ− ^ϕn(|η|)

|η| ^η

·(ξ−η)≥kn

[_Φn(|ξ−η|)]

ϕ− n+2 ϕ−

n+1

[_Φn(|ξ|) +_Φn(|η|)]^{1/(ϕ−n+1),} ∀ ξ,η∈_R^N, ξ 6= η.

In our case, we established that there exist constantkn>0 so that Z

Ω

ϕn(|∇vm|)

|∇v_m| ∇v_m− ^ϕn(|∇vn|)

|∇v_n| ∇v_n

(∇v_m− ∇v_n)dx

≥kn

Z

Ω

[_Φn(|∇vm− ∇vn|)]

ϕ− n+2 ϕ−

n+1

[_Φn(|∇vm|) +_Φn(|∇vn|)]^{1/(ϕ−n+1) dx.}

Due to relation (3.15) we deduce that

mlim→_∞ Z

ΩΦn(|∇(vm−vn)|)

Φn(|∇(vm−vn)|) Φn(|∇vm|) +_Φn(|∇vn|)

1/(ϕ⁻_n+₁)

dx=0.

SinceΦnis a convex function we obtain by relation (2.5) that Φn(|∇(vm−vn)|)≤ ^Φn(₂|∇v_m|) +_Φn(₂|∇v_n|)

2 ≤2^ϕ+n−1[_Φn(|∇vm|) +_Φn(|∇vn|)] . Using assumption (1.4), the last two relations require

mlim→_∞ Z

ΩΦn(|∇(v_m−v_n)|)dx=0 , and (2.9) generates

mlim→_∞kv_m−v_nk

W₀^1,Φn =0 .

That being the case,{v_m}_m converges strongly to v_n inW₀^1,Φn(_Ω)as m→ ∞. Hence, relation (3.12) contribute to

J_n,λ(vn) =γn<0 and J_n,λ⁰ (vn) =0 . (3.16) As a result,vnis the minimizer of J_n,λ on B₁(0), and alsovnis a critical point of the functional J_n,λ. Of course, v_n is really a weak solution of (1.1). Finally, note that J_n,λ(|v|) ≤ J_n,λ(v) for anyv ∈W₀^1,Φn(_Ω)and for this reasonv_nis a nonnegative function onΩ.

The proof of Theorem3.4 is complete.

4 The asymptotic behavior of the sequence of solutions { v

}

of problem (1.1) given by Theorem 3.4 as n → _∞

The goal of this section is to prove the following result.

Theorem 4.1. Let λ^? > 0 be given by (3.8). For eachλ ∈ (0,λ^?)and each n ∈ _N^? we denote by v_nthe nonnegative weak solution of problem(1.1)given by Theorem3.4. The sequence{v_n}converges uniformly inΩtodist(·,∂Ω), the distance function to the boundary ofΩ.

In order to prove Theorem4.1we first establish the uniform Hölder estimates for the weak solutions of (1.1).

Lemma 4.2. Letλ^? > 0be given by(3.8). Fixλ ∈ (0,λ^?)and let vnbe the nonnegative solution of problem(1.1) given by Theorem3.4. Then there is a subsequence{v_n}which converges uniformly in Ω, as n→_∞, to a continuous function v_∞ ∈C(_Ω)with v_∞ ≥₀inΩand v_∞ =₀on∂Ω.

Proof. Letq ≥ N be an arbitrary real number. By (1.3) we can choose q< ϕ⁻_n for sufficiently large positive integern. Using Hölder’s inequality, relation (3.6), recalling that v_n ∈ B₁(0)⊂ W₀^1,Φn(_Ω)and taking into account (2.9) we have

_Z

Ω|∇v_n|^qdx 1/q

≤ _Z

Ω|∇v_n|^ϕ−n dx 1/ϕ⁻_n

|_Ω|^{1/q−1/ϕ−n}

≤

|_Ω|+ ¹ Φn(1)

Z

ΩΦn(|∇v_n|)dx 1/ϕ⁻_n

|_Ω|^{1/q−1/ϕ−n}

≤

|_Ω|+ ¹

Φn(1)kv_nk^ϕ−n

W₀^1,Φn

1/ϕ⁻_n

|_Ω|^{1/q−1/ϕ−n}

≤

|_Ω|+ ¹ Φn(₁)

1/ϕ⁻_n

|_Ω|^{1/q−1/ϕ−n} .

Thereupon, using (3.9) we find that sequence{|∇v_n|} is uniformly bounded inL^q(_Ω). It is clear thatq> N ensures that the embedding ofW₀^1,q(_Ω)intoC(_Ω)is compact. Keeping in mind the reflexivity of the Sobolev space W₀^1,q(_Ω)we deduce that there exists a subsequence (not relabelled) of {vn}and a functionv_∞ ∈ C(_Ω)such thatvn *v_∞ weakly inW₀^1,q(_Ω)and v_n → v_∞ uniformly in Ω as n → ∞. In addition, the facts that v_n ≥ 0 in Ω and v_n = 0 on

∂Ωfor each ϕ⁻_n > N hint that v_∞ ≥ 0 in Ωand v_∞ = 0 on ∂Ω. The proof of Lemma 4.2 is complete.

In Theorem4.5below we show that functionv_∞ given by Lemma4.2 is the solution in the viscosity sense (see, Crandall, Ishii & Lions [11]) of a certain limiting problem. Accordingly, we adopt the usual strategy of first proving that continuous weak solutions of problem (1.1) at leveln are indeed solutions in the viscosity sense. Before recalling the definition of viscosity solutions for this type of problems, let us note that if we assume for a moment that the solutionsvnof problem (1.1) are sufficiently smooth so that we can perform the differentiation in the PDE

−div

ϕn(|∇vn|)

|∇v_n| ∇vn

=λe^vn, inΩ, we get

− ^ϕn(|∇v_n|)

|∇vn| ^∆vn−|∇v_n|ϕ⁰_n(|∇v_n|)−ϕ_n(|∇v_n|)

|∇vn|^{3 ∆∞vn}=λe^vn, inΩ, (4.1)

where∆stands for the Laplace operator,∆v:=Trace(D²v) =_∑i^N₌₁^∂2v

∂x²_i and∆_∞ stands for the

∞-Laplace operator,

∆∞v:= hD²v∇v,∇vi=

∑

∂v

∂x_i

∂v

∂x_j

∂²v

∂x_i∂x_j , whileD²v denotes the Hessian matrix ofv.

Remark that (4.1) can be reformulated as

Hn(vn,∇vn,D²vn) =0, inΩ with functionHndefined as follows

Hn(y,z,S):=−^ϕn(|z|)

|z| ^TraceS−|z|ϕ⁰_n(|z|)−ϕn(|z|)

|z|³ hSz,zi −λe^y, wherey∈_R,z is a vector inR^N andSstands for a real symmetric matrix inM^N×N.

Since our main objective in this section is the asymptotic analysis of solutions {vn} as n → ∞, we are now ready to give the definition of viscosity solutions for the homogeneous Dirichlet boundary value problem associated to degenerate elliptic PDE of the type

(H_n(v,∇v,D²v) =_{0 inΩ,}

v=0 on∂Ω. (4.2)

Definition 4.3.

i) An upper semicontinuous functionvis a viscosity subsolution of problem (4.2) ifv≤0 on

∂Ωand, whenever x₀ ∈ _ΩandΨ∈ C²(_Ω)are such that v(x₀) = _Ψ(x₀)andv(x)<_Ψ(x) ifx ∈B(x₀,r)\ {x₀}for somer>0, we have H_n(_Ψ(x₀),∇_Ψ(x₀),D²Ψ(x₀))≤0.

ii) A lower semicontinuous functionvis a viscosity supersolution of problem (4.2) ifv ≥0 on

∂Ωand, wheneverx₀ ∈ _ΩandΥ ∈ C²(_Ω)are such thatv(x₀) =_Υ(x₀)andv(x)> _Υ(x) ifx ∈B(x₀,r)\ {x₀}for somer>0, we have H_n(_Υ(x₀),∇_Υ(x₀),D²Υ(x₀))≥0.

iii) A continuous function v is a viscosity solution of problem (4.2) if it is both viscosity supersolution and viscosity subsolution of problem (4.2).

In the sequel, functions Ψ and Υ stand for test functions touching the graph of v from above and below, respectively.

Our goal now is to prove that any continuous weak solution of (1.1) is also viscosity solution of (1.1) and in order to establish this result we follow the approach by Juutinen, Lindqvist & Manfredi in [27, Lemma 1.8] (see also [35, Lemma 1] for a similar approach but in the framework of inhomogeneous differential operators).

Lemma 4.4. A continuous weak solution of problem(1.1)is also a viscosity solution of (1.1).

Proof. Firstly, we prove that if v_n is a continuous weak solution of problem (1.1) for a fixed positive integern, then it is a viscosity subsolution of problem (1.1). We begin by considering x⁰_n ∈ _Ω and a test function Ψn ∈ C²(_Ω) such thatv_n(x⁰_n) = _Ψn(x⁰_n)and v_n−_Ψn has a strict local maximum atx⁰_n, that isv_n(y)<_Ψn(y)_if y∈B(x⁰_n,ρ)\ {x⁰_n}_{for some}ρ>_0.

Next, we have to show that

−div ϕn(|∇_Ψn(x⁰_n)|)

|∇_Ψn(x⁰_n)| ∇_Ψn(x⁰_n)

!

≤λe^Ψn(x0n)

or

−^ϕn(|∇_Ψn(x⁰_n)|)

|∇_Ψn(x⁰_n)| ^∆Ψn(x⁰_n)−|∇_Ψn(x⁰_n)|ϕ⁰_n(|∇_Ψn(x⁰_n)|)−ϕ_n(|∇_Ψn(x⁰_n)|)

|∇_Ψn(x⁰_n)|^{3 ∆∞Ψn}(x⁰_n)≤λe^Ψn(x0n). Arguingad contrarium, suppose that this is not the case of the above assertion. In other words, we admit that there exists a radius ρ_n > 0 such thatB(x⁰_n,ρ_n) ⊂ _Ωfrom the Euclidean space R^N such that

−^ϕn(|∇_Ψn(y)|)

|∇_Ψn(y)| ^∆Ψn(y)−|∇_Ψn(y)|ϕ⁰_n(|∇_Ψn(y)|)−ϕ_n(|∇_Ψn(y)|)

|∇_Ψn(y)|^{3 ∆∞Ψn}(y)> λe^Ψn(y) for all y ∈ B(x⁰_n,ρn). For ρn small enough, we may presume that vn−_Ψn has a strict local maximum atx⁰_n, that isv_n(y)<_Ψn(y)if y∈B(x⁰_n,ρ_n)\ {x⁰_n}. This fact implies that actually

sup

∂B(x⁰_n,ρn)

(v_n−_Ψn)<0.

Thus, we may consider a perturbation of the test functionΨndefined as w_n(y):= _Ψn(y) +¹

2 sup

y∈∂B(x⁰_n,ρn)

[v_n−_Ψn](y) that has the properties

•wn(x⁰_n)<vn(x⁰_n);

•w_n>v_non ∂B(x_n⁰,ρ_n);

• −div ^ϕn_|∇^(|∇_w^wn|)

n| ∇w_n

>λe^Ψn in B(x⁰_n,ρ_n).

Multiplying the above inequality by the positive part of the function v_n− w_n, i.e.

(v_n−w_n)⁺, that vanishes on the boundary of the ball B(x⁰_n,ρ_n), and integrating on B(x⁰_n,ρ_n), we get

Z

Mn

ϕ_n(|∇w_n(x)|)

|∇wn(x)| ∇w_n(x) [∇v_n(x)− ∇w_n(x)] dx>λ Z

Mn

e^Ψn(x)[v_n(x)−w_n(x)] dx, (4.3) where the setM_n:= {x∈ B(x⁰_n,ρ_n)_; w_n(x)<v_n(x)}_.

On the other hand, taking the test function in relation (3.1) to be w:Ω→_R, w(x) =

((v_n−w_n)⁺(x), ifx∈ B(x⁰_n,ρ_n), 0, ifx∈ _Ω\B(_x⁰_n,ρ_n)_, we obtain

Z

B(x⁰_n,ρn)

ϕ_n(|∇v_n(x)|)

|∇v_n(x)| ∇v_n(x)∇(v_n−w_n)⁺(x)dx=λ Z

B(x⁰_n,ρn)e^vn(x)(v_n−w_n)⁺(x)dx or

Z

Mn

ϕ_n(|∇v_n(x)|)

|∇v_n(x)| ∇v_n(x)∇(v_n−w_n)(x)dx=λ Z

Mn

e^vn(x)(v_n−w_n)(x)dx