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

**an Orlicz–Sobolev space approach**

**Andrei Grecu**

^{1,2}

### and **Denisa Stancu-Dumitru**

^{B}

^{2, 3}

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

*ϕ*n(|∇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 C

^{1}satisfyingLieberman-type condition

N−1< *ϕ*^{−}_{n} −1≤ ^{t}^{ϕ}

0n(t)

*ϕ*_{n}(t) ≤ *ϕ*^{+}_{n} −1< _{∞,} ∀ t≥0 (1.2)
for some constants *ϕ*^{−}_{n} and*ϕ*^{+}_{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 constant*β*>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 when*ϕ*_{n}(t) =|t|^{n}^{−}^{2}t,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

(−_{∆}_{n}v=*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}∈W_{0}^{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 →_{∞}, for*λ* > 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) when*λ* 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 from**R**onto**R**of classC^{1}. We defineΦ:[0,∞)→** _{R}**by

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,∞)→** _{R}**given 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≤ ^{t}^{ϕ}

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*^{−}≤ ^{t}* ^{ϕ}*(t)

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

−_{ln}s^{ϕ}^{−} =

Z _{t}

st

*ϕ*^{−}
*τ* dτ≤

Z _{t}

st

*ϕ*(*τ*)

Φ(*τ*) ^{dτ}=_{ln}_{Φ}(t)−_{ln}_{Φ}(st)≤

Z _{t}

st

*ϕ*^{+}

*τ* dτ= −_{ln}s^{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})

*ϕ*

^{0}(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 *ϕ*which are odd, increasing homeomor-
phism from **R** onto**R, and** *ϕ* and the corresponding primitive Φsatisfy condition (2.2) (see
[10, Examples 1–3, p. 243]):

1. *ϕ*(t) =|t|^{p}^{−}^{2}t,Φ(t) = ^{|}^{t}_{p}^{|}^{p} with p>1 and*ϕ*^{−} = *ϕ*^{+}= p.

2. *ϕ*(t) = log(1+|t|^{r})|t|^{p}^{−}^{2}t, Φ(t) = log(1+|t|^{r})^{|}^{t}_{p}^{|}^{p} − _{p}^{r}R|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}^{−}_{+|}^{2}^{t}_{t}_{|)} for t 6= 0, *ϕ*(0) = 0, Φ(t) = _{p}_{log}^{|}_{(}^{t}^{|}_{1}^{p}_{+|}_{t}_{|)}+ ^{1}_{p}R|t|
0

s^{p}

(1+s)(log(1+s))^{2} ds with
p>2 and*ϕ*^{−} = p−1, *ϕ*^{+}= p=lim inft→_{∞} ^{log}^{Φ}(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→** _{R}**such that

kuk_{L}Φ(_{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_{L}Φ(_{Ω}) is a Banach space and
its Orlicz normk · k_{L}Φ(_{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}.

ByW_{0}^{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.

W_{0}^{1,Φ}(_{Ω}):=C^{∞}_{0} (_{Ω})^{k·k}^{1,Φ}.
Note that the norms k · k_{1,Φ} and k · k_{W}1,Φ

0 := k |∇ · | k_{Φ} are equivalent on the Orlicz–Sobolev
spaceW_{0}^{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 W_{0}^{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}^{−}^{2}t, 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
W_{0}^{1,Φ}(_{Ω})become theclassical Sobolev spaces W^{1,p}(_{Ω})andW_{0}^{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 constant*λ*^{?} >0 (independent ofn) such
that for each*λ*∈ (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,∞)→** _{R}**by

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} ∈ W_{0}^{1,Φ}^{n}(_{Ω}) and the
following relation holds true

Z

Ω

*ϕ*_{n}(|∇v_{n}|)

|∇vn| ∇v_{n}∇w dx=*λ*
Z

Ωe^{v}^{n}w dx, ∀w∈W_{0}^{1,Φ}^{n}(_{Ω}). (3.1)
Note that the integral from the right-hand side of relation (3.1) is well-defined since
the Orlicz–Sobolev spaceW_{0}^{1,Φ}^{n}(_{Ω}) is continuously embedded in the classical Sobolev space
W_{0}^{1,ϕ}^{−}^{n}(_{Ω})(see, e.g. [2, Theorem 8.12]) and for*ϕ*^{−}_{n} > Nwe haveW_{0}^{1,ϕ}^{−}^{n}(_{Ω})⊂L^{∞}(_{Ω}). Moreover,
we recall that Morrey’s inequality holds true, i.e. there exists a positive constantC_{n}such that

kvk_{L}∞(_{Ω})≤Cnk |∇v| k

L^{ϕ}^{−}^{n}(_{Ω}), ∀ v∈W_{0}^{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} [*λ*_{1}(*ϕ*^{−}_{n})]

N−*ϕ*−
n
(*ϕ*−

n)^{2}, (3.3)
where|B(0, 1)|is the volume of the unit ball in**R**^{N}and for each real numberp∈ (1,∞),*λ*_{1}(p)
denotes the first eigenvalue for thep-Laplace operator with homogeneous Dirichlet boundary
conditions, i.e.

*λ*_{1}(p):= inf

u∈C^{∞}_{0}(_{Ω})\{0}

R

Ω|∇u|^{p}dx
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)_{:}=_{inf}_{y}_{∈}* _{∂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} :W_{0}^{1,}^{Φ}^{n}(_{Ω})→** _{R}**defined by
J

_{n,λ}(v)

_{:}=

Z

ΩΦn(|∇v|)dx−*λ*
Z

Ωe^{v} dx, ∀v ∈W_{0}^{1,Φ}^{n}(_{Ω})_{.}
Standard arguments can be used in order to show thatJ_{n,λ} ∈C^{1}(W_{0}^{1,Φ}^{n}(_{Ω}),**R**)and

hJ_{n,λ}^{0} (v),wi=

Z

Ω

*ϕ*_{n}(|∇v|)

|∇v| ∇v∇w dx−*λ*
Z

Ωe^{v}w dx, ∀ v, w∈W_{0}^{1,Φ}^{n}(_{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}|+_{Φn}^{1}_{(}_{1}_{)}^{i}^{1/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 let*λ*^{?}_{n} be given by relation (3.5). Then for each *λ* ∈ (0,*λ*^{?}_{n})
we have

J_{n,λ}(v)≥ ^{1}

2, ∀ v∈W_{0}^{1,Φ}^{n}(_{Ω}) with kvk_{W}1,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+ ^{Φ}^{n}(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∈W_{0}^{1,}^{Φ}^{n}(_{Ω}). (3.6)
By the above inequality, and since for each v∈W_{0}^{1,Φ}^{n}(_{Ω})_{with}kvk

W_{0}^{1,Φn} := k |∇v| k_{Φ}_{n} =_{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 ∈W_{0}^{1,Φ}^{n}(_{Ω}) withkvk_{W}1,Φn
0

=1. (3.7)
Next, taking into account that W_{0}^{1,}^{Φ}^{n}(_{Ω}) is continuously embedded in W_{0}^{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^{k}^{v}^{k}^{L}^{∞}^{(}^{Ω}^{)}

≥1−*λ*|_{Ω}|e^{C}^{n}^{k |∇}^{v}^{| k}^{L}^{ϕ}^{−}^{n}^{(}^{Ω}^{)}, ∀v∈W_{0}^{1,Φ}^{n}(_{Ω})withkvk_{W}1,Φn

0 =1.

Then for each*λ*∈ (0,*λ*^{?}_{n}), combining the above estimates with relation (3.7) we get
J_{n,λ}(v)≥1−*λ*|_{Ω}|e^{C}^{n}

h|_{Ω}|+_{Φn}^{1}_{(}_{1}_{)}^{i}^{1/ϕ}

−n

≥1−*λ*^{?}_{n}|_{Ω}|e^{C}^{n}

h|_{Ω}|+_{Φn}^{1}_{(}_{1}_{)}^{i}^{1/ϕ}

−n

= ^{1}
2,
for all v∈W_{0}^{1,}^{Φ}^{n}(_{Ω})with kvk_{W}1,Φn

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

**Lemma 3.3.** For each positive integer n let*λ*^{?}_{n}be given by relation(3.5). Define
*λ*^{?} := inf

n∈_{N}^{∗}*λ*^{?}_{n}. (3.8)

Then*λ*^{?} >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→_{∞}*ϕ*n(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*^{+}_{n}2^{ϕ}^{−}^{n} ≤ *βϕ*^{−}_{n}2^{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*^{−}_{n}2^{ϕ}^{−}^{n}i1/*ϕ*^{−}_{n}

≤^{}*βϕ*^{−}_{n}2^{ϕ}^{−}^{n}^{+}^{1}1/*ϕ*^{−}_{n}

.
Now, taking into account the fact that lim_{n}→_{∞} *βϕ*^{−}_{n}2^{ϕ}^{−}^{n}^{+}^{1}1/ϕ^{−}_{n}

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

Next, using (3.9) and the expression of*λ*^{?}_{n}we infer that
*λ*^{?}_{n} = ^{1}

2|_{Ω}|^{e}

−Cn

h

|_{Ω}|+_{Φn}^{1}_{(}_{1}_{)}^{i}^{1/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 −→*λ*_{1}(p)is continuous (see, Lindqvist [29] or Huang [23]) we conclude from the
above estimates that*λ*^{?} =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 each*λ* ∈ (0,*λ*^{?})and each n ∈ _{N}^{?}, problem
(1.1)has a nonnegative solution v_{n} ∈B_{1}(_{0})⊂W_{0}^{1,Φ}^{n}(_{Ω})identified by J_{n,λ}(v_{n}) =_{inf}

B1(0)J_{n,λ}, where
B_{1}(0)is the unit ball centered at the origin in the Orlicz–Sobolev space W_{0}^{1,Φ}^{n}(_{Ω}).

Proof. We consider *λ* ∈ (0,*λ*^{?}) and n ∈ _{N}^{?} arbitary fixed. For each v ∈ W_{0}^{1,Φ}^{n}(_{Ω}) with
kvk_{W}1,Φn

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

W_{0}^{1,Φn} ≥

Z

ΩΦn(|∇v|)dx≥ kvk^{ϕ}^{+}^{n}

W_{0}^{1,Φn}. (3.10)

Thus, taking into account (3.10), Morrey’s inequality (3.2) and relation (3.6), for each v ∈
B_{1}(0)⊂W_{0}^{1,}^{Φ}^{n}(_{Ω})we obtain

J_{n,λ}(v) =

Z

ΩΦn(|∇v|)dx−*λ*
Z

Ωe^{v} dx

≥ kvk^{ϕ}^{+}^{n}

W_{0}^{1,Φn} −*λ*|_{Ω}|e^{k}^{v}^{k}^{L}^{∞}^{(}^{Ω}^{)}

≥ −*λ*|_{Ω}|e^{C}^{n}^{k |∇}^{v}^{| k}^{L}^{ϕ}^{−}^{n}^{(}^{Ω}^{)}

≥ −*λ*|_{Ω}|e^{C}^{n}

h

|_{Ω}|+_{Φn}^{1}_{(}_{1}_{)}^{i}^{1/ϕ}

−n

.
Computing J_{n,λ}(0) =−*λ*|_{Ω}|we deduce that

J_{n,λ}(0)<0
while by Lemma3.2we get

*∂B*inf1(0)J_{n,λ} ≥ ^{1}
2 >0,
which imply that

*γ*_{n}:= inf

B1(0)

J_{n,λ} ∈(−_{∞}, 0).

We consider*e*>0 such that

*e*< inf

*∂B*1(0)J_{n,λ}− inf

B1(0)J_{n,λ}. (3.11)

Ekeland’s variational principle applied to J_{n,λ} restricted to B_{1}(0)provides the existence of
v* _{e}*∈ B

_{1}(

_{0})having the properties

i) J_{n,λ}(v*e*)< inf

B1(0)

J_{n,λ}+*e,*

ii) J_{n,λ}(v* _{e}*)< J

_{n,λ}(v) +

*e*kv−v

*k*

_{e}W_{0}^{1,Φn} for all v6=v* _{e}*.
Since inf

_{B}

1(0)J_{n,λ} ≤inf_{B}_{1}_{(}_{0}_{)}J_{n,λ} and*e*is chosen small such that (3.11) holds true, using relation
i)above we arrive at

J_{n,λ}(v*e*)< inf

B1(0)

J_{n,λ}+*e*≤ inf

B_{1}(0)J_{n,λ}+*e*< inf

*∂B*_{1}(0)J_{n,λ},

from which we deduce that v*e* is not an element on the boundary of the unit ball of space
W_{0}^{1,Φ}^{n}(_{Ω}), v* _{e}* ∈/

*∂B*

_{1}(0), and consequently, v

*is an element in the interior of this ball, that meansv*

_{e}*∈ B*

_{e}_{1}(

_{0})

_{.}

Next, we focus on the functional F_{n,λ} : B_{1}(0) → ** _{R}** defined by F

_{n,λ}(v) = J

_{n,λ}(v) +

*e*kv−v

*k*

_{e}_{W}1,Φn

0 . Obviously,v* _{e}*is a minimum point ofF

_{n,λ}(viaii)) that infers F

_{n,λ}(v

*e*+tw)−F

_{n,λ}(v

*e*)

t ≥0

for smallt>0 and any w∈ B_{1}(0). Computing the above relation we find
J_{n,λ}(v*e*+tw)−J_{n,λ}(v*e*)

t +*e*kwk_{W}1,Φn

0 ≥0

and then passing to the limit ast→0^{+}it yields thathJ_{n,λ}^{0} (v* _{e}*)

_{,}wi+

*e*kwk

_{W}1,Φn

0 ≥0 that implies
kJ_{n,λ}^{0} (v* _{e}*)k

_{(}

_{W}1,Φn

0 (_{Ω}))^{?} ≤*e, where*(W_{0}^{1,Φ}^{n}(_{Ω}))^{?} is the dual space ofW_{0}^{1,Φ}^{n}(_{Ω}).

In consideration of that, we draw to the conclusion that there exists a sequence {vm}_{m} ⊂
B_{1}(0)such that

mlim→_{∞}J_{n,λ}(v_{m}) =*γ*_{n} and lim

m→_{∞}J_{n,λ}^{0} (v_{m}) =0 . (3.12)
The sequence{v_{m}}_{m}is certainly bounded inW_{0}^{1,Φ}^{n}(_{Ω})sincev_{m} ∈ B_{1}(0)for allm∈_{N}^{?}and
this fact induces the existence of vn ∈ W_{0}^{1,}^{Φ}^{n}(_{Ω})such that, up to a subsequence, {vm}_{m} con-
verges weakly tov_{n}inW_{0}^{1,Φ}^{n}(_{Ω})and uniformly inΩ, since*ϕ*^{−}_{n} >N, asm→∞. Furthermore,
we infer that

mlim→_{∞}
Z

Ωe^{v}^{m}(vm−vn)dx=0
and

mlim→_{∞}hJ_{n,λ}^{0} (v_{m})_{,}v_{m}−v_{n}i=_{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 W_{0}^{1,Φ}^{n}(_{Ω}), as m → _{∞}, we have
that

mlim→_{∞}hJ_{n,λ}^{0} (v_{n}),v_{m}−v_{n}i=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}− ^{ϕ}^{n}(|∇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_{n}such that

*ϕ*_{n}(|*ξ*|)

|*ξ*| * ^{ξ}*−

^{ϕ}^{n}(|

*η*|)

|*η*| ^{η}

·(*ξ*−*η*)≥kn

[_{Φ}_{n}(|*ξ*−*η*|)]

*ϕ*−
n+2
*ϕ*−

n+1

[_{Φ}_{n}(|*ξ*|) +_{Φ}_{n}(|*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

*ϕ*n(|∇vm|)

|∇v_{m}| ∇v_{m}− ^{ϕ}^{n}(|∇vn|)

|∇v_{n}| ∇v_{n}

(∇v_{m}− ∇v_{n})dx

≥kn

Z

Ω

[_{Φ}_{n}(|∇vm− ∇vn|)]

*ϕ*−
n+2
*ϕ*−

n+1

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

Due to relation (3.15) we deduce that

mlim→_{∞}
Z

ΩΦn(|∇(vm−vn)|)

Φn(|∇(vm−vn)|)
Φn(|∇vm|) +_{Φ}_{n}(|∇vn|)

1/(*ϕ*^{−}_{n}+_{1})

dx=0.

SinceΦnis a convex function we obtain by relation (2.5) that
Φn(|∇(vm−vn)|)≤ ^{Φ}^{n}(_{2}|∇v_{m}|) +_{Φ}_{n}(_{2}|∇v_{n}|)

2 ≤2^{ϕ}^{+}^{n}^{−}^{1}[_{Φ}_{n}(|∇vm|) +_{Φ}_{n}(|∇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_{n}k

W_{0}^{1,Φn} =0 .

That being the case,{v_{m}}_{m} converges strongly to v_{n} inW_{0}^{1,Φ}^{n}(_{Ω})as m→ ∞. Hence, relation
(3.12) contribute to

J_{n,λ}(vn) =*γ*n<0 and J_{n,λ}^{0} (vn) =0 . (3.16)
As a result,vnis the minimizer of J_{n,λ} on B_{1}(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 ∈W_{0}^{1,Φ}^{n}(_{Ω})and for this reasonv_{n}is a nonnegative function onΩ.

The proof of Theorem3.4 is complete.

**4** **The asymptotic behavior of the sequence of solutions** { **v**

_{n}### }

_{n}**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 each*λ* ∈ (0,*λ*^{?})and each n ∈ _{N}^{?} we denote by
v_{n}the 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.** Let*λ*^{?} > 0be given by(3.8). Fix*λ* ∈ (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_{∞} ≥_{0}inΩand v_{∞} =_{0}on*∂*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} ∈ B_{1}(0)⊂
W_{0}^{1,Φ}^{n}(_{Ω})and taking into account (2.9) we have

_{Z}

Ω|∇v_{n}|^{q}dx
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_{n}k^{ϕ}^{−}^{n}

W_{0}^{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 ofW_{0}^{1,q}(_{Ω})intoC(_{Ω})is compact. Keeping in
mind the reflexivity of the Sobolev space W_{0}^{1,q}(_{Ω})we deduce that there exists a subsequence
(not relabelled) of {vn}and a functionv_{∞} ∈ C(_{Ω})such thatvn *v_{∞} weakly inW_{0}^{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

*ϕ*n(|∇vn|)

|∇v_{n}| ∇vn

=*λe*^{v}^{n}, inΩ,
we get

− ^{ϕ}^{n}(|∇v_{n}|)

|∇vn| ^{∆}^{v}^{n}−|∇v_{n}|*ϕ*^{0}_{n}(|∇v_{n}|)−*ϕ*_{n}(|∇v_{n}|)

|∇vn|^{3} ^{∆}^{∞}^{v}^{n}=*λe*^{v}^{n}, inΩ, (4.1)

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

*∂x*^{2}_{i} and∆_{∞} stands for the

∞-Laplace operator,

∆∞v:= hD^{2}v∇v,∇vi=

### ∑

N i,j=1*∂v*

*∂x*_{i}

*∂v*

*∂x*_{j}

*∂*^{2}v

*∂x*_{i}*∂x*_{j} ,
whileD^{2}v denotes the Hessian matrix ofv.

Remark that (4.1) can be reformulated as

Hn(vn,∇vn,D^{2}vn) =0, inΩ
with functionHndefined as follows

Hn(y,z,S):=−^{ϕ}^{n}(|z|)

|z| ^{Trace}^{S}−|z|*ϕ*^{0}_{n}(|z|)−*ϕ*n(|z|)

|z|^{3} hSz,zi −*λe*^{y},
wherey∈** _{R,}**z is a vector in

**R**

^{N}andSstands for a real symmetric matrix in

**M**

^{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,D^{2}v) =_{0} _{in}_{Ω,}

v=0 on*∂Ω.* (4.2)

**Definition 4.3.**

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

*∂Ω*and, whenever x_{0} ∈ _{Ω}andΨ∈ C^{2}(_{Ω})are such that v(x_{0}) = _{Ψ}(x_{0})andv(x)<_{Ψ}(x)
ifx ∈B(x_{0},r)\ {x_{0}}for somer>0, we have H_{n}(_{Ψ}(x_{0}),∇_{Ψ}(x_{0}),D^{2}Ψ(x_{0}))≤0.

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

*∂Ω*and, wheneverx_{0} ∈ _{Ω}andΥ ∈ C^{2}(_{Ω})are such thatv(x_{0}) =_{Υ}(x_{0})andv(x)> _{Υ}(x)
ifx ∈B(x_{0},r)\ {x_{0}}for somer>0, we have H_{n}(_{Υ}(x_{0}),∇_{Υ}(x_{0}),D^{2}Υ(x_{0}))≥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
x^{0}_{n} ∈ _{Ω} and a test function Ψn ∈ C^{2}(_{Ω}) such thatv_{n}(x^{0}_{n}) = _{Ψ}_{n}(x^{0}_{n})and v_{n}−_{Ψ}_{n} has a strict
local maximum atx^{0}_{n}, that isv_{n}(y)<_{Ψ}_{n}(y)_{if} y∈B(x^{0}_{n},*ρ*)\ {x^{0}_{n}}_{for some}*ρ*>_{0.}

Next, we have to show that

−div *ϕ*n(|∇_{Ψ}_{n}(x^{0}_{n})|)

|∇_{Ψ}_{n}(x^{0}_{n})| ∇_{Ψ}_{n}(x^{0}_{n})

!

≤*λe*^{Ψ}^{n}^{(}^{x}^{0}^{n}^{)}

or

−^{ϕ}^{n}(|∇_{Ψ}_{n}(x^{0}_{n})|)

|∇_{Ψ}_{n}(x^{0}_{n})| ^{∆Ψ}^{n}(x^{0}_{n})−|∇_{Ψ}_{n}(x^{0}_{n})|*ϕ*^{0}_{n}(|∇_{Ψ}_{n}(x^{0}_{n})|)−*ϕ*_{n}(|∇_{Ψ}_{n}(x^{0}_{n})|)

|∇_{Ψ}_{n}(x^{0}_{n})|^{3} ^{∆}^{∞}^{Ψ}^{n}(x^{0}_{n})≤*λe*^{Ψ}^{n}^{(}^{x}^{0}^{n}^{)}.
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(x^{0}_{n},*ρ*_{n}) ⊂ _{Ω}from the Euclidean space
**R**^{N} such that

−^{ϕ}^{n}(|∇_{Ψ}_{n}(y)|)

|∇_{Ψ}_{n}(y)| ^{∆Ψ}^{n}(y)−|∇_{Ψ}_{n}(y)|*ϕ*^{0}_{n}(|∇_{Ψ}_{n}(y)|)−*ϕ*_{n}(|∇_{Ψ}_{n}(y)|)

|∇_{Ψ}_{n}(y)|^{3} ^{∆}^{∞}^{Ψ}^{n}(y)> *λe*^{Ψ}^{n}^{(}^{y}^{)}
for all y ∈ B(x^{0}_{n},*ρ*n). For *ρ*n small enough, we may presume that vn−_{Ψ}_{n} has a strict local
maximum atx^{0}_{n}, that isv_{n}(y)<_{Ψ}_{n}(y)if y∈B(x^{0}_{n},*ρ*_{n})\ {x^{0}_{n}}. This fact implies that actually

sup

*∂B*(x^{0}_{n},ρn)

(v_{n}−_{Ψ}_{n})<0.

Thus, we may consider a perturbation of the test functionΨndefined as
w_{n}(y):= _{Ψ}_{n}(y) +^{1}

2 sup

y∈*∂B*(x^{0}_{n},ρn)

[v_{n}−_{Ψ}_{n}](y)
that has the properties

•wn(x^{0}_{n})<vn(x^{0}_{n});

•w_{n}>v_{n}on *∂B*(x_{n}^{0},*ρ*_{n});

• −div ^{ϕ}^{n}_{|∇}^{(|∇}_{w}^{w}^{n}^{|)}

n| ∇w_{n}

>*λe*^{Ψ}^{n} in B(x^{0}_{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(x^{0}_{n},*ρ*_{n}), and integrating on B(x^{0}_{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^{Ψ}^{n}^{(}^{x}^{)}[v_{n}(x)−w_{n}(x)] dx, (4.3)
where the setM_{n}_{:}= {x∈ B(x^{0}_{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(x^{0}_{n},*ρ*_{n}),
0, ifx∈ _{Ω}\B(_{x}^{0}_{n}_{,}*ρ*_{n})_{,}
we obtain

Z

B(x^{0}_{n},ρn)

*ϕ*_{n}(|∇v_{n}(x)|)

|∇v_{n}(x)| ∇v_{n}(x)∇(v_{n}−w_{n})^{+}(x)dx=*λ*
Z

B(x^{0}_{n},ρn)e^{v}^{n}^{(}^{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^{v}^{n}^{(}^{x}^{)}(v_{n}−w_{n})(x)dx