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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Calc Var Partial Differ Equ. Author manuscript; available in PMC 2010 August 9.
Published in final edited form as:
Calc Var Partial Differ Equ. 2005 May 1; 212(1): 208–233.
doi:  10.1016/j.jde.2004.07.021
PMCID: PMC2917850
NIHMSID: NIHMS171496

On the structure of solutions to a class of quasilinear elliptic Neumann problems. Part II

Abstract

We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208–233, 2005) to study the structure of positive solutions to the equation εm Δmu − um−1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of RN (N ≥ 2). First, we study subcritical case for 2 < m < N and show that after passing by a sequence positive solutions go to a constant in C1, α sense as ε → ∞. Second, we study the critical case for 1 < m < N and prove that there is a uniform upper bound independent of ε [set membership] [1, ∞) for the least-energy solutions. Third, we show that in the critical case for 1 < m ≤ 2 the least energy solutions must be a constant if ε is sufficiently large and for 2 < m < N the least energy solutions go to a constant in C1, α sense as ε → ∞.

1 Introduction

Let ΩRN (N ≥ 2) be a smooth bounded domain. We continue our former work [8] to investigate the structure of positive solutions to a class of quasilinear elliptic Neumann problems as follows:

{εmΔmuum1+f(u)=0inΩ,u>0inΩ,un=0onΩ,}
(1) (2) (3)

where Δmu = div ([mid ][nabla]u[mid ]m−2[nabla]u), 1 < m < N and n is the unit outer normal of [partial differential]Ω. The coefficient ε > 0 is a parameter.

The problem (1)–(3) appears in the study of non-Newtonian fluids, chemotaxis, and biological pattern formation. For example, in the study of non-Newtonian fluids, the quantity m is a characteristic of the medium. Media with m > 2 are called dilatant fluids and those with m < 2 are called pseudoplastics. If m = 2, they are Newtonian fluids (see [2] and its bibliography). In this special case m = 2, (1)–(3) is also known as the stationary equation of the Keller–Segal system in chemotaxis (see [11]) or the limiting stationary equation of the so-called Giener–Meinhardt system in biological pattern formation [19].

For the case m = 2, the asymptotic behavior of mountain-pass solutions to the system (1)–(3) as ε → 0+ was studied by Lin, Ni, and Takagi in a series of remarkable papers [11,13,14]. First, Lin et al. [11] applied the mountain-pass lemma [1] to show the existence of a mountain-pass solution uε to (1)–(3). Under the assumption that f(t)t is increasing on R+ it was shown [14] that every mountain-pass solution is a least-energy solution, by which it is meant that uε has the least energy among all the solutions to (1)–(3), with the energy functional defined by

Iε(u)=Ω(ε22u2+12u2F(u+))dx,

where u+ = max {u, 0} and F(u+)=0u+f(s)ds. They also gave some important preliminary results about the mountain-pass solutions. In [13,14], Ni and Takagi investigated the asymptotic behavior of the least-energy solution uε as ε → 0+ and showed that the least-energy solution has a single-spike which is achieved at a point on [partial differential]Ω and moreover for ε > 0 sufficiently small this point stays at a point where maximum of mean curvature of [partial differential]Ω is attained. For the general case 1 < m < N, it was shown [9,10] that as ε → 0+ the global maximum point of the least-energy solution uε approaches a point where maximum of mean curvature of [partial differential]Ω is attained at a rate of o(ε). Also see [37] for existence and properties of multiple-peaks solutions for the case m = 2.

In our former work [8] we gave some a-priori estimates of mountain-pass solutions and investigated uniform upper bound and Harnack inequalities etc for all positive solutions of (1)–(3) under assumption 1 < m < N and the following assumptions on f.

  • (f1): f:RR is a continuous function and f : (0, ∞) → (0, ∞) is C1.
  • (f2): f(t) [equivalent] 0 for t ≤ 0 and f(t)tm10 as t → 0+ and f(t) > 0 for t > 0.
  • (f3): limtf(t)tp=B>0 with m − 1 < p < ν − 1 and ν=NmNm.
  • (f4): Let F(t)=0tf(s)ds ds. There exists a constant θ > m such that 0 ≤ θF (t) ≤ tf (t) for t ≥ 0.

Especially we proved the following results (see [11,12] for the case m = 2).

Theorem A

([8], Theorem 1.3, P. 210) Under assumptions (f1)–(f4), for any positive solution uε [set membership] W1, m (Ω) to (1)–(3), there exists a constant C independent of ε [set membership] (0, +∞) such that

supxΩuε(x)C.

Moreover if 1 < m ≤ 2, there exists a positive constant ε1 sufficiently large such that for ε > ε1, any positive solution to the problem (1)–(3) that belongs to W1, m (Ω) must be a constant.

In this paper we continue to investigate asymptotic behaviors of all solutions of (1)–(3) under assumptions of (f1)–(f4) with 2 < m < N, and least-energy solutions of (1)–(3) with 1 < m < N and f(u)=u+N(m1)+mNm (i.e., the critical case) as ε → ∞. For 1 < m < N and f(u)=u+N(m1)+mNm associated with (1)–(3) is the functional Jε:W1,m(Ω)R defined by

Jε(v)=1mΩ(εmvm+vm)dxNmNmΩv+NmNmdx.
(4)

Let

Γset={hC([0,1];W1,m(Ω))h(0)=0,h(1)=e},
(5)

where e [not identical with] 0 is a nonnegative function in W1, m (Ω) with Jε (e) ≤ 0 (e.g., a sufficiently large constant). Later we will see

cε=infhΓsetmax0t1Jε(h(t))
(6)

is a positive critical value of Jε.

Moreover, since tN(m1)+mNmtm1=tm2Nm is increasing for t > 0 it is known [9] that cε can also characterized as

cε=inf{M[v]vW1,m(Ω),v0andv0inΩ}
(7)

with

M[v]=supt0Jε(tv).

Hence cε is the least among all positive critical values of Jε. Therefore we call such a critical point uε of Jε with Jε (uε) = cε a least-energy solution. Now our results can be stated as follows:

Theorem 1

Under assumptions 2 < m < N and (f1)–(f4) on f, for any positive solution uε [set membership] W1,m with ε [set membership] [1, ∞) there exist two positive constants C independent of ε or uε and α = α(m, N, Ω) [set membership] (0, 1) such that

uεuεC1,α(Ω)Cεβ,

where β=min{m2N(m2),mm1} and uε is a positive constant defined as the average of uε over, Ω i.e., uε=1ΩΩuεdx. Especially we have uεuεC1,α(Ω)0 as ε → ∞. Moreover, after passing by a sequence uε converges to a positive zero of tm−1 − f(t) as ε → ∞.

Next is our result about uniform upper bound of the least-energy solution uε for ε [set membership] [1, ∞), 1 < m < N and f(u)=u+N(m1)+mNm.

Theorem 2

Suppose 1 < m < N and uε. For ε [set membership] [1, ∞) there exists a constant C > 0 independent of ε such that any least-energy solution uε to (1)–(3) satisfies

supxΩuε(x)C.

For asymptotic behaviors of the least-energy solutions uε as ε → ∞, we have

Theorem 3

Let 1 < m < N, f(u)=u+N(m1)+mNm and uε be a least energy solution. Then we have

  • (f1): For 1 < m ≤ 2 there exists a positive constant ε1 sufficiently large such that for ε > ε1, uε [equivalent] 1.
  • (f2): For 2 < m < N, uε1C1,α(Ω)0 as ε → ∞, where α is stated as in Theorem 1.

The organization of this paper is as follows. Some a-priori estimates and proof of Theorem 1 will be presented in Sect. 2. In Sect. 3 we will discuss existence of least energy solutions and prove Theorem 2. The proof of Theorem 3 will be given in Sect. 4.

2 Some a-priori estimates and proof of Theorem 1

In this section we will give some a-priori estimates related to uε and use them to prove Theorem 1. From now on C, Ci (i = 0, 1,…) are generic positive constants, we will specify them whenever it is necessary.

Let uε [set membership] W1, m (Ω) be a positive solution to (1)–(3) with 2 < m < N and assumptions (f1)–(f4) on f. Then we know uεC1(Ω) and uε is uniformly bounded due to Theorem A, i.e.,

supxΩuε(x)C.
(8)

Decompose uε as uε=uε+vε, where

uε=1ΩΩuεdxandΩvεdx=0.
(9)

Then it follows from (1)–(3) that vε satisfies the following equations

{εmΔmvεuεm1+f(uε)=0inΩ,vεn=0onΩ.}
(10)

Note that

uεm1f(uε)uεm1+f(uε)=(uε+vε)m1uεm1f(uε+vε)+f(uε)=[(m1)01(uε+tvε)m2dt01f(uε+tvε)dt]vε.

Then the first equation in (10) can be expressed as

εmΔmvε[(m1)01(uε+tvε)m2dt01f(uε+tvε)dt]vε=uεm1f(uε).
(11)

Multiplying both sides of (11) by vε, integrating over Ω and using integration by parts we have

εmΩvεmdx+Ω[(m1)01(uε+tvε)m2dt01f(uε+tvε)dt]vε2dx=0.

Since 0<uε+tvε2maxΩuε(x)C and limt0+f(t)tm2=0 for m > 2 it follows that

((m1)(uε+tvε)m2f(uε+tvε))Cfor(x,t)Ω×[0,1],

where (v) = min{v, 0}. Thus we get

εmΩvεmdxCΩvε2dx.
(12)

On the other hand, (9) implies that the following Poincaré inequality holds for vε with some positive constant γ:

ΩvεmdxγmΩvεmdx.

. Thus by Hölder inequality we have

(εγ)mΩvεmdxC(Ωvεmdx)2m,

which yields

vεLm(Ω)Cεmm2.

Therefore by interpolation and the fact that ||vε||L(Ω)C it follows that for all qm

vεLq(Ω)Cεm2q(m2).
(13)

Especially we have

vεm2vεLNm1(Ω)Cεm2(m1)N(m2).
(14)

Lemma 1

The following estimate holds for vε:

vεL(Ω)Cεβ,

where β=min{m2N(m2),mm1} and C is a positive constant depending only on N, m, Ω and the uniform upper bound as stated in Theorem A.

Proof We can rewrite (10) as follows.

{Δmvεvεm2vε=vεm2vε+1εm(uεm1f(uε))inΩ,vεn=0onΩ.}
(15)

For convenience, let g=vεm2vε+1εm(uεm1f(uε)). It is easy to check gLNm1(Ω) due to (14) and uniform boundedness of uε as stated in Theorem A. Without loss of generality we assume gLNm1(Ω)0, or else the lemma holds trivially. Let v~ε=vεgLNm1(Ω)1m1 and g~=ggLNm1(Ω). We can find that v~ε satisfies the following equations

{Δmv~εv~εm2v~ε=g~inΩ,v~εn=0onΩ.}
(16)

Next we use Moser iteration method to prove this lemma. For convenience, write v instead of vε.

If we multiply both sides of the first equation in (16) by [mid ]v[mid ]m(s−1)v (s ≥ 1) and integrate over Ω, then by virtue of the homogeneous Neumann boundary condition we have

sm[m(s1)+1]Ωvsmdx+Ωvmsdx=Ωg~vm(s1)vdx.
(17)

Since s−m [m(s − 1) + 1] ≤ m + 1 for s ≥ 1 and g~LNm1(Ω)=1 and we obtain from (17) that

sm[m(s1)+1]Ω(vsm+vms)dx(m+1)(Ωv[m(s1)+1]NNm+1dx)Nm+1N.
(18)

Recall the Sobolev embedding theorem

(Ωwvdx)mvΓmΩ(wm+wm)dx
(19)

for w [set membership] W1, m (Ω), where Γ > 0 is a constant depending on Ω.

It follows from (18) that

sm[m(s1)+1]Γm(Ωvvsdx)mv(m+1)(Ωv[m(s1)+1]NNm+1dx)Nm+1N.

Noting that s−m[m(s − 1) + 1] ≥ s−(m−1) for s ≥ 1 thus we have

(Ωvvsdx)mNv(Nm+1)((m+1))Γmsm1)NNm+1Ωv[m(s1)+1]NNm+1dx
(20)

for s ≥ 1. Here we remark that if we take s = 1 in (17) and take care of g~LNm1(Ω)=1 we have

Ω(vm+vm)dxΩ(m1)(Nm)+mNmvLv(Ω),

which yields from the Sobolev embedding theorem that

ΩvνdxC0
(21)

with C0=(ΓmΩ(m1)(Nm)+mNm)νm1.

Next we define two sequences {sj} and {Mj} by

[m(s01)+1]N(Nm+1)=ν,[m(sj+11)+1]N(Nm+1)=νsjforj=0,1,2,,
(22)

M0=C0,Mj+1=([(m+1)Γmsjm1]NNm+1Mj)ν(Nm+1)mNforj=0,1,2,,
(23)

We note that sj can be explicitly given by

sj=(1Nm+(m1)(Nm)+mm)[(Nm+1Nm)j1]+Nm+1Nm
(24)

for j = 0, 1, 2, …, and it follows that sj > 1 for all j ≥ 0 and sj → ∞ as j → ∞.

We shall show that

Ωv[m(sj1)+1]NNm+1dxMjforj0
(25)

and

MjeKsj1
(26)

for some constant K > 0.

Clearly (25) holds for j = 0 due to (21). Suppose that we have proved (25) for j ≥ 0. Then by (20) we have

Ωv[m(sj+11)+1]NNm+1dx([(m+1)Γmsjm1]NNm+1Ωv[m(sj1)+1]NNm+1dx)ν(Nm+1)mN([(m+1)Γmsjm1]NNm+1Mj)ν(Nm+1)mN=Mj+1,

so that (25) is true also for j + 1. Therefore it remains to show (26).

Put σj=νmlog((m+1)Γmsjm1) and μj = log Mj. Hence

μj+1=Nm+1Nmμj+σj.

Then from (24) it follows that

σj=νmlog((m+1)Γm)+ν(m1)mlog((1Nm+(m1)(Nm)+mm))×([(Nm+1Nm)j1]+Nm+1Nm)

and therefore we can find a constant C* such that

σjC(j+1)

for all j ≥ 0. We now define {τj} by τ0 = μ0 and

τj+1=Nm+1Nmτj+C(j+1)forj0.

Clearly μjτj for all j ≥ 0. Moreover

τj=(Nm+1Nm)j[μ0+C(Nm)(Nm+1)]C(j+Nm+1)(Nm),

we know there is a constant K > 0 such that

τjKsj1

and then we obtain (26). Here we note that K depends only on μ0, m, N and C*, whereas μ0 depends only on, Γ, m, N and [mid ]Ω[mid ], and C* depends only on m, N and Γ. Thus (25) and (26) are true. So (22), (25) and (26) tell us that

vLνsj1(Ω)νsj1eKsj1,

and hence letting j → ∞ we obtain

vL(Ω)eKν.

Note that v stands for vε. Therefore till now we have shown that there is a constant C depends only on m, N, and Ω such that

v~εL(Ω)C.

From the relation v~ε=vεgLNm1(Ω)1m1 it follows that

vεL(Ω)CgLNm1(Ω)1m1.
(27)

Note that (14) and Theorem A tell us that

gLNm1(Ω)vεm2LNm1(Ω)+1εm(uεm1f(uε))LNm1(Ω)Cεm2(m1)N(m2)+CεmΩm1NCεβ~

with β~=min{m2N(m2),m}. Thus it follows from (27) that

vεL(Ω)Cεβ

with β=β~m1=min{m2N(m2),mm1}. Proof of Lemma 1 is complete.

Next is a lemma related to C1,α-estimate of vε.

Lemma 2

Let w [set membership] W1, m (Ω) [intersection operator] L (Ω) (1 < m < ∞) be a weak solution to the following problem:

{Δmw=hinΩ,wn=0onΩ.}
(28)

Then there exist two constants C > 0 and α [set membership] (0, 1) depending only on m, N and Ω such that

wC1,α(Ω)C(wL(Ω)+hL(Ω)1m1).

Proof In [16] the author proved the follows: Suppose w [set membership] W1,m (B1 (0)) [intersection operator] L (B1 (0)) (hereafter BR(x) denotes the ball centered at x with radius R) is a solution of the differential equation

B1(0)j=1N{aj(x,w,w)ψxj}hψdx=0,ψCc(B1(0))

with ||w||L(B1(0)), ||h||L(B1(0)) ≤ 1 and for j = 1, …, N,

ajC0(B1(0)×R×RN)C1(B1(0)×R×(RN{0}))

and satisfies the following ellipticity and growth conditions

aj(x,μ,0)=0,i,j=1Najηi(x,μ,η)ξiξjγ1ηm2ξ2,i,j=1Najηi(x,μ,η)Γ1ηm2,i,j=1Najxi(x,μ,η)+ajμ(x,μ,η)Γ1ηm1,
(29)

for some positive constants γ1, Γ1 and x [set membership] B1(0), μR, ηRN{0} and ξRN. Then there exist constants α [set membership] (0, 1), C depending only on m, N, γ1 and Γ1 such that

wC1,α(B12(0))C.

Here we remark that the proof also works well if C1 C1(B1(0)×R×(RN{0})) is replaced by locally Lipschitz for (x,μ,η)B1(0)×R×(RN{0}).

For any compact set KΩ [subset or is implied by] [subset or is implied by] Ω and x0 [set membership] KΩ let ρ = min {dist (KΩ, [partial differential]Ω), 1}. Applying above interior estimate to

w~(x)=w(x0+ρx)(wL(Ω)+ρmhL(Ω)1(m1))

and using standard covering argument we have a constant α* [set membership] (0, 1) depending only on m and N such that

w~C1,α(KΩ)C

with C depending only on m, N, KΩ and ρ, which tells us that

w~C1,α(KΩ)C(m,N,KΩ,ρ)(wL(Ω)+hL(Ω)1(m1)).
(30)

For the boundary regularity we can use the local reflection method. For P [set membership] [partial differential]Ω, without loss of generality, we may assume that P is the coordinate origin and the xN-axis is normal to [partial differential]Ω at P, hence there exists a smooth function h* (x’), x’ = (x1, …, xN−1) defined for [mid ]x[mid ] < δ satisfying

  • (f1): h*(0) = 0 and hxj(0)=0 for j = 1,…, N − 1 and
  • (f2): ΩB={(x,xN)xN>h(x)} and ΩB={(x,xN)xN=h(x)} in a neighborhood B of P.

For yRN and [mid ]y[mid ] sufficiently small, we define a mapping straightening the boundary portion around P as in [11]: x = Ф(y) = (ϕ1y), …,ϕ(y)) by

ϕj(y)=yjyN(hyj(y))for=1,,N1,ϕN(y)=yN+h(y).

Since in virtue of property f1 the differential map DФ of Ф satisfies DФ (0) = IN × N, the identity matrix, we can assume Ф is defined on Bκ (0) for some sufficiently small constant κ > 0 such that

DΦ(y)IN×N12
(31)

for all y [set membership] Bκ (0) where ||A|| = max||x|| = 1 (Ax, x) for a N × N matrix A. Then Ф has an inverse mapping y = Ψ (x) = Ф−1 (x) in the neighborhood (Bκ (0)) of x = 0. We write Ψ (x) = (ψ (x),…,ψN (x). Set w^(y)=w(Φ(y)) and

w(y,yN)={w^(y,yN)for(y,yN)Bκ+(0),w^(y,yN)for(y,yN)Bκ(0).}
(32)

Let

gij=k=1Nϕkyiϕkyi(y),gij=k=1Nψixkψjxk(Φ(y)).

Here we note that giN|Bκ(0)[intersection operator]{yN=0} = 0 for i = 1,…,N − 1. Next denote

gij(y,yN)={gij(y,yN)for(y,yN)Bκ+(0),(1)δiN+δjNgij(y,yN)for(y,yN)Bκ(0),}gij(y,yN)={gij(y,yN)for(y,yN)Bκ+(0),(1)δiN+δjNgij(y,yN)for(y,yN)Bκ(0),}

where δij is the Kronecker symbol. Denote g(y)=det(gij(y)). Then from (31) we know

g(y)C(N),yBκ(0)
(33)

for some absolute constant C(N) > 0. Then w~W1,m(Bκ(0))L(Bκ(0)) satisfies

{j=1Nyjaj(y,w)=g(hΦ)inBκ(0),wyN=0onBκ(0){yN=0}}
(34)

with

aj(y,w)=i=1N[s,lNgslwyswyl]m22ggjiwyi,forj=1,,N.
(35)

It is not hard to check that aj (y, η)(j = 1, …, N) satisfies all inequalities in (29) with some positive constants γ1 depending only on m and N due to (31) and (33) and Γ1 depending only on m, N, D2Фi and D2ψi (i = 1, …, N) via gslyj and gslyj (s, l, j = 1, …, N) (hence Γ1 depends only on m, N and Ω).

Applying the interior estimate to

w˜(y)=w(κy)wL(Bκ(0))+κmg(hΦ)L(Bκ(0))1m1

we know there exist two constants α0 [set membership] (0, 1) and C depending only on m, N and Ω such that

w˜C1,α0(B12(0))C,

which tells us that

wC1,α0(Φ(Bκ2(0)))C(m,N,κ,Ω)(wL(Ω)+fL(Ω)1m1).
(36)

Especially, we have

wC1,α0(Φ(Bκ2+(0)))C(m,N,κ,Ω)(wL(Ω)+fL(Ω)1m1).
(37)

Since [partial differential]Ω is compact Lemma 2 follows from standard covering argument to patch the interior estimate (30) and the boundary estimate (37) together. Proof of Lemma 2 is complete.

With helps of Lemma 1 and Lemma 2 we can give proof of Theorem 1 as follows.

Proof of Theorem 1 It follows from (10), Lemma 1 and Lemma 2 that there exist a positive constant α [set membership] (0, 1) depending only on m, N, Ω such that

vεC1,α(Ω)C(vεL(Ω)+1εm(uεm1f(uε))L(Ω)1m1)C(εβ+εmm1)Cεβ,
(38)

with β=min{m2N(m2),mm1} as defined in Lemma 1. that we have used uniform boundedness of uε as stated in Theorem A. Therefore the constant C depends only on m, N, Ω, f and the uniform bound of uε as stated in Theorem A.

Let Zf = min{t > 0[mid ]tm–1 = f(t)}. It follows from assumption (f2) that Zf > 0. The maximum principle tells us that supΩ uεZf ([8]). Thus it follows from (38) and Theorem A that for sufficiently large ε,

12ZfuεC,
(39)

which implies there exist a sequence {εj} with limj→∞εj = ∞ and a constant uc > 0 such that limjuεj=uc. Note that from (1)–(3) it follows that

εjmΩuεjmdx=Ωuεj(f(uεj)uεjm1)dx.
(40)

Note that (12) and (38) imply

εjmΩuεjmdx=εjmΩvεjmdxCΩvεj2dxCεj2βΩ.
(41)

Taking limit at both sides of (40) as j → ∞ and using (38) we obtain

Ωuc(f(uc)ucm1)dx=0

which implies f(uc)ucm1=0, i.e., the constant uc is a positive zero of tm–1f(t). Proof of Theorem 1 is complete.

3 Uniform upper bound for least energy solutions of the critical case

In this section we first discuss the existence of least energy solutions of the critical case f(u)=u+N(m1)+mNm(1<m<N) and then show that for ε [set membership] [1, ∞) there is an upper bound independent of ε (hence uniform) for the least energy solutions.

Note that for this critical case the problem (1)–(3) is as follows.

{εmλmuum1+uN(m1)+mNm=0inΩu>0inΩ,un=0onΩ,}
(42) (43) (44)

Let u=εNmmv. Then v solves the following problem:

{Δmv1εmvm1+vN(m1)+mNm=0inΩu>0inΩ,un=0onΩ,}
(45) (46) (47)

Associated with (45)–(47) is functional Jε:W1,m(Ω)R defined by

Jε(v)=1mΩ(vm+1εmvm)dxNmNmΩv+νdx.
(48)

Recall that Jε:W1,m(Ω)R was defined by (4) and also note that for u=εNmmvW1,m(Ω) we have

Jε(u)=εNJε(v).
(49)

Thus existence of least energy solutions of (45)–(47) implies the existence of least energy solution of (42)–(44).

Recall Γset defined by (5) and let

c~ε=infhΓsetmax0t1Jε(h(t)).
(50)

The following result was proved in [18] (see also [15]) for 1 < m < N and also see [17] for m = 2.

Lemma 3 0<c~ε12NSmNm is a critical value of the functional Jε defined by (48), where

Sm=infwCc(RN){RNwmdx:RNwν=1}

is the best Sobolev constant for the embedding D1,m(RN)Lν(RN), and D1,m(RN)={wLm(RN):wLν(RN)} with norm (RNwmdx)1m.

From above discussions we know there is a least energy solution uε to the problem (42)–(44) with

Jε(uε)12NεNSmNm.
(51)

Note that due to the characterization (7) we also know

Jε(uε)M[1]=1NΩ.
(52)

Therefore (51) and (52) together imply

Jε(uε)=1NΩuενdxC1.
(53)

with C1=min{12NεNSmNm,1NΩ}. Also note that

Ω(εmuεm+uεm)dx=ΩuενdxC1N.
(54)

Next we give proof of Theorem 2.

Proof of Theorem 2 We need several steps to prove this theorem.

Claim 1

There exists a constant C* > 0 such that

supΩuεCεNmm.
(55)

Suppose not. Then there exist a sequence {εk} (k ≥ 1), a sequence of constants {ck} and a sequence of points {Pk}Ω with εk, Ck → ∞ such that

M~k=supΩuεk=uεk(Pk)ckεkNmm.
(56)

Since Ω is compact, passing by a subsequence if it is necessary we can assume PkP0Ω. For convenience, we write uk instead of uεk.

Case 1

P0 [set membership] [partial differential]Ω. Without loss of generality we may assume that P0 is the coordinate origin and the xN-axis is normal to [partial differential]Ω at P0. As in the proof of Lemma 2 we introduce a mapping Φ straightening the boundary portion around P0 which is defined on Bκ(0) for some constant κ > 0. Let uk be defined as in the proof of Lemma 2 with w replaced by uk. We know ukW1,m(Bk(0)) satisfies

{εmj=1Nyjaj(y,uk)=g(ukm1ukN(m1)+mNm)inBκ(0)ukyN=0onBκ(0){yN=0}}
(57)

with

aj(y,uk)=i=1N[s,lNgslukysukyl]m22ggjiukyi,
(58)

for j = 1, … , N, where g−sl is defined as in the proof of Lemma 2.

Put Qk = Ψ(Pk) with Ψ being the inverse of Φ as defined in the proof of Lemma 2 and write Qk=(qk,αk), αk ≥ 0. Since Qk → 0 as k → ∞ we may assume that Qkκ2 for all k with κ being chosen as in the proof of Lemma 2.

Let λk=εkM~kmNm. From (56) it follows that λk1ckmNm0 as k → ∞. Then the proof can be divided to treat two cases according to the behavior of αkλk as k → ∞.

Subcase (i). The sequence {αkλk} remain bounded. By passing to a subsequence if it is necessary we may assume

αkλkβ0ask.

Define a scaled function by

vk(z)=M~k1uk(λkz+qk,λkzN).

Note that vk is well-defined in the ball Bκλk and that 0 < vk(z) ≤ 1 for all k. From (57) we see that vk satisfies

{Σj=1Nzja~j(z,vk)=g~(M~kmνvkm1vkN(m1)+mNm)inBκλk(0),vkzN=0onBκλk(0){zn=0}}
(59)

with

a~j(z,vk)=i=1N[s,lNg˜sl(z)vkzsvkzl]m22g~(z)g˜ji(z)vkzi,
(60)

for j = 1, …, N, where g˜sl(z)=gsl(λkz+qk,λkzN), g~(z)=g(λkz+qk,λkzN) for s, l = 1, …, N. Note that a~j(z,vk) satisfies the structure conditions (29) with two positive constants γ1 = γ1 (m, N) and Γ1 = Γ1(m, N, Ω) (see the proof of Lemma 2). Next choose a sequence {Rl} such that Rl → ∞ as l → ∞. For fixed l, B4Rl(0)Bκλk(0) provided k is sufficiently large. Thus by the interior estimate we mentioned at the beginning of the proof of Lemma 2 it follows that {vk} is uniformly bounded in C1,α (B2Rl (0)) for some α [set membership] (0, 1) which is independent of k. By standard arguments using a diagonal process, we can find a subsequence {vkj} which converge uniformly to v~C1,α#(RN)(0<α#<α) on any compact subset of RN. Note that g˜ij(z)δij as k → ∞ due to the fact that λk → 0 and DΨ(x) → DΨ(0) =IN×N as x → 0. Also note that M~k as k → ∞. Thus we conclude that v~ satisfies

{Δmv~+v~N(m1)+mNm=0inRN,0v~1,v~(0,,0,β)=1.}
(61)

On the other hand, for any fixed R > β* + 1 we have

BR(0)vmdz=limjBR(0)vkjmdzlimjBkλkj(0)vkjmdz=limjεkjmNBk(0)ukjmdy=limj2εkjmNBk+(0)ukjmdy=limj2εkjmNΦ(Bk+(0))ukjmdet(DΨ)dxlimjεkjN(C(N)Ωεkjmukjmdx)=0(by(54)).

Hence v~0 in BR(0), which implies v [not identical with] 1 in BR(0) since v~(0,,0,β)=1, and this contradicts, (61).

Subcase (ii). The sequence {αkλk} is unbounded. We may assume that αkλk as k → ∞. In this case we define

vk(z)=M~k1uk(λkz+Qk).

Then vk is well defined in the ball Bκλk(0) and vk(0) = 1 for all k = 1, 2, … From (58) we see that vk satisfies

{Σj=1Nzja~j(z,vk)=g~(M~kmνvkm1vkN(m1)+mNm)inBκλk(0),vkzN=0onBκλk(0){zn=αkλk}}
(62)

with

a~j(z,vk)=i=1N[s,lNg˜sl(z)vkzsvkzl]m22g~(z)g˜ji(z)vkzi,
(63)

for j = 1, … , N, where g˜sl(z)=gsl(λkz+Qk), g~(z)=g(λkz+Qk) in (63) for s, l = 1, … , N. For any R > 0 we have αkλk>2R if k is sufficiently large, so that the entire ball B2R(0) is contained in Bκλk(0){zN>αkλk}. Repeating the argument as in the subcase (i), we obtain a subsequence vk which converges uniformly to C1,α# function v on any compact set of RN, and v satisfies

{Δmv+vN(m1)+mNm=0inRN,0v1,v(0)=1.}
(64)

Nevertheless, for any fixed R > 0 we know

BR(0)vmdz=limjBR(0)vkjmdzlimjBkλkj(0){zN>αkλk}vkjmdz=limjεkjmNBk+(0)ukjmdy=limjεkjmNΦ(Bk+(0))ukjmdet(DΨ)dxlimjεkjN(C(N)Ωεkjmukjmdx)=0(by(54)).

Hence v~0 in BR(0), which implies v~1 in BR(0) since v~(0,,0,β)=1, and this contradicts (64).

Case 2

P0 [set membership] Ω. In this case we define

vk(z)=M~k1uk(λkz+Pk).

Then vk(0) = 1 for all k = 1, 2,…; vk satisfies

{ΔmvkM~kmνvkm1+vkN(m1)+mNm=0inΩk,vknk=0onΩk,}
(65)

where nk is the unit outer normal of Ωk = {zkz + Pk [set membership] Ω}. For any R > 0 we have αkλk>2R if k is sufficiently large, so that the entire ball B2R(0) is contained in Ωk Repeating the argument as in the subcase (i), we obtain a subsequence such that {vkj is uniformly convergent to a C1,α# function v^ on any compact set of RN and moreover it follows from (65) that v^ satisfies (64) with v replaced by v^. On the other hand, for any fixed R > 0 we have

BR(0)v^mdz=limjBR(0)vkjmdzlimjΩkjvkjmdz=limjεkjmNΩukjmdx=limjεkjN(Ωεkjmukjmdx)=0(by(54)).

Hence v^0 in BR(0), which implies v^1 in BR(0) since v^(0)=1, and this contradicts (64). Hence Claim 1 is proved by combining the above cases together.

Claim 2

There exists a constant 0 < C < ∞ independent of ε such that uε ≤ C.

If we multiply both sides of (42) by um(s – 1)+1ε (s ≥ 1) and integrate over Ω, then by virtue of the homogeneous Neumann boundary condition (44) it follows that

sm[m(s1)+1]εmΩuεsmdx+Ωuεmsdx=Ωuεm(s1)+νdx.
(66)

Note that Claim 1 tells us that

Ωuεm(s1)+νdxsupΩuενmΩuεmsdxCεmΩuεmsdx.
(67)

Since s−m [m(s – 1)+ 1] ≤ m + 1 for ≥ 1 we obtain from (66) and (67)that

sm[m(s1)+1]Ω(uεsm+uεms)dxCΩuεmsdx.
(68)

Note that smm(s1)+1Nm2+(Nm)sm1 for sνm. Thus it follows from the Sobolev embedding theorem (19) and (68) that for sνm,

(Ωuενsdx)mν(Cs)m1Ωuεmsdx.
(69)

Next we define two sequences {s^j} and {M^j} by

s^j=(νm)j1ν,M^j=Ωuεs^jdxforj=1,2,.
(70)

Note that s^jν for all j ≥ 0 and s^j as j →∞. The inequality (69) yields that

(M^j+1)1s^j+1(Cs^j)m1mν1s^j+1M^j1s^j.

If we let μ^j=1s^jlogM^j we get

μ^j+1μ^j+m1ν(νm)(j1)[log(Cν)+(j1)logνm].

Thus we have

μ^j+1μ^1+j=1m1ν(νm)(j1)[log(Cν)+(j1)logνm]

Note that

j=1m1ν(νm)(j1)[log(Cν)+(j1)logνm]C2

for some constant C2 independent of ε. Therefore we get

uεL(Ω)limjeμ^jeC2eμ^1=C3(Ωuεν)1ν.
(71)

(54) tells us that

(Ωuεν)1νΩ1ν.

Hence we have

uεL(Ω)C,
(72)

for some positive constant independent of ε. Proof of Theorem 2 is complete.

By applying Lemma 2 to uε we have the following uniform C1,α-estimate of uε as a corollary of Theorem 2.

Corollary 1

Suppose 1 < m < N. For ε [set membership] [1, ∞) there exist two positive constants α [set membership] (0, 1) and C depending only on m, N and Ω such that for any least-energy solution uε to (42)–(44) we have

uεC1,α(Ω)C.

4 Asymptotic behaviors of least energy solutions of the critical case as ε → ∞

In this section we study asymptotic behaviors of least energy solution uε of the critical case as ε → ∞ and give Proof of Theorem 3. We decompose uε as we did before in (9), i.e., uε as uε=uε+vε, where

uε=1ΩΩuεdxandΩvεdx=0.
(73)

Then (42)–(44) tell us that vε satisfies the following equations

{εmΔmvεuεm1+uεN(m1)+mNm=0inΩ,vεn=0onΩ.}
(74)

Since

uεm1uεN(m1)+mNmuεm1+uεN(m1)+mNm=[(m1)(01(uε+tvε)m2dt)N(m1)+mNm01(uε+tvε)N(m2)+2mNmdt]vε

we have

εmΔmvε[(m1)01(uε+tvε)m2dtN(m1)+mNm01(uε+tvε)N(m2)+2mNmdt]vε=uεm1uεN(m1)+mNm.
(75)

Multiplying both sides of (75) by vε, integrating over Ω and using integration by parts we have

εmΩvεmdx+Ω[(m1)01(uε+tvε)m2dtN(m1)+mNm01(uε+tvε)N(m2)+2mNmdt]vε2dx=0.
(76)

Theorem 2 tells us

0<uε+tvε2maxΩuε(x)C.

Since

limt0+tN(m2)+2mNmtm2=0

it follows from (76) that

((m1)(uε+tvε)m2N(m1)+mNm(uε+tvε)N(m2)+2mNm)C

for (x,t)Ω×[0,1]. Thus we get by the Poincaré inequality

εmγmΩvεmdxεmΩvεmdxCΩvε2dx.
(77)

Proof of Theorem 3 Case 1. 1 < m ≤ 2. It follows from (77) that

εmγmΩvεmCΩvε2dxCsupΩvε2mΩvεmdxCΩvεmdx(by Theorem2).

If we take ε>ε1=Cmγ, we know Ωvεmdx=0, which yields that vε [not identical with] 0. Hence

uε1ΩΩuεdxonΩ.

Maximum Principle tells us that supΩ uε ≥ 1 ([8]). Therefore uε [not identical with] 1 for ε > ε1.

Case 2. 2 < m < N. For this case, by using Hölder inequality, interpolation and the fact that vεL(Ω)C as we did in Sect. 2 we obtain that for all qm

vεLq(Ω)Cεm2q(m2).

By repeating the proof of Lemma 1 we have

vεL(Ω)Cεβ
(78)

with β as defined in Lemma 1. Applying Lemma 2 to vε and taking care of (78) yield

vεC1,α(Ω)Cεβ
(79)

for some constants α [set membership] (0, 1) depending only on m,> N and Ω, and C depending only on m, N, Ω and the uniform upper bound in Theorem 2. Note that tm1tN(m1)+mNm has only 1 as its positive zero. Next we show that uε1 as ε → ∞. Since maximum principle tells us that supΩ ≥ 1 ([8]) it follows from (78) and Theorem 2 that for sufficiently large ε,

12uεC.
(80)

To show uε1 as ε → ∞ it is equivalent to show that for any sequence {ε j} with lim j→∞ εj = ∞, lim limjuεj=1.

Suppose not. Then it follows from (80) that there exist a sequence {εj} with lim j→ ∞ εj = ∞ and a constant 12ucC such that lim limjuεj=uc1 Note that from (42)–(44) it follows that

εjmΩuεjmdx=Ω(uεjνuεjm)dx.
(81)

Note that (77) and (78) imply

εjmΩuεjmdx=εjmΩvεjmdxCΩvεj2dxCεj2βΩ
(82)

Taking limit at both sides of (81) as j → ∞ and using (79) we obtain

Ω(ucνucm)dx=0

which implies uνcumc = 0, i.e., the constant uc must be 1, contradiction. Therefore we have

uε1C1,α(Ω)vεC1,α(Ω)+uε1C1,α(Ω)0

as ε →∞. Proof of Theorem 3 is complete.

Contributor Information

Chunshan Zhao, Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA.

Yi Li, Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA ; ude.awoiu.htam@ily; Department of Mathematics, Xian Jiaotong University, Xian, Shaanxi, China.

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