Applications of maximum modulus method and Phragmén–Lindelöf method for second-order boundary value problems with respect to the Schrödinger operator

(1)

R E S E A R C H

Open Access

Applications of maximum modulus method

and Phragmén–Lindelöf method for

second-order boundary value problems with

respect to the Schrödinger operator

Zhen Liu

1*

*_{Correspondence:}

[email protected]

1_{School of Mathematics and}

Statistics, Kashi University, Kashi, China

Abstract

In this paper, we present a reliable combination of the maximum modulus method with respect to the Schrödinger operator (Meng in J. Syst. Sci. Complex. 16:446–452,

2003) and Phragmén–Lindelöf method (Shehu in Matematiche 64:57–66,2015) to investigate the solution of a second-order boundary value problem with respect to the Schrödinger operator. We establish the uniqueness of the solution for this problem. The results reveal that this method is eﬀective and simple.

Keywords: Maximum modulus; Schrödinger operator; Uniqueness; Boundary value problem

1 Introduction

Letג= (0,W)×(0,w) be a bounded rectangular domain inR2_{, which represents a porous}

medium with Lipschitz boundary∂ג=1∪2, where

2=

{0} ×[0,w]∪[0,W]× {w}∪{W} ×[0,w]

is the part in contact with air or covered by ﬂuid, and

1= [0,W]× {0}

is the impervious part of∂ג. Let P=ג×(0,M), whereM> 0. Let φ be a nonnegative Lipschitz function deﬁned in the closure ofP, which we denote byP.

Deﬁne

Σ1=1×(0,M), Σ2=2×(0,M), Σ3=Σ2∩ {φ> 0},

and

Σ4=Σ2∩ {φ= 0}.

(2)

Letχbe a function of the variabletsatisfying

c1≤χ(t)≤c2, a.e.t∈(0,W), (1.1)

for two positive constantsc1andc2, and letω0be a function of the variabletsatisfying

(see [3,4])

0≤ω0(t)≤1, a.e.t∈ג. (1.2)

The governing equation in terms of the deﬂection functionf(t) is the elliptic boundary value problem with respect to the Schrödinger operator (see [2,5,9,11,16,24]): Find (f,ω)∈L2_(0,_T_;_H1₍_ג₎₎_×_L∞₍_P_{) such that}

f ≥0, 0≤ω≤1, f(1 –ω) = 0 a.e. inP,

f =φ onΣ2,

P

χ(t1)(ft2+ω)ςt2–ωςs

dt ds≤

גω0(t)ς(t, 0)dt,

∀ς∈H1(P), ς= 0 onΣ3,

ς≥0 onΣ4,

ς(t,M) = 0 for a.e.t∈ג.

(1.3)

For the existence of a solution of problem (1.3) in the homogeneous case, we refer to [10] and [19] in the incompressible and compressible cases, respectively. For the heterogeneous case, we refer to [6] in a more general framework under Assumptions (1.1) and (1.2) for both incompressible and compressible cases. For the incompressible case with nonlinear Darcy’s law, we refer to [18] for Dirichlet, Neumann, and generalized boundary condi-tions. Extensions to the quasilinear and incompressible case were obtained in [10,18] in both homogeneous and nonhomogeneous frameworks. Meng [20] was able to extend the above regularity result to a more general framework under weaker assumptions on the data. The uniqueness of a solution for a homogeneous dam with general geometry was established by the method of doubling variables in [8,24], but it is not obvious whether it works in the heterogeneous situation. Extensions to the Schrödinger operator modeling incompressible ﬂuid ﬂow governed by the nonlinear Darcy law with Dirichlet or Neumann boundary conditions were obtained in [15,20], respectively.

On the other hand, the following boundary value problem with respect to the Schrödin-ger operator corresponding to (1.3) is given by [7,14,18,29,33,36]:

f ≥0, 0≤ω≤1, u(1 –ω) = 0 inP, χ(t1)(ft2+ω)t2–ωs= 0 inP,

u=φ onΣ2,

ω(·, 0) =ω0 inג,

χ(t1)(ft2+ω)·ν= 0 onΣ1,

χ(t1)(ft2+ω)·ν≤0 onΣ4.

(3)

Regarding the existence of a solution of the problem with respect to the Schrödinger op-erator (1.4), we refer to [15] and [37], respectively, for the evolutionary dam problem with homogeneous coeﬃcients and for a class of free boundary problem with nonlocal bound-ary condition in heterogeneous domain. The regularity of the solution of the problem with nonlocal boundary condition was discussed in [40] (also see, e.g., [13,32]), where it was proved thatω∈C0_([0,_T_];_Lp_{(ג)) for all}_p_∈_{[1, +}_∞_{) in the free boundary problems with} respect to the Schrödinger operator of types

divχ(t)∇f+H(t)ω–ωs

and

divχ(t)∇f+H(t)ω– (f +ω)s,

and thatf ∈C0_([0,_M_];_Lp_{(ג)) for all}_p_∈_{[1, 2] in the second-order class.}

The uniqueness of a solution of the Schrödinger equation in the homogeneous case for both incompressible and compressible ﬂuids was obtained in [20] by using the method of doubling variables. In the case of a rectangular dam wet at the bottom and dry near to the top, the uniqueness was obtained in [17] and [35] by the ﬁxed point theory (as one of the main tools in this subject; see [21,25]) with respect to the Schrödinger operator, respec-tively, in homogeneous and heterogeneous porous media (see [28]). For the evolution free boundary problem in theory of the Schrödinger equation, we refer to [8,15].

As an application of the Phragmén–Lindelöf method related to a second-order bound-ary value problem with respect to the Schrödinger operator, in this paper, we consider the weak formulation of an evolution dam problem with heterogeneous coeﬃcients (1.3) in ג. We establish the uniqueness of the solution for this problem with respect to the Schrödinger operator. This uniqueness result is new in the general framework of a het-erogeneous and bounded rectangular domain.

2 Main result and its proof

In this section, we obtain our main results that a solution of problem with respect to the Schrödinger operator (1.3) is unique. We assume that

χ∈C1[0,W]. (2.1)

Theorem 2.1 Let(f1,ω1)and(f2,ω2)be two solutions of (1.3).Then

P χ(t1)

f1(t,s) –f2(t,s)

+

t2+

1 –ω2(t,s)

ω{f1>f2}

+1 –ω2(t,s)

+1 –f2t2(t,s)

ω_{f1>0}ιςt2dt ds≤0, (2.2)

where

(4)

Proof Note that

(f1,ω1) : (t,s,x,y)→

f1(t,s),ω1(x,y)

,

(f2,ω2) : (t,s,x,y)→

f2(t,s),ω2(x,y)

.

Deﬁne

ψ(t,s,x,y) =ς t1+x1 3 ,

t2+x2

3

ι y+s 3

ρ1,σ

t1–x1

3

ρ2,σ

t2–x2

3

ρ3,σ

y–s

3

for (t,s,x,y)∈P×P, where

ρ1,σ(r) = 1 σρ1

r

σ

,

ρ2,σ(r) = 1 σρ2

r

σ

,

ρ3,σ(r) = 1 σρ3

r

σ

withρ1,ρ2,ρ3∈D(R),ρ1,ρ2,ρ3≥0,supp(ρ1),supp(ρ2),supp(ρ3)⊂(–1, 1).

So

ψ(·,·,x,y)∈D(P), (2.3) ψ(t,s,·,·)∈D(P), (2.4) where (x,y)∈Pand (t,s)∈P.

Deﬁne

ϑ(t,s,x,y) =max (f1(t,s) –f2(t,s))

+

,ψ(t,s,x,y)

, (2.5)

whereis a positive real number (see [22]).

By applying the ﬁxed point theory to (f1,ω1) withk=f2(x,y) andς(t,s) =ϑ(t,s,x,y) for

almost every (x,y)∈Pwe know that

P

χ(t1)(f1t2+ω1)ϑt2dt ds= 0, (2.6)

which, together withf1.(1 –ω1) = 0 a.e. inP, gives that

ω1χ(t1) max

(f1–f2)+

,ψ

t2

=χ(t1) max

(f1–f2)+

,ψ

t2

a.e. inP.

Meanwhile, (2.6) also yields that

P

(5)

and

P×P

χ(t1)(f1t2+ 1)ϑt2d= 0, (2.7)

whered=dt ds dx dy.

Now we apply the ﬁxed point theory to (f2,ω2) with

k=f1(t,s), ς(x,y) =ϑ(t,s,x,y)

to obtain that

P

χ(x1)(f2x2+ω2) ϑ–min f1

,ψ

x2

dy ds= 0.

So

P×P

χ(x1)(f2x2+ω2) ϑ–min f1

,ψ

x2

dy ds dt ds= 0. (2.8)

By subtracting (2.8) from (2.7) we have that

P×P

χ(t1)f1t2ϑt2–χ(x1)f2x2ϑx2

+χ(t1)ϑt2–ω2χ(x1)ϑx2

d

–

P×P

χ(x1)(f2x2+ω2)min f1

,ψ

x2

d= 0. (2.9)

It follows from (2.3)–(2.5) that

P×P

χ(t1)f1t2ϑx2d= 0, (2.10)

P×P

χ(x1)f2x2ϑt2d= 0, (2.11)

P×P

χ(t1)(∂t2+∂x2)ϑd= 0, (2.12)

P×P

ω2χ(x1)ϑt2d= 0, (2.13)

P×P

χ(x1)(f2x2+ω2)min f1

,ψ

t2

d= 0, (2.14)

P×P

χ(t1)(∂t2+∂x2)min f1

,ψ

(6)

Combining (2.10)–(2.15) and (2.9), we get that

P×P

χ(t1)(∂t2+∂x2)f1–χ(x1)(∂t2+∂x2)f2

(∂t2+∂x2)ϑ

+χ(t1) –ω2χ(x1)

(∂t2+∂x2)ϑ

d

+

P×P

χ(t1) –χ(x1)(∂t2+∂x2)f2

(∂t2+∂x2)min f1 ,ψ d +

P×P

χ(t1) –ω2χ(x1)

(∂t2+∂x2)min f1

,ψ

d= 0. (2.16)

Put

t+x

3 =z,

t–x

3 =ξ,

y+s

3 =κ,

y–s

3 =τ. (2.17)

Note that (z,κ)∈Pand

(ξ,τ)∈ –W 3 ,

W

3

× –w 3,

w

3

× –M 3 ,

M

3

:=ג1× – M

3,

M

3

:=P1.

We have that

P×P1

χ(z1+ξ1)f1z2(z+ξ,κ+τ)

–χ(z1–ξ1)f2z2(z–ξ,κ–τ)

ϑz2dμ

+

P×P1

χ(z1+ξ1) –ω2(z–ξ,κ–τ)χ(z1–ξ1)

ϑz2dμ

+

P×P1

χ(z1+ξ1) –ω2(z–ξ,κ–τ)χ(z1–ξ1)

×min f1 ,ψ z2 dμ +

P×P1

χ(z1+ξ1) –f2z2(z–ξ,κ–τ)χ(z1–ξ1)

×min f1 ,ψ

z2

dμ= 0 (2.18)

from (2.16)–(2.17), wheredμ=dz dκdξdτ. Put

I,σ=

P×P1

ϑz2dz dκdξdτ,

J,σ=

P×P1

χ(z1+ξ1) –ω2(z–ξ,κ–τ)χ(z1–ξ1)

(7)

K1

,σ =

P×P1

χ(z1+ξ1) –ω2(z–ξ,κ–τ)χ(z1–ξ1)

×min f1 ,ψ

z2 dμ,

K2

,σ =

P×P1

χ(z1+ξ1) –f2z2(z–ξ,κ–τ)χ(z1–ξ1)

×min f1 ,ψ

z2 dμ.

Lemma 2.2

lim σ→0

lim →0J,σ

=

P

ιω{f1>f2}χ(z1)

1 –ω2(z,κ)

ςz2dz dκ, (2.19)

lim σ→0

lim →0K

1 ,σ = P

ιω{f1>0}χ(z1)

1 –ω2(z,κ)

ςz2dz dκ, (2.20)

lim σ→0

lim →0K

2 ,σ = P

ιω_{f1>0}χ(z1)

1 –f2z2(z,κ)

ςz2dz dκ. (2.21)

Proof Set

J,σ=

P×P1

χ(z1+ξ1) –χ(z1–ξ1)

ϑz2dμ

+

P×P1

χ(z1–ξ1)

1 –ω2(z–ξ,κ–τ)

ϑz2dμ

:=J1,σ+J2,σ. (2.22)

Combining (2.3)–(2.5) and the fact thatχ(z1+ξ1) –χ(z1–ξ1) does not depend onz2,

we have that

J1,σ = 0.

So

lim σ→0

lim →0J

1

,σ

= 0. (2.23)

Put

lim σ→0

lim →0J

2 ,σ = P ιχ(z1)

1 –ω2(z,κ)

ςz2dz dκ. (2.24)

Set

A=

(f1–f2)+>ψ

and

(8)

We know that

J2

,σ=

Bχ(z1–ξ1)

1 –ω2(z–ξ,κ–τ)

f1–f2

z2 dμ +

Aχ(z1–ξ1)

1 –ω2(z–ξ,κ–τ)

ψz2dμ

:=J2,1,σ +J2,2,σ. (2.25)

By applying (2.17) and that (1 –ω2)f2= 0 andf1x2= 0 a.e. inP(see [1,41]), we have that

J2,1

,σ =

B

χ(x1)

1 –ω2(x,y)

f1t2

d.

By applying (2.3) and that the function (x,y)→χ(x1)(1 –ω2(x,y)) does not depend on t2we have that

J2,1

,σ =

P×P χ(x1)

1 –ω2(x,y) min f1 ,ψ t2 d –

Aχ(x1)

1 –ω2(x,y)

ψt2dt ds dy ds

=

P×P χ(x1)

1 –ω2(x,y) min f1 ,ψ t2 d –

P×P χ(x1)

1 –ω2(x,y)

ψt2d

+

Bχ(x1)

1 –ω2(x,y)

ψt2d

=

Bχ(x1)

1 –ω2(x,y)

ψt2d.

Taking into account the fact thatlim→0|B|= 0 (see [39]), we know that lim

→0J 2,1

,σ = 0,

which yields that

lim σ→0

lim →0J

2,1

,σ

= 0. (2.26)

To estimateJ2,2

,σ, we obtain that

lim σ→0

lim →0J

2,2 ,σ = P

χ(z1)ω{f1>f2}

1 –ω2(z,κ)

ςz2ιdz dκ (2.27)

by passing to the limit as→0 and then asσ→0.

Hence, combining (2.26)–(2.27), we obtain (2.24) by letting→0 andσ→0 in (2.25). Now we pass successively to the limit in (2.22) as→0 and then asσ→0, and using (2.23)–(2.24), we obtain (2.19). Finally, arguing as in the proof (2.19), we obtain (2.20) and

(9)

Lemma 2.3

lim σ→0

lim →0I,σ

≥

P ιχ(z1)

f1(z,κ) –f2(z,κ)

+

z2ςz2dz dκ. (2.28)

Proof Set

I,σ=

A

ψz2dμ

+

B

f1–f2

z2 dμ

:=I1_,_σ+I2_,_σ. (2.29)

Note thatI2

,σ can be rewritten as

I2

,σ= 1

B

χ(z1+ξ1)f1z2(z+ξ,κ+τ)f1z2(z+ξ,κ+τ)

+χ(z1–ξ1)f2z2(z–ξ,κ–τ)f2z2(z–ξ,κ–τ)

dz dκdξdτ

–

Bχ(z1–ξ1)f2z2(z–ξ,κ–τ)f1z2(z+ξ,κ+τ)dμ

–

B(χ(z1+ξ1)f1z2(z+ξ,κ+τ)f2z2(z–ξ,κ–τ)dμ

:=I2,1,σ–I2,2,σ –I2,3,σ.

It follows from (1.1) that

I2

,σ≥–I

2,2

,σ–I

2,3

,σ. (2.30)

Let us estimateI2,2

,σ. It follows from (2.3) and (2.5) that

I2,2

,σ=

Bχ(x1)f2x2(x,y)

f1(t,s) –f2(x,y)

t2

dt ds dy ds

=

P×P

χ(x1)f2x2(x,y)ϑt2d

–

Aχ(x1)f2x2(x,y)ψt2d

=

P×P

χ(x1)f2x2(x,y)ϑt2d

–

P×P

(10)

+

Bχ(x1)f2x2(x,y)ψt2d

=

B

χ(x1)f2x2(x,y)ψt2d.

Taking into account thatlim→0|B|= 0 (see [39]), we obtain that

lim →0

I2,2

,σ

= 0. (2.31)

Similarly,

lim →0

I2,3

,σ

= 0. (2.32)

Combining (2.31)–(2.32), we obtain that

lim →0

I2

,σ

≥0 (2.33)

by passing to the limit as→0 in (2.30). Let us estimateI1

,σ. We obtain that

lim →0

I1

,σ

=

P×P1

ω{f1>f2}

ψz2dμ,

which yields that

lim →0

I1

,σ

=

P×P1

ω{f1>f2}χ(z1+ξ1)

f1(z+ξ,κ+τ)

–f2(z–ξ,κ–τ)

z2ψz2dμ

+

P×P1

ω{f1>f2}

χ(z1+ξ1)

–χ(z1–ξ1)

f2z2(z–ξ,κ–τ)ψz2dμ

=Iσ1,1+Iσ1,2. (2.34)

By applying (2.1) and taking into account the fact thatsupp(ρ1,σ)⊂(–σ,σ) (see [39]), we obtain that

_I1,2

σ ≤C

P×P1

|ξ1||f2z2||ςz2|ιρ1,σ(ξ1)ρ2,σ(ξ1)ρ3,σ(τ)dμ

≤σC

P×P1

|f2z2||ςz2|ιρ1,σ(ξ1)ρ2,σ(ξ1)ρ3,σ(τ)dμ

:=σCWσ

(11)

So

lim σ→0I

1,2

σ = 0. (2.35)

ForI1,1

σ , we have that

lim σ→0I

1,1

σ =

P ιχ(z1)

f1(z,κ) –f2(z,κ)

+

z2ςz2dz dκ (2.36)

by passing to the limit asσ→0.

Combining (2.35)–(2.36), we obtain that

lim σ→0

lim →0I

1

,σ

=

P ιχ(z1)

f1(z,κ) –f2(z,κ)

+

z2ςz2dz dκ (2.37)

by lettingσ→0 in (2.34).

Finally, we pass successively to the limit in (2.29) as→0 and then asσ→0, and using (2.33) and (2.37), we obtain (2.28).

Now, using Lemmas2.2and3.1and letting successively→0 andσ→0 in (2.18), we obtain (2.2). This completes the proof of Theorem2.1.

3 An application

In this section, we establish the existence of nontrivial positive solutions of (1.3) withς= ς∗by Theorem2.1. To this end, we ﬁrst construct a pair of proper super- and subsolutions of (1.3) withς=ς∗(see [23,27]). We deﬁne the continuous functions (see [26,30,34,38])

J+(ζ) =J0,

J–(ζ) =

⎧ ⎨ ⎩

J0–peλ3ζ, ζ<ζ1,

e–λ4ζ_, _ζ_≥_ζ

1,

K+(ζ) =

⎧ ⎨ ⎩

– ζeσ∗ζ_, _ζ _<_ζ

2,

(ϕ_γ – 1)J0, ζ ≥ζ2,

K–(ζ) =

⎧ ⎨ ⎩

– ζeσ∗ζ_–_L_(–_ζ₎1/2_eσ∗ζ_, _ζ _<_ζ

3,

0, ζ ≥ζ3,

where

τ1=min

σ∗ 2 ,

α∗ 2d1

, τ2=

ϕ/d1,

andζ1,ζ2,ζ3∈R,p>J0,> 0,> 0,L> 0 will be clariﬁed later.

Lemma 3.1 Assume thatϕ>γ.Then(1.3)withς=ς∗admits a solution satisfying

(12)

Proof Deﬁne

m:=

_∞

0

RI(t,s)e

λ∗(t–c∗s)_{dt ds}_,

n:=

∞

0

RI(t,s)

t–α∗seλ∗(t–c∗s)_{dt ds}_.

It is easy to see thatm,nare bounded and

ζm+n≤(ζ+K)m–α∗

∞

0

RI(t,s)se

λ∗(t–c∗s)_{dt ds}_,

and soζm+n< 0 for allζ< –K. Moreover, Lemma2.3indicates that

d2λ∗2–α∗λ∗+ϕm–γ = 0, 2d2λ∗–c∗+ϕn= 0.

Note that ifh1≥0,h2≥0 withh1+h2> 0, then h1h2

h1+h2 ≤

min{u,v}.

Ifζ<ζ2, thenI+(ζ) = – ζeσ ∗_ζ

, and it suﬃces to prove that

α∗K₊(ζ)≥d2K+(ζ) +ϕ(I∗K+)(ζ) –γK+(ζ). (3.1)

Note that

(I∗K+)(ζ) =

∞

0

RI(t,s)K+

ζ –t–c∗sdt ds

≤

_∞

0

+∞

ζ–ζ∗–α∗s

I(t,s)I+

ζ–t–α∗sdt ds

= –

∞

0

RI(t,s)

ζ–t–c∗seλ∗(ζ–t–c∗s)_{dt ds}

= –

∞

0

RI(t,s)

ζ+t–c∗seλ∗(ζ+t–c∗s)dt ds

= – ζeλ∗ζ

∞

0

RI(t,s)e

λ∗(t–c∗s)_{dt ds}

–eλ∗ζ

∞

0

RI(t,s)

t–c∗seλ∗(t–c∗s)dt ds

= – ζeλ∗ζ_m_–_eλ∗ζ_n_.

It is obvious that (3.1) holds in the case where the following inequality holds.

α∗K₊(ζ)≥d2K+(ζ) –ϕ ζe

σ∗ζ

m–ϕeσ∗ζn–γK+(ζ).

It follows that

K

+(ζ) = –e

σ∗ζ

1 +σ∗ζ,

K

+(ζ) = –e

σ∗ζ

(13)

for anyζ <ζ2by Lemma2.2and the deﬁnition ofI+(ζ).

So

α∗K₊(ζ) –d2K+(ζ) +ϕ ζeσ ∗_ζ

m+ϕeσ∗ζ_n₊_γ_K

+(ζ)

=Λλ

σ∗,α∗eσ∗ζ₊_Λ_σ∗_,_α∗_ζ_eσ∗ζ _{= 0,}

which yields (3.1). Putζ>ζ2. Then

ϕJ+(ζ)(I∗K+)(ζ) J+(ζ) + (I∗K+)(ζ)

≤ ϕJ0(

ϕ γ – 1)J0

J0+ (ϕγ – 1)J0 =γ ϕ

γ – 1

J0.

Denote

p=J0eτ1(K–ζ ∗₎

+sup ζ<0

–ϕe(σ∗–τ1)ζ₍_ζ_m₊_n₎ α∗τ1–d1τ12

and

J0–peλ3ζ=e–λ4ζ.

Ifζ<ζ1, then

J–(ζ) =J0–peτ1ζ> 0

and

α∗J_–(ζ)≤d1J–(ζ) –ϕ(I∗I+)(ζ),

whereζ<ζ1.

It follows that

(I∗K+)(ζ) =

∞

0

RI(t,s)I+

≤

∞

0

+∞

ζ–ζ∗–α∗s

I(t,s)I+

= – ζeσ∗ζ_m_–_eσ∗ζ_n

sinceζ<ζ1≤ζ∗–K.

To verify that

α∗J_–(ζ)≤d1J–(ζ) +ϕ ζeσ ∗_ζ

m+ϕeσ∗ζ_n_,

by simple calculations, we have

–α∗τ1peλ3ζ ≤–d1λ23peλ3ζ +ϕ ζeσ ∗_ζ

m+ϕeσ∗ζ_n_, _ζ_<_ζ

1,

(14)

It suﬃces to conﬁrm that

α∗J_–(ζ)≤d1J–(ζ) –ϕJ–(ζ),

which is equivalent to

–α∗λ4e–λ4ζ ≤d1λ24e–

λ4ζ _–_ϕ_e–λ4ζ_,

and this is also evident by the deﬁnition ofλ4.

LetL≥M1≥√–ζ2be such that J–(ζ)≥J0/2,

ζ <ζ3:= – L

2

≤ζ2

for> 0 andζ2< 0.

It is obvious that the deﬁnition ofζ3implies that

K–(ζ)≤K+(ζ)

for allζ ∈R. Ifζ<ζ3, then

K–(ζ) = – ζeλ ∗_ζ

–L(–ζ)1/2eλ∗ζ≤– ζeλ∗ζ=K+(ζ),

which yields that

(I∗K–)(ζ) =

∞

0

RI(t,s)K–

ζ–t–c∗sdt ds

≤

_∞

0

RI(t,s)K+

ζ–t–c∗sdt ds

= – ζeλ∗ζm–eλ∗ζn.

So

ϕJ–(ζ)(I∗K–)(ζ) J–(ζ) + (I∗K–)(ζ)

–ϕ(I∗K–)(ζ)

≥ ϕJ20(I∗K–)(ζ)

J0

2 + (I∗K–)(ζ)

–ϕ(I∗K–)(ζ)

≥–2ϕ

J0

(I∗K–)(ζ)

2

≥–2ϕ

2

J0

e2λ∗ζ(ζm+n)2,

which gives that

c∗K_–(ζ)≤d2K–(ζ) +ϕ(I∗K–)(ζ) –γK–(ζ) –

2ϕ2

J0

(15)

By a simple calculation we have that

K

–(ζ) =K+(ζ) +Le

σ∗ζ

1 2(–ζ)

–1/2_–_σ∗_(–_ζ₎1/2

,

K

–(ζ) =K+(ζ) +Leσ ∗_ζ1

4(–ζ)

–3/2₊_σ∗_(–_ζ₎–1/2_–_σ∗2

(–ζ)1/2

.

Since –ζ+y+α∗s≥0 for anyζ <ζ3,|y|<K, ands≥0, there existsθ∈(0, 1) such that

–ζ+t+α∗s1/2

= (–ζ)1/2+1 2(–ζ)

–1/2_t₊_α∗_s_–1

8

–ζ+θt+α∗s–3/2t+α∗s2

≤(–ζ)1/2+1 2(–ζ)

–1/2_t₊_α∗_s

by applying the Taylor theorem (see [12,31]), which yields that

(I∗K–)(ζ) =

∞

0

RI(t,s)I–

=

∞

0

K

–K

I(t,s)I–

≥

∞

0

K

–K

I(t,s)–ζ–t–α∗seσ∗(ζ–t–α∗s)

–L–ζ –t–α∗s1/2eσ∗(ζ–t–α∗s)_{dt ds}

≥–

_∞

0

K

–K

I(t,s)ζ –t–α∗seσ∗(ζ–t–α∗s)dt ds

–L

∞

0

K

–K

I(t,s)

(–ζ)1/2+1 2(–ζ)

–1/2_t₊_α∗_s_eσ∗(ζ–t–α∗s)_{dt ds}

= – ζeσ∗ζm–eσ∗ζn–L(–ζ)1/2eσ∗ζm+1 2L(–ζ)

–1/2_eσ∗ζ

n.

So

α∗K₊(ζ) +Lα∗eσ∗ζ

1 2(–ζ)

–1/2_–_σ∗_(–_ζ₎1/2

≤d2K+(ζ) +d2Leσ ∗_ζ1

4(–ζ)

–3/2₊_σ∗_(–_ζ₎–1/2_–_σ∗2_(–_ζ₎1/2

–ϕ ζeσ∗ζm–ϕeσ∗ζn–ϕL(–ζ)1/2eσ∗ζm+ϕ 2L(–ζ)

–1/2_eσ∗ζ

n

–γK+(ζ) +γL(–ζ)1/2eσ ∗_ζ

–2ϕ

2

J0

e2σ∗ζ(ζm+n)2,

which is true provided that

d2Leσ ∗_ζ1

4(–ζ)

–3/2_–2ϕ2 J0

(16)

Put

M2:=sup

ζ<0

8ϕ2(ζm+n)2_(–_ζ₎3/2_eσ∗ζ d2J0

+ 1

for anyζ <ζ3, whereL:=M1+M2. Whenζ>ζ3, it is straightforward to show the required

result. The proof is complete.

4 Conclusions

In this paper, we further studied the Phragmén–Lindelöf method related to a second-order order boundary value problem with respect to the Schrödinger operator. We also pre-sented some mathematical consequences of the method including a stability result. The main technical tools used to develop the mathematical analysis are local and global bifur-cation, monotonicity techniques, the augmented Phragmén–Lindelöf method, blow-up arguments, and some techniques used in the previous works. As an application, we proved the uniqueness of a solution for the deﬁnite problem of a parabolic variational inequality.

Acknowledgements

The work was supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region of China (No. 2019D01A04).

Funding Not applicable.

Abbreviations Not applicable.

Availability of data and materials Not applicable.

Competing interests

The author declares that there are no competing interests.

Authors’ contributions

The author completed the paper and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 16 July 2019 Accepted: 24 September 2019

References

1. Acerbi, E., Fusco, N.: A transmission problem in the calculus of variations. Calc. Var. Partial Diﬀer. Equ.2, 1–16 (1994) 2. Amann, H.: Nonlinear elliptic equations with nonlinear boundary conditions. In: Eckhaus, W. (ed.) New Developments

in Diﬀerential Equations. Math Studies, vol. 21, pp. 43–63. North-Holland, Amsterdam (1976) 3. Anderson, R.M., May, R.M.: Population Biology of Infectious Diseases. Springer, Berlin (1982)

4. Anderson, R.M., May, R.M., Anderson, B.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1992)

5. Andreu, F., Mazón, J.M., Toledo, J., Rossi, J.D.: Porous medium equation with absorption and a nonlinear boundary condition. Nonlinear Anal.49, 541–563 (1992)

6. Carrillo, J.: On the uniqueness of the solution of the evolution dam problem. Nonlinear Anal., Theory Methods Appl.

22, 573–607 (1994)

7. Carrillo, J., Gilardi, G.: La vitesse de propagation dans le problème de la digue. In: Ann. Fac. Sci. Toulouse Math., vol. XI, 3, pp. 7–28 (1990)

8. Chen, X.F., Friedman, A.: A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal.32, 778–800 (2000)

9. Chipot, M., Fila, M., Quittner, P.: Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Math. Univ. Comen.60, 35–103 (1991)

10. Dibenedetto, E., Friedman, A.: Periodic behaviour for the evolutionary dam problem and related free boundary problems. Commun. Partial Diﬀer. Equ.11, 1297–1377 (1986)

(17)

12. Duca, D.I.: Properties of the intermediate point from the Taylor’s theorem. Math. Inequal. Appl.12(4), 763–771 (2009) 13. Gala, S., Liu, Q., Ragusa, M.A.: Logarithmically improved regularity criterion for the nematic liquid crystal ﬂows inB–1

∞,∞

space. Comput. Math. Appl.65(11), 1738–1745 (2013)

14. Gilardi, G.: A new approach to evolution free boundary problems. Commun. Partial Diﬀer. Equ.4, 1099–1123 (1979) 15. Grigor’yan, A.A., Losev, A.G.: On the dimension of the solution spaces of the stationary Schrödinger equation on

noncompact Riemannian manifolds. Mat. Fiz. Komp’yut. Model.40, 34–42 (2017)

16. Kermack, W.O., M’Kendrick, A.G.: The design and interpretation of experiments based on a four-fold table: the statistical assessment of the eﬀect of treatment. Proc. R. Soc. Edinb.60, 362–375 (1940)

17. Lin, Z.: A free boundary problem for a predator–prey model. Nonlinearity20, 1883–1892 (2007) 18. Lions, J.L.: Remarks on Darcy’s law, the Brooke Benjamin special issue. IMA J. Appl. Math.46, 29–38 (1991) 19. López-Gómez, J., Marquez, V., Wolanski, N.: Blow up results and localization of blow up points for the heat equations

with a nonlinear boundary conditions. J. Diﬀer. Equ.92, 384–401 (1991)

20. Meng, F.: Maximum modulus estimation to the solution of quasi-linear equations with variable exponents. J. Syst. Sci. Complex.16, 446–452 (2003)

21. Mokhtarzadeh, M.R., Pournaki, M.R., Razani, A.: A note on periodic solutions of Riccati equations. Nonlinear Dyn.

62(1–2), 119–125 (2010)

22. Moser, J.: A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic diﬀerential equations. Commun. Pure Appl. Math.13, 457–468 (1960)

23. Murray, J.D.: Mathematical Biology. I. An Introduction, 3rd edn. Interdisciplinary Applied Mathematics, vol. 17. Springer, New York (2002)

24. Polidoro, S., Ragusa, M.A.: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam.24(3), 1011–1046 (2008)

25. Pournaki, M.R., Razani, A.: On the existence of periodic solutions for a class of generalized forced Liénard equations. Appl. Math. Lett.20(3), 248–254 (2007)

26. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal.7, 487–513 (1971) 27. Rass, L., Radcliﬀe, J.: Spatial Deterministic Epidemics. Mathematical Surveys and Monographs, vol. 102. Am. Math.

Soc., Providence (2003)

28. Razani, A.: Subsonic detonation waves in porous media. Phys. Scr.94, 8 (2019)

29. Rossi, J.D.: The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition. Acta Math. Univ. Comen.67, 343–350 (1998)

30. Rossi, J.D.: Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem. In: Stationary Partial Diﬀerential Equations. Vol. II. Handb. Diﬀer. Equ., pp. 311–406. Elsevier, Amsterdam (2005)

31. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

32. Russo, A., Tartaglione, A.: Strong uniqueness theorems and the Phragmén–Lindelöf principle in nonhomogeneous elastostatics. J. Elast.102(2), 133–149 (2011)

33. Sattinger, D.H.: Topics in Stability and Bifurcation Theory. Lecture Notes in Mathematics, vol. 309. Springer, Berlin (1973)

34. Serrin, J.: The problem of Dirichlet for quasilinear elliptic diﬀerential equations with many independent variables. Philos. Trans. R. Soc. Lond. Ser. A264, 413–496 (1969)

35. Shehu, I.: An application of Phragmén–Lindelöf’s theorem in an abstract elliptic equation of second order. Matematiche64, 57–66 (2015)

36. Stein, E.M.: Singular Integrals and Diﬀerentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)

37. Turaev, R.N.: An internal boundary value nonlocal problem with a free boundary for a quasilinear parabolic equation. Uzbek. Mat. Zh.4, 95–99 (2012)

38. Walter, W.: On existence and nonexistence in the large of solutions of parabolic diﬀerential equations with a nonlinear boundary condition. SIAM J. Math. Anal.6, 85–90 (1975)

39. Willem, M.: Minimax Theorems. Progress in Nonlinear Diﬀerential Equations and Their Applications, vol. 24. Birkhäuser, Boston (1996)

40. Yao, Y., Zhang, W., Kang, J., Ahmed, A.: Stationary solutions of a class of nonlinear Schrödinger equations. Monatshefte Math.173, 593–603 (2014)

41. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR, Ser. Mat.