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The Singular Perturbation Problem for a Class of Generalized Logistic Equations Under Non-classical Mixed Boundary Conditions

  • Sergio Fernández-Rincón ORCID logo and Julián López-Gómez ORCID logo EMAIL logo
Published/Copyright: October 31, 2018
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Advanced Nonlinear Studies
From the journal Advanced Nonlinear Studies Volume 19 Issue 1

Abstract

This paper studies a singular perturbation result for a class of generalized diffusive logistic equations, d⁢ℒ⁢u=u⁢h⁢(u,x), under non-classical mixed boundary conditions, ℬ⁢u=0 on ∂⁡Ω. Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To overcome the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of ∂⁡Ω through the regularity of the associated conormal projections and conormal distances. This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, [39], Gilbarg and Trudinger, [21], Krantz and Parks, [27], Foote, [18] and Li and Nirenberg [28] concerning the regularity of the inner distance function to the boundary.

Keywords: Singular Perturbation; Positive Solution; Generalized Logistic Equation; Non-classical Mixed Boundary Conditions; Conormal Vector Field; Conormal Projection; Conormal Distance; Regularity
MSC 2010: 35B25; 35B09; 35J25; 35Q92

1 Introduction

The main goal of this paper is to analyze the limiting behavior as d↓0 of the positive solutions of

(1.1) { d ⁢ ℒ ⁢ u = u ⁢ h ⁢ ( u , x ) in  ⁢ Ω , ℬ ⁢ u = 0 on  ⁢ ∂ ⁡ Ω ,

where Ω is a bounded domain of ℝN, N≥1, d>0 is a positive constant, and the differential operator ℒ is uniformly elliptic in Ω and has the form

(1.2) ℒ = - div ( A ∇ ⋅ ) + b ∇ + c ,

with A∈ℳNsym⁢(𝒞1⁢(Ω¯)), b∈ℳ1×N⁢(𝒞⁢(Ω¯)) and c∈𝒞⁢(Ω¯). Given any Banach space X and two integers n,m≥1, ℳn×m⁢(X) stands for the vector space of the matrices with n rows and m columns with entries in X. Naturally, we set ℳn⁢(X):=ℳn×n⁢(X), and ℳnsym⁢(X) denotes the subset of symmetric matrices.

Except in Section 2, ∂⁡Ω is assumed to be an (N-1)-dimensional manifold of class 𝒞2 consisting of finitely many (connected) components

Γ D j , 1 ≤ j ≤ n D , Γ R k , 1 ≤ k ≤ n R ,

for some integers nD, nR≥1, and we denote by

Γ D := ⋃ j = 1 n D Γ D j , Γ R := ⋃ j = 1 n R Γ R j ,

the Dirichlet and Robin portions of ∂⁡Ω=ΓD∪ΓR. It should be noted that either ΓD or ΓR might be empty. Associated with this decomposition of ∂⁡Ω, arises in a rather natural way the boundary operator ℬ defined by

ℬ u = { D ⁢ u := u on  ⁢ Γ D , R ⁢ u := ∂ ⁡ u ∂ ⁡ 𝝂 + β ⁢ u on  ⁢ Γ R ,    for every  u ∈ W 2 , p ( Ω ) , p > N ,

where β∈𝒞⁢(∂⁡Ω), 𝐧 stands for the outward normal vector field along ∂⁡Ω, and 𝝂:=A⁢𝐧 is the conormal vector field associated to ℒ.

As far as concerns the nonlinearity of (1.1), the function h⁢(u,x) is assumed to satisfy the following:

  1. h : ℝ × Ω ¯ → ℝ is of class 𝒞1 in u≥0 and continuous in x∈Ω¯.

  2. ∂ u ⁡ h ⁢ ( u , x ) < 0 for all u>0 and x∈Ω¯.

In addition, throughout this paper, we will impose that, for some d>0,

  1. there exists a positive constant M>0 such that h⁢(M,x)<d⁢c⁢(x) for all x∈Ω¯.

In particular, we eventually can assume (H3) with d=0, i.e., that

  1. there exists a positive constant M>0 such that h⁢(M,x)<0 for all x∈Ω¯.

Note that (H4) implies (H3) for sufficiently small d>0, regardless the sign of c⁢(x). A prototypic example of admissible h, for which (1.1) becomes a generalized diffusive logistic equation, is given by

h ⁢ ( u , x ) = ℓ ⁢ ( x ) - a ⁢ ( x ) ⁢ f ⁢ ( u ) , u ∈ ℝ , x ∈ Ω ¯ ,

where ℓ∈𝒞⁢(Ω¯) can change of sign, a∈𝒞⁢(Ω¯) and f∈𝒞1⁢(ℝ) satisfy minΩ¯⁡a>0, f⁢(0)=0, f′⁢(u)>0 for all u>0 and limu↑∞⁡f⁢(u)=+∞. For this choice, it is easily seen that (H1) and (H2) hold. As far as concerns (H3), note that, for every d>0,

h ⁢ ( M , x ) = ℓ ⁢ ( x ) - a ⁢ ( x ) ⁢ f ⁢ ( M ) ≤ max Ω ¯ ⁡ ℓ - f ⁢ ( M ) ⁢ min Ω ¯ ⁡ a < d ⁢ min Ω ¯ ⁡ c , x ∈ Ω ¯ ,

provided M=M⁢(d)>0 is sufficiently large, because f⁢(M)↑∞ as M↑∞. Thus, (H3) holds for all d>0. Moreover, by taking a sufficiently large M>0 so that f⁢(M)>maxΩ¯⁡ℓ/minΩ¯⁡a, it is clear that (H4) also holds.

Under the general conditions (H1), (H2) and (H4), it is easily seen that the maximal non-negative solution of the non-spatial equation u⁢h⁢(u,x)=0,

Θ h ⁢ ( x ) := { 0 if  ⁢ h ⁢ ( ξ , x ) ⁢ < 0 ⁢  for all  ⁢ ξ > ⁢ 0 , ξ if  ⁢ ξ > 0 ⁢  exists such that  ⁢ h ⁢ ( ξ , x ) = 0 ,

is continuous in Ω¯. Actually, for every x∈Ω¯, Θh⁢(x) is the unique non-negative linearly stable, or linearly neutrally stable, steady state of the ordinary differential equation

u ′ ⁢ ( t ) = u ⁢ ( t ) ⁢ h ⁢ ( u ⁢ ( t ) , x ) , t ≥ 0 ,

which is throughout referred as the kinetic model associated to (1.1).

According to the next theorem, which is the main existence result of this paper, for sufficiently small d>0, (1.1) possesses, at most, one positive solution. Throughout this paper, for any given V∈𝒞⁢(Ω¯), we will denote by σ1⁢[d⁢ℒ+V;ℬ,Ω] the principal eigenvalue of the linear eigenvalue problem

{ d ⁢ ℒ ⁢ φ + V ⁢ ( x ) ⁢ φ = σ ⁢ φ in  ⁢ Ω , ℬ ⁢ φ = 0 on  ⁢ ∂ ⁡ Ω ,

i.e., its lowest real eigenvalue. According to [31, Theorem 7.7], it is algebraically simple and it provides us with the unique eigenvalue that is associated with a positive (principal) eigenfunction.

Theorem 1.1.

Assume that h⁢(u,x) satisfies (H1), (H2) and (H3) for some d>0. Then problem (1.1) has a positive solution u∈⋂p>NW2,p⁢(Ω) if and only if

σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] < 0 .

Moreover, it is unique if it exists.

Note that it complements [19, Lemma 3.4]. The main goal of this paper is to establish the next singular perturbation result, where θ{d,h} stands for the maximal non-negative solution of (1.1).

Theorem 1.2.

Assume that h satisfies (H1), (H2) and (H4), and let ΓR+ denote the union of the components of ΓR, where Θh is everywhere positive. Then, for any compact subset K of Ω∪ΓR+∪Θh-1⁢(0),

lim d ↓ 0 ⁡ θ { d , h } = Θ h   uniformly in  ⁢ K .

In other words, the maximal non-negative solution of (1.1) approximates Θh as d↓0 uniformly on compact subsets of Ω∪ΓR+∪Θh-1⁢(0).

To the best of our knowledge, the most pioneering version of this result goes back to [5], where the singular perturbation problem

(1.3) { - d ⁢ Δ ⁢ u = u ⁢ ( 1 - a ⁢ ( x ) ⁢ u 2 ) in  ⁢ Ω , u = 0 , on  ⁢ ∂ ⁡ Ω ,

with Ω and a⁢(x) of class 𝒞∞ and minΩ¯⁡a>0, was analyzed in dimension N≤3. Precisely, in [5], Berger and Fraenkel established that for sufficiently small d>0, problem (1.3) possesses a unique smooth positive solution ud⁢(x), which converges to 1/a⁢(x) as d↓0, outside a boundary layer of width O⁢(d). Moreover, a global continuation of ud in d was performed up to the critical value of the diffusion, where ud bifurcates from u=0. The main technical tool of [5] relies on the method of matched asymptotic expansions, applied to approximate the positive solution. The global existence of the positive solution was derived from some classical results in critical point theory. An abstract version of this singular perturbation result for autonomous equations was given by the same authors in [6]. Two years later, De Villiers [10] sharpened these findings up to cover a general class of 𝒞∞ functions, g⁢(u,x), instead of u-a⁢(x)⁢u3. Almost simultaneously, Fife [16], and Fife and Greenlee [17] extended these results to a general class of nonhomogeneous Dirichlet boundary value problems, including

(1.4) { - d ⁢ div ⁡ ( A ⁢ ( x , d ) ⁢ ∇ ⁡ u ) = g ⁢ ( u , x , d ) in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω ,

with Ω, A⁢(x,d) and g⁢(u,x,d) of class 𝒞∞ and such that, for every x∈Ω¯, the equation g⁢(u,x,0)=0 has a solution u0⁢(x), for which ∂u⁡g⁢(u0⁢(x),x,0)<0. This negativity entails the linearized stability of the equilibrium solution u0⁢(x) of the associated kinetic model

(1.5) u ′ ⁢ ( t ) = g ⁢ ( u ⁢ ( t ) , x , 0 ) , t ≥ 0 ,

for all x∈Ω¯. Much like in [5], the singular perturbations results of [16, 17] are based on a bound for the inverse of the linearization about the formal solution constructed with the matched asymptotic expansion. Fife and Greenlee [17] also analyzed the more general case when g⁢(u,x,0)=0 possesses two 𝒞∞-curves of solutions, u0,1⁢(x) and u0,2⁢(x), x∈Ω¯, which are linearly stable as steady-state solutions of (1.5) and separated away from each other.

Essentially, all these monographs adapted the former asymptotic expansion methods developed in the context of ODEs by the Russian School (e.g., see [7, 40]) to a PDE’s framework. Naturally, working with ODEs many of the underlying technicalities can be easily overcome.

The first papers where some intrinsic techniques of the theory of PDEs, like the method of sub and supersolutions, were used to obtain singular perturbation results were those of Howes [23, 22, 24]. As a result, the previous restrictive regularity assumptions were relaxed. Precisely, Howes [23] considered a general class of problems, including (1.4) with A=I and g⁢(u,x,d)=g⁢(u,x) of class 𝒞m for sufficiently large m≥1. Essentially, assuming that Ω is sufficiently smooth and that, for every x∈Ω¯, g⁢(u0⁢(x),x)=0 for some smooth u0⁢(x) which is linearly stable as an equilibrium of (1.5), Howes found some sufficient conditions for the existence of a classical solution ud of (1.4) such that

lim d ↓ 0 ⁡ u d = u 0   uniformly on compact subsets of  ⁢ Ω .

Almost simultaneously, Howes [22] extended these results to cover the following very special class of Robin problems:

(1.6) { - d ⁢ Δ ⁢ u = g ⁢ ( u , x ) in  ⁢ Ω , ∂ ⁡ u ∂ ⁡ 𝝂 ⁢ ( x ) + β ⁢ ( x ) ⁢ u ⁢ ( x ) = 0 on  ⁢ ∂ ⁡ Ω ,

where β≥0 on ∂⁡Ω and β∈𝒞2,μ⁢(∂⁡Ω) for some μ∈(0,1). As a consequence, e.g., of [22, Theorem 2.1], Howes could infer in [22, Example 2.2] that in the special case when g⁢(u)=u-u3, u0,±≡±1 are I0-stable zeroes of g⁢(u,x)=0, because g′⁢(±1)=-2<0, and therefore (1.6) has two solutions ud,±⁢(x), such that

lim d ↓ 0 ⁡ u d , ± ⁢ ( x ) = ± 1   uniformly in  ⁢ Ω ¯ .

In these papers, the regularity of the support domain Ω is imposed through the existence of a function F∈𝒞2,μ⁢(ℝN;ℝ) such that |∇⁡F⁢(x)|=1 for all x∈∂⁡Ω and

(1.7) Ω = { x ∈ ℝ N : F ⁢ ( x ) < 0 } , ∂ ⁡ Ω = F - 1 ⁢ ( 0 ) .

Incidentally, in the papers of Howes the problem of ascertaining whether, or not, a function F satisfying (1.7) exists, with the required regularity, remained open. Except for some pioneering results of Oleĭnik [35, 36, 37] for linear problems with transport terms, [22] seems to be the first paper dealing with the singular perturbation problem for a semilinear equation under Neumann or (classical) Robin boundary conditions with β≥0. The singular perturbation results of Howes for essential nonlinearities involving transport terms, like those of [22, Sections 3 and 4] and [24], remain outside the general scope of this paper.

Some time later, these pioneering findings were slightly, and occasionally substantially, improved by Angenent [4], De Santi [11], Clément and Sweers [9], and Kelley and Ko [26], among many others, who dealt with the singular perturbation problem under Dirichlet boundary conditions through some comparison techniques based on the synthesis of Amann [1, 2], Sattinger [38] and Matano [33].

As shown by the simplest examples of truly spatially heterogeneous semilinear elliptic equations in the context of population dynamics, the most serious shortcoming of the classical singular perturbation theory is caused by the fact that the curves, u0,j⁢(x), 1≤j≤q, q=1,2, solving the equation g⁢(u,x)=0 must preserve their stability character for all x∈Ω¯, regarded as steady-state solutions of (1.5). For example, even in the simplest case situation when g⁢(u,x) inherits a logistic structure,

g ⁢ ( u , x ) = ℓ ⁢ ( x ) ⁢ u - a ⁢ ( x ) ⁢ u 2

for some functions ℓ,a∈𝒞⁢(Ω¯) such that ℓ⁢(x) changes sign in Ω and minΩ¯⁡a>0, most of the assumptions imposed in the previous references fail to be true. Indeed, although u0,1⁢(x)≡0 and u0,2⁢(x):=ℓ⁢(x)/a⁢(x), x∈Ω¯, might provide us with two smooth curves of g⁢(u,x)=0 for sufficiently smooth ℓ⁢(x) and a⁢(x), it becomes apparent from ∂u⁡g⁢(u,x)=ℓ⁢(x)-2⁢a⁢(x)⁢u that

  1. u 0 , 1 ⁢ ( x ) = 0 is linearly stable, as a steady-state solution of (1.5) if and only if ℓ⁢(x)<0,

  2. u 0 , 2 ⁢ ( x ) = ℓ ⁢ ( x ) / a ⁢ ( x ) is linearly stable if and only if ℓ⁢(x)>0.

Figure 1 Plots of u↦g⁢(u,xi):=ℓ⁢(xi)⁢u-a⁢(xi)⁢u2{u\mapsto g(u,x_{i}):=\ell(x_{i})u-a(x_{i})u^{2}}, i∈{1,2,3}{i\in\{1,2,3\}}, for a function ℓ∈𝒞⁢(Ω¯){\ell\in\mathcal{C}(\bar{\Omega})} that changes sign in Ω with ℓ⁢(x1)>0{\ell(x_{1})>0}, ℓ⁢(x2)=0{\ell(x_{2})=0} and ℓ⁢(x3)<0{\ell(x_{3})<0}. In the central case, (B), u=0{u=0} must be a double zero of g⁢(⋅,x2){g(\,\cdot\,,x_{2})}. In each of these plots we have superimposed the 1-dimensional dynamics of (1.5) on the horizontal axis.
Figure 1

Plots of u↦g⁢(u,xi):=ℓ⁢(xi)⁢u-a⁢(xi)⁢u2, i∈{1,2,3}, for a function ℓ∈𝒞⁢(Ω¯) that changes sign in Ω with ℓ⁢(x1)>0, ℓ⁢(x2)=0 and ℓ⁢(x3)<0. In the central case, (B), u=0 must be a double zero of g⁢(⋅,x2). In each of these plots we have superimposed the 1-dimensional dynamics of (1.5) on the horizontal axis.

Therefore, the curves u0,i⁢(x), i=1,2, cannot satisfy the requirements of the previous references, because they have a different stability character if ℓ⁢(x)≠0. Even considering the ‘mixed interlaced branches’  constructed from u0,1⁢(x) and u0,2⁢(x) through

u ~ 0 , 1 ⁢ ( x ) := max ⁡ { u 0 , 1 ⁢ ( x ) , u 0 , 2 ⁢ ( x ) } , u ~ 0 , 2 ⁢ ( x ) := min ⁡ { u 0 , 1 ⁢ ( x ) , u 0 , 2 ⁢ ( x ) } , x ∈ Ω ¯ ,

it is apparent that u~0,1⁢(x) is linearly stable if and only if ℓ⁢(x)≠0, and hence the classical theory cannot be applied neither, because the linearized stability fails at ℓ-1⁢(0) and, in general, these curves are far from smooth. In these degenerate situations, not previously considered in the specialized literature, Furter and López-Gómez [20] established that the unique positive solution ud of

{ - d ⁢ Δ ⁢ u = u ⁢ ( ℓ ⁢ ( x ) - a ⁢ ( x ) ⁢ u ) in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω ,

satisfies

lim d ↓ 0 ⁡ u d = ℓ + / a = max ⁡ { 0 , ℓ / a } = u ~ 0 , 1   uniformly on compact subsets of  ⁢ Ω

(see [20, Theorem 3.5]), which suggests the validity of the next general principle in the context of (1.1):

Principle of Singular Perturbation (PSP). If for every x∈Ω¯, the associated kinetic problem possesses a unique linearly stable, or linearly neutrally stable, non-negative steady-state solution Θ⁢(x), which is somewhere positive in Ω, then, for sufficiently small d>0, the associated parabolic problem possesses a unique positive steady-state solution θd. Moreover, limd↓0⁡θd=Θ uniformly on any compact subset of Ω, where Θ⁢(x) is continuous.

This is widely confirmed by Theorems 1.1 and 1.2. The same principle was already shown to hold under homogeneous Neumann boundary conditions by Hutson et al. [25, Lemma 2.4], as well as in the context of competitive systems (see [25, Theorem 4.1] and [13, Theorem 1], [15, Theorem 1.2], [14, Theorem 4.1] for some special cases when ℒ=-Δ or b=0).

A problem of a different nature was studied by Nakashima, Ni and Su [34] for the special case when ℒ=-Δ and g⁢(u,x)=a⁢(x)⁢f⁢(u), for the appropriate choices of the functions a⁢(x) and f⁢(u), under Neumann boundary conditions. In such case, the steady-state solutions of (1.5) are spatially homogeneous, though their linearized stabilities, regarded as equilibria of (1.5), vary with the location of x in Ω¯ according to the sign of a⁢(x). In spite of these differences, it turns out that this model also satisfies the Principle of Singular Perturbation formulated above (see [34, Theorem 1.3]).

Our Theorem 1.2 provides us with an extremely general version of all previous existing singular perturbation results for Kolmogorov nonlinearities of the form g⁢(u,x)=u⁢h⁢(u,x), where h⁢(u,x) satisfies (H1), (H2) and (H4). Actually, it is the first general result available for second order uniformly elliptic operators, ℒ, under general mixed boundary conditions of non-classical type. As the general linear existence theory developed in [31, Section 4.6] is only available for operators of the form (1.2), in this paper the principal part of ℒ is required to be in divergence form. Nevertheless, even imposing this restriction, Theorem 1.2 is substantially sharper than most of the previous singular perturbation results for the generalized logistic equation.

The proof of Theorem 1.2 is based on the method of sub and supersolutions, which relies on the theorem of characterization of the Strong Maximum Principle of López-Gómez and Molina-Meyer [32, 30], and Amann and López-Gómez [3]. A comparison argument provides us with a global uniform supersolution of (1.1) on Ω¯, while the construction of the appropriate local subsolutions, combined with a compactness argument, provides us with the necessary lower estimates to get Theorem 1.2. The main technical difficulties that we must overcome in the proof of Theorem 1.2 come from the following facts:

  1. The principal eigenfunctions associated to ℒ in interior balls do not enjoy the nice symmetry properties of the principal eigenfunctions of -Δ, which take the maximum on the center of these balls. This difficulty is overcome through a technical device introduced in [29], which facilitates the construction of local subsolutions in the general non-autonomous case.

  2. A more subtle difficulty relies on the construction of a global supersolution of (1.1) sufficiently close to Θh, which is far from obvious when dealing with general mixed boundary conditions. As no previous singular perturbation result is available under mixed boundary conditions, these difficulties have been overcomed for the first time here.

  3. In our general setting, the coefficient function β⁢(x) can change sign. Thus, we must perform a preliminary change of variables for transforming (1.1) into an equivalent problem of the same nature with β≥0.

The resolution of the technical difficulties sketched in (II) and (III) relies on the next theorem, which might be of independent interest in differential geometry.

Theorem 1.3.

Assume that Ω is an open subdomain of RN such that ∂⁡Ω is a topological (N-1)-manifold. Then, for every integer r≥2, the next assertions are equivalent:

  1. ∂ ⁡ Ω is of class 𝒞 r .

  2. ∂ ⁡ Ω admits an outward vector field 𝝂 ∈ 𝒞 r - 1 ⁢ ( ∂ ⁡ Ω ; ℝ N ) and, for each of these vector fields, there exists an open subset 𝒰 of ℝ N , with ∂ ⁡ Ω ⊂ 𝒰 , and a function Π 𝝂 ∈ 𝒞 r - 1 ⁢ ( 𝒰 ; ∂ ⁡ Ω ) such that

    1. Π 𝝂 ⁢ ( x ) = x for all x ∈ ∂ ⁡ Ω ,

    2. Π 𝝂 ⁢ ( x - λ ⁢ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) ) = Π 𝝂 ⁢ ( x ) for every x ∈ 𝒰 and λ ∈ ℝ such that x - λ ⁢ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) ∈ 𝒰 . Thus,

      ∂ ⁡ Π 𝝂 ∂ ⁡ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) ⁢ ( x ) = 0   for all  ⁢ x ∈ 𝒰 .

    Moreover, the function 𝔡 𝝂 : 𝒰 → ℝ defined by

    (1.8) 𝔡 𝝂 ⁢ ( x ) := { | x - Π 𝝂 ⁢ ( x ) | / | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | if  ⁢ x ∈ 𝒰 ∩ Ω , - | x - Π 𝝂 ⁢ ( x ) | / | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | if  ⁢ x ∈ 𝒰 ∖ Ω ,

    is of class 𝒞 r .

  3. Property (b) holds for some outward vector field 𝝂∈𝒞r-1⁢(∂⁡Ω;ℝN).

  4. ∂ ⁡ Ω admits an outward vector field 𝝂 ∈ 𝒞 r - 1 ⁢ ( ∂ ⁡ Ω ; ℝ N ) and, for each of these vector fields, there exists an open subset 𝒰 of ℝ N , with ∂ ⁡ Ω ⊂ 𝒰 , and a function ψ ∈ 𝒞 r ⁢ ( 𝒰 ; ℝ ) such that ψ ⁢ ( x ) < 0 for all x ∈ Ω ∩ 𝒰 , ψ⁢(x)>0 for all x∈𝒰∖Ω¯ and

    min x ∈ ∂ ⁡ Ω ⁡ ∂ ⁡ ψ ∂ ⁡ 𝝂 ⁢ ( x ) > 0 .

  5. Property (d) holds for some outward vector field 𝝂∈𝒞r-1⁢(∂⁡Ω;ℝN).

  6. There exist an open subset 𝒰 of ℝ N , with ∂ ⁡ Ω ⊂ 𝒰 , and a function Ψ ∈ 𝒞 r ⁢ ( 𝒰 ; ℝ ) such that Ω = { x ∈ 𝒰 : Ψ ⁢ ( x ) < 0 } , ∂⁡Ω=Ψ-1⁢(0), and |∇⁡Ψ⁢(x)|=1 for all x∈∂⁡Ω.

A vector field 𝝂 on ∂⁡Ω is said to be an outward vector field if there exists ε0>0 such that

x + ε ⁢ 𝝂 ⁢ ( x ) ∈ ℝ N ∖ Ω ¯   and   x - ε ⁢ 𝝂 ⁢ ( x ) ∈ Ω

for all x∈∂⁡Ω and 0<ε<ε0. The function Π𝝂, whose existence is established by part (b), will be throughout called the projection onto the boundary along the vector field𝝂, or simply coprojection when 𝝂 is the conormal vector field. Naturally, the distance to the boundary along 𝝂, or conormal distance, is defined through

dist 𝝂 ⁢ ( x , ∂ ⁡ Ω ) := | x - Π 𝝂 ⁢ ( x ) | | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | , x ∈ 𝒰 ,

where |⋅| stands for the Euclidean norm in ℝN. According to Theorem 1.3, ∂⁡Ω is of class 𝒞r if and only if for some outward vector field 𝝂∈𝒞r-1, the function dist𝝂 is of class 𝒞r in 𝒰∖∂⁡Ω. In part (d), the function ψ is given essentially by -dist𝝂 in 𝒰∩Ω. Note that, by the continuity of ψ on 𝒰, ψ⁢(x)=0 for all x∈∂⁡Ω.

According to [31, Lemma 2.1], using a partition of the unity of class 𝒞r, or a cut-off function, the function ψ⁢(x) in part (d), as well as Ψ in part (f), can be assumed to be globally defined in a neighborhood of Ω¯, or even in ℝN, and in such case ψ⁢(x)<0 (resp. Ψ⁢(x)<0) for all x∈Ω and ψ⁢(x)>0 (resp. Ψ⁢(x)>0) for all x∈ℝN∖Ω¯.

Note that (f) is the condition used in some of the classical papers discussed above, with F:=Ψ. It is astonishing that, in spite of the equivalence between (a) and (f), yet the existence of Ψ of class 𝒞r satisfying (f) is far from adopted in the specialized literature as the most natural, and simple, definition for a bounded domain of class 𝒞r. Indeed, the usual definition in the most paradigmatic textbooks, like [21] or [12], involves local charts at any point of the boundary, instead of the minimal requirements of (e). Theorem 1.3 might help to clarify all these – always very delicate – regularity issues, though, as pointed out to the authors by the reviewer: “It is legitimate to ask why a smooth domain is not defined through a smooth embedding. But it seems to me that to define a differential structure on a manifold you need the concept of local chart and atlas. So you cannot escape from the definition with local charts”.

Nevertheless, to the best of our knowledge, the existence of the conormal projection and the conormal distance constructed in Theorem 1.3, as well as the proof of the fact that they inherit the regularity of ∂⁡Ω, seem completely new findings. Astonishingly, the Math. Sci. Net. of the Amer. Math. Soc. was unable to capture any entry with the words conormal distance, or conormal projection, though a huge list was given with conormal. Thus, Theorem 1.3 might be introducing these concepts into the debate of the characterization of the regularity of ∂⁡Ω in terms of the regularity of the associated distance function. Note that 𝒞2 is the minimal regularity of ∂⁡Ω, required to guarantee that the distance function through the ‘nearest point’ is well defined (see [27, Example 4]).

Actually, although Gilbarg and Trudinger [21, Lemma 14.16] show that the distance function to the boundary, dist⁡(x,∂⁡Ω), is of class 𝒞r, r≥2, if ∂⁡Ω is of class 𝒞r, and this result was later sharpened up to cover the case r=1 by Krantz and Parks [27], even the problem of establishing the regularity of ∂⁡Ω from the regularity of dist⁡(x,∂⁡Ω) remains open. These results actually sharpened a pioneering finding of Serrin [39], which established the 𝒞r-1-regularity of dist⁡(x,∂⁡Ω) from the 𝒞r-regularity of ∂⁡Ω. Some time later, Foote [18] generalized some of the results of [27] by establishing that, for every compact submanifold M of class 𝒞k, k≥2, there exists a neighborhood U such that the distance function d⁢(x,M) is 𝒞k in U∖M. Under these assumptions, the fact that M has a neighborhood U with the unique nearest point property, as well as the fact that the projection map Π:U→M is 𝒞k-1, relies on the tubular neighborhood theorem with the added observation that Π factors through the map that creates the neighborhood. More recently, almost twenty years later, Li and Nirenberg [28] established that if Ω is a domain in a smooth complete Finsler manifold, and G stands for the largest open subset of Ω with the nearest point property in the Finsler metric, then the distance function from ∂⁡Ω is in 𝒞lock,a⁢(G∪∂⁡Ω), k≥2 and 0<a≤1, if ∂⁡Ω is of class 𝒞k,a. But no converse result, within the vain of the characterization provided by Theorem 1.3, seems to be available in the literature.

This paper is distributed as follows. Section 2 proves Theorem 1.3, Section 3 uses Theorem 1.3 to reduce the general case when β changes sign to the classical case when β≥0. This simplifies substantially the underlying analysis and, in particular, the proof of Theorem 1.2. Section 4 establishes some important monotonicity properties of the associated principal eigenvalues with respect to the domain and the potential, Section 5 proves Theorem 1.1 and derives from it some important monotonicity properties, and Section 6 delivers the proof of Theorem 1.2.

2 Proof of Theorem 1.3

It suffices to prove the following implications: (a) implies (b), (b) implies (c), (d) and (f), (c), or (d), or (f), implies (e), and (e) implies (a). First, we will prove that (a) implies (b). Note that the normal vector field is of class 𝒞r-1 as soon as ∂⁡Ω is of class 𝒞r. Now, consider a field 𝝂 satisfying the requirements of part (b). For each ε>0, let us denote by Q𝝂∈𝒞r-1⁢((-ε,ε)×∂⁡Ω;ℝN) the function defined by

Q 𝝂 : ( - ε , ε ) × ∂ Ω → 𝒰 ε := Im Q 𝝂 ⊂ ℝ N , ( s , x ) ↦ x - s 𝝂 ( x ) ,

which establishes a bijection over its image for sufficiently small ε>0. Moreover, shortening ε>0, if necessary, Q𝝂-1 also is of class 𝒞r-1-regularity. Indeed, the proof of the injectivity proceeds by contradiction.

Figure 2 Scheme for the realization of Q𝝂{Q_{\boldsymbol{\nu}}} and Q𝝂-1{Q_{\boldsymbol{\nu}}^{-1}} and their relationships with the projection Π𝝂{\Pi_{\boldsymbol{\nu}}} and the distance function 𝔡𝝂{\mathfrak{d}_{\boldsymbol{\nu}}}.
Figure 2

Scheme for the realization of Q𝝂 and Q𝝂-1 and their relationships with the projection Π𝝂 and the distance function 𝔡𝝂.

Suppose that Q𝝂 is not injective for sufficiently small ε>0. Then there exist {sn1}n≥1,{sn2}n≥1⊂ℝ, with sn1→0 and sn2→0 as n↑∞, and {xn1}n≥1,{xn2}n≥1⊂∂⁡Ω, such that

( s n 1 , x n 1 ) ≠ ( s n 2 , x n 2 )   and   Q 𝝂 ⁢ ( s n 1 , x n 1 ) = Q 𝝂 ⁢ ( s n 2 , x n 2 )   for all  ⁢ n ≥ 1 .

In other words,

(2.1) x n 1 - s n 1 ⁢ 𝝂 ⁢ ( x n 1 ) = x n 2 - s n 2 ⁢ 𝝂 ⁢ ( x n 2 )   for all  ⁢ n ≥ 1 .

Moreover, without lost of generality, we can assume that sn1⁢sn2>0 for all n≥1. Otherwise, since 𝝂 is an outward vector field, for sufficiently large n≥1, we have that

x n 1 - s n 1 ⁢ 𝝂 ⁢ ( x n 1 ) = x n 2 - s n 2 ⁢ 𝝂 ⁢ ( x n 2 )

should lie, simultaneously, in Ω and in ℝN∖Ω¯, which is impossible.

Suppose xn1=xn2 for some n≥1. Then, since 𝝂⁢(xn1)≠0, (2.1) implies that sn1=sn2, which cannot hold. Hence, xn1≠xn2 for all n≥1. Since ∂⁡Ω is compact, along some subsequences of {xn1} and {xn2}, relabeled by n, we have that

lim n → ∞ ⁡ x n j = x ∞ j , j = 1 , 2 ,

for some x∞1,x∞2∈∂⁡Ω. Subsequently, we are renaming by {sn1}n≥1, {sn2}n≥1, {xn1}n≥1 and {xn2}n≥1 the new subsequences. Letting n↑∞ in (2.1) yields x∞1=x∞2=:x∞. Now, for each j=1,2, we consider the sequence {ςnj}n≥1 defined through

ς n j := s n i ⁢ | 𝝂 ⁢ ( x n j ) | , n ≥ 1 .

Then, by the continuity of 𝝂, the new sequences still satisfy

(2.2) lim n → ∞ ⁡ ς n 1 = lim n → ∞ ⁡ ς n 2 = 0 ,

and, setting ξ:=𝝂/|𝝂| for the unitary outward vector field, (2.1) can be equivalently expressed as

(2.3) x n 1 - x n 2 = ς n 1 ⁢ ξ ⁢ ( x n 1 ) - ς n 2 ⁢ ξ ⁢ ( x n 2 ) = ( ς n 1 - ς n 2 ) ⁢ ξ ⁢ ( x n 1 ) + ς n 2 ⁢ ( ξ ⁢ ( x n 1 ) - ξ ⁢ ( x n 2 ) )

for all n≥1. On the other hand, since xn1≠xn2, we have that

x n 1 - x n 2 | x n 1 - x n 2 | ∈ 𝕊 N - 1 ⊂ ℝ N   for all  ⁢ n ≥ 1 ,

where 𝕊N-1 stands for the (N-1)-dimensional sphere. As the sphere is compact, we can extract subsequences of {ςn1}n≥1, {ςn2}n≥1, {xn1}n≥1 and {xn2}n≥1, again labeled by n, such that

τ ∞ := lim n → ∞ ⁡ x n 1 - x n 2 | x n 1 - x n 2 | ∈ T x ∞ ⁢ ∂ ⁡ Ω ,

where Tx∞⁢∂⁡Ω stands for the tangent hyperplane of ∂⁡Ω at x∞. Note that |τ∞|=1. Moreover, by construction, we have that

| ς n j | = | x n j - Q 𝝂 ⁢ ( s n j , x n j ) | , Q 𝝂 ⁢ ( s n 1 , x n 1 ) = Q 𝝂 ⁢ ( s n 2 , x n 2 ) , j = 1 , 2 , n ≥ 1 .

Thus, since sn1⁢sn2>0 for all n≥1, the triangular inequality yields

| ς n 1 - ς n 2 | = | | ς n 1 | - | ς n 2 | | = | | x n 1 - Q 𝝂 ⁢ ( s n 1 , x n 1 ) | - | x n 2 - Q 𝝂 ⁢ ( s n 2 , x n 2 ) | | ≤ | x n 1 - x n 2 |

for all n≥1. Consequently, by the Bolzano–Weierstrass theorem, there exist η∈[-1,1] and subsequences of {ςn1}n≥1, {ςn2}n≥1, {xn1}n≥1 and {xn2}n≥1, relabeled by n, such that

(2.4) lim n → ∞ ⁡ ς n 1 - ς n 2 | x n 1 - x n 2 | = η .

Now we will show that, as a consequence of the regularity of ξ, the limit

lim n → ∞ ⁡ ξ ⁢ ( x n 1 ) - ξ ⁢ ( x n 2 ) | x n 1 - x n 2 |

is well defined in ℝN. Indeed, since ∂⁡Ω is a 𝒞r-manifold, there exist δ>0 and a local chart of ∂⁡Ω on a neighborhood of x∞, Φ∈𝒞r⁢(Bδ⁢(0);ℝN) with Φ⁢(0)=x∞. Subsequently, we set ynj:=Φ-1⁢(xnj) for j=1,2 and sufficiently large n≥1. By the continuity of Φ-1,

lim n → ∞ ⁡ y n j = 0 , j = 1 , 2 .

Since xn1≠xn2 and Φ is a local diffeomorphism, yn1≠yn2 and hence

y n 1 - y n 2 | y n 1 - y n 2 | ∈ 𝕊 N - 2 , n ≥ n 0 .

Thus, by compactness, we can extract subsequences, relabeled by n, such that

(2.5) τ ~ ∞ := lim n → ∞ ⁡ y n 1 - y n 2 | y n 1 - y n 2 | ∈ 𝕊 N - 2 .

Then, for every φ∈𝒞1⁢(Bδ⁢(0);ℝN), we have that

lim n → ∞ ⁡ φ ⁢ ( y n 1 ) - φ ⁢ ( y n 2 ) | y n 1 - y n 2 | = 𝒟 ⁢ φ ⁢ ( 0 ) ⁢ τ ~ ∞ = ∂ ⁡ φ ∂ ⁡ τ ~ ∞ ⁢ ( 0 ) .

Indeed,

| φ ⁢ ( y n 1 ) - φ ⁢ ( y n 2 ) | y n 1 - y n 2 | - 𝒟 ⁢ φ ⁢ ( 0 ) ⁢ τ ~ ∞ | = | φ ⁢ ( y n 2 + ( y n 1 - y n 2 ) ) - φ ⁢ ( y n 2 ) | y n 1 - y n 2 | - 𝒟 ⁢ φ ⁢ ( 0 ) ⁢ τ ~ ∞ |
= | 1 | y n 1 - y n 2 | ⁢ ∫ 0 1 𝒟 ⁢ φ ⁢ ( y n 2 + t ⁢ ( y n 1 - y n 2 ) ) ⁢ ( y n 1 - y n 2 ) ⁢ 𝑑 t - ∫ 0 1 𝒟 ⁢ φ ⁢ ( 0 ) ⁢ τ ~ ∞ ⁢ 𝑑 t |
= | ∫ 0 1 ( 𝒟 ⁢ φ ⁢ ( y n 2 + t ⁢ ( y n 1 - y n 2 ) ) ⁢ y n 1 - y n 2 | y n 1 - y n 2 | - 𝒟 ⁢ φ ⁢ ( 0 ) ⁢ τ ~ ∞ ) ⁢ 𝑑 t |
≤ ∫ 0 1 | 𝒟 φ ( y n 2 + t ( y n 1 - y n 2 ) ) ( y n 1 - y n 2 | y n 1 - y n 2 | - τ ~ ∞ ) | d t
+ ∫ 0 1 | ( 𝒟 φ ( y n 2 + t ( y n 1 - y n 2 ) ) - 𝒟 φ ( 0 ) ) τ ~ ∞ | d t ,

which, thanks to (2.5) and the uniform continuity of 𝒟⁢φ in Bδ/2⁢(0), converges to 0 as n↑∞. Hence, by the regularity of 𝝂, and so of ξ, we have that

lim n → ∞ ⁡ ξ ⁢ ( x n 1 ) - ξ ⁢ ( x n 2 ) | x n 1 - x n 2 | = lim n → ∞ ⁡ ( ξ ∘ Φ ) ⁢ ( y n 1 ) - ( ξ ∘ Φ ) ⁢ ( y n 2 ) | Φ ⁢ ( y n 1 ) - Φ ⁢ ( y n 2 ) |
= lim n → ∞ ⁡ | y n 1 - y n 2 | | Φ ⁢ ( y n 1 ) - Φ ⁢ ( y n 2 ) | ⁢ ( ξ ∘ Φ ) ⁢ ( y n 1 ) - ( ξ ∘ Φ ) ⁢ ( y n 2 ) | y n 1 - y n 2 |
(2.6) = 1 | 𝒟 ⁢ Φ ⁢ ( 0 ) ⁢ τ ~ ∞ | ⁢ 𝒟 ⁢ ( ξ ∘ Φ ) ⁢ ( 0 ) ⁢ τ ~ ∞ ∈ ℝ N .

Therefore, thanks to (2.2), (2.4) and (2.6), dividing by |xn1-xn2| in (2.3) and letting n↑+∞ yields

τ ∞ = η ⁢ ξ ⁢ ( x ∞ ) = η ⁢ 𝝂 ⁢ ( x ∞ ) | 𝝂 ⁢ ( x ∞ ) | .

Since τ∞∈𝕊N-1, taking norms in both sides provides us with |η|=1. However, since τ∞∈Tx∞⁢∂⁡Ω and ξ is an outward unit vector field along ∂⁡Ω, we have that

〈 τ ∞ , 𝐧 ⁢ ( x ∞ ) 〉 = 0   and   〈 ξ ⁢ ( x ∞ ) , 𝐧 ⁢ ( x ∞ ) 〉 > 0 ,

respectively, which implies η=0, driving to a contradiction. Thus, there exists ε>0 such that Q𝝂:(-ε,ε)→𝒰ε is bijective. Note that Q𝝂 inherits the regularity of 𝝂. So, it is of class 𝒞r-1⁢((-ε,ε)×∂⁡Ω;𝒰ε), and Q𝝂⁢(0,x)=x for all x∈∂⁡Ω.

It remains to show the regularity of Q𝝂-1:𝒰ε→(-ε,ε)×∂⁡Ω for sufficiently small ε>0. This is a consequence of the inverse function theorem. By continuity and compactness, it suffices to establish that 𝒟⁢Q𝝂 is non-degenerate on {0}×∂⁡Ω. Indeed, since ∂⁡Ω is a class 𝒞r manifold, for each x∈∂⁡Ω, there exist δx>0 and a homeomorphism onto its image Φx∈𝒞r(Bδx(0)⊂ℝN-1;ℝN), with Φx⁢(0)=x and Φx⁢(Bδx⁢(0))⊂∂⁡Ω. Actually, Φx parameterizes ∂⁡Ω in a neighborhood of x. Consider the function Q~𝝂:(-ε,ε)×Bδx⁢(0)→𝒰ε defined by

Q ~ 𝝂 ⁢ ( s , y ) := Q 𝝂 ⁢ ( s , Φ x ⁢ ( y ) ) = Φ x ⁢ ( y ) - s ⁢ 𝝂 ⁢ ( Φ x ⁢ ( y ) ) .

Then, for every s∈(-ε,ε) and y∈Bδx⁢(0), 𝒟⁢Q𝝂⁢(s,Φx⁢(y)) is represented by

𝒟 ⁢ Q ~ 𝝂 ⁢ ( s , y ) = [ - 𝝂 ⁢ ( Φ x ⁢ ( y ) ) , 𝒟 ⁢ Φ x ⁢ ( y ) - s ⁢ 𝒟 ⁢ ( 𝝂 ∘ Φ x ) ⁢ ( y ) ] .

In particular,

𝒟 ⁢ Q ~ 𝝂 ⁢ ( 0 , y ) = [ - 𝝂 ⁢ ( Φ x ⁢ ( y ) ) , 𝒟 ⁢ Φ x ⁢ ( y ) ] .

Since Φx is a local chart of a 𝒞r (N-1)-dimensional manifold, rank⁡𝒟⁢Φx⁢(y)=N-1 for all y∈Bδx⁢(0), and hence it generates the tangent space at Φx⁢(y). Thus, since 𝝂⁢(Φx⁢(y)) is a non-tangential vector field, it becomes apparent that

rank ⁡ 𝒟 ⁢ Q ~ 𝝂 ⁢ ( 0 , y ) = N .

Consequently, 𝒟⁢Q~𝝂⁢(0,y) is an isomorphism. Therefore, Q𝝂 establishes a 𝒞r-1-diffeomorphism onto its image for sufficiently small ε>0. In order to complete the proof of (a) implies (b), it remains to construct the projection Π𝝂 and show that the function 𝔡𝝂 defined in (1.8) is of class 𝒞r. Let P1:ℝ×∂⁡Ω→ℝ and P2:ℝ×∂⁡Ω→∂⁡Ω denote the projections on the first and the second component, respectively, i.e.,

P 1 : ℝ × ∂ ⁡ Ω → ℝ , ( s , x ) ↦ s ,
P 2 : ℝ × ∂ ⁡ Ω → ∂ ⁡ Ω , ( s , x ) ↦ x .

Obviously, P1 and P2 are of class 𝒞∞ and, by construction, it is easily seen that the map

Π 𝝂 := P 2 ∘ Q 𝝂 - 1 : 𝒰 ε → ∂ ⁡ Ω ⊂ ℝ N

satisfies all the requirements of part (b). Indeed, Π𝝂 also is of class 𝒞r-1, as Q𝝂-1 and P2. Moreover, for every x∈∂⁡Ω, we have that

Π 𝝂 ⁢ ( x ) = P 2 ∘ Q 𝝂 - 1 ⁢ ( x ) = P 2 ⁢ ( 0 , x ) = x .

Since Q𝝂 is a diffeomorphism, for every x∈𝒰ε, there exists s∈(-ε,ε) such that

x = Q 𝝂 ⁢ ( s , Π 𝝂 ⁢ ( x ) ) = Π 𝝂 ⁢ ( x ) - s ⁢ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) .

Hence, if λ∈ℝ satisfies x-λ⁢𝝂⁢(Π𝝂⁢(x))∈𝒰ε, we find that

Π 𝝂 ⁢ ( x - λ ⁢ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) ) = Π 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) - s ⁢ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) - λ ⁢ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) ) = P 2 ∘ Q 𝝂 - 1 ⁢ ( Q 𝝂 ⁢ ( s + λ , Π 𝝂 ⁢ ( x ) ) ) = Π 𝝂 ⁢ ( x ) .

In particular, this entails that ∂⁡Π𝝂∂⁡𝝂⁢(Π𝝂⁢(x))⁢(x)=0 for all x∈𝒰ε. By the definition of Q𝝂, 𝔡𝝂=P1∘Q𝝂-1, and so it is of class 𝒞r-1⁢(𝒰ε). Moreover, for every x∈𝒰ε,

x = Π 𝝂 ⁢ ( x ) - 𝔡 𝝂 ⁢ ( x ) ⁢ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) .

Thus,

𝔡 𝝂 ⁢ ( x ) = 1 | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | 2 ⁢ 〈 Π 𝝂 ⁢ ( x ) - x , 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) 〉 ,

and hence, combining the Leibniz rule with the properties of the projection Π𝝂, we find that, for every x∈𝒰ε,

𝒟 ⁢ 𝔡 𝝂 ⁢ ( x ) = - 2 〈 𝝂 ( Π 𝝂 ( x ) ) , 𝒟 ( 𝝂 ∘ Π 𝝂 ) ( x ) ) 〉 | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | 4 ⁢ 〈 Π 𝝂 ⁢ ( x ) - x , 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) 〉
+ 1 | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | 2 ⁢ ( 〈 𝒟 ⁢ Π 𝝂 ⁢ ( x ) - Id , 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) 〉 + 〈 Π 𝝂 ⁢ ( x ) - x , 𝒟 ⁢ ( 𝝂 ∘ Π 𝝂 ) ⁢ ( x ) 〉 )
= 1 | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | 2 ⁢ ( 〈 𝒟 ⁢ Π 𝝂 ⁢ ( x ) , 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) 〉 - 𝔡 𝝂 ⁢ ( x ) ⁢ 〈 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) , 𝒟 ⁢ ( 𝝂 ∘ Π 𝝂 ) ⁢ ( x ) 〉 - 〈 Id , 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) 〉 )
= 1 | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | 2 ⁢ ( ∂ ⁡ Π 𝝂 ∂ ⁡ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) ⁢ ( x ) - 𝔡 𝝂 ⁢ ( x ) ⁢ ∂ ⁡ ( 𝝂 ∘ Π 𝝂 ) ∂ ⁡ 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) ⁢ ( x ) - 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) )
= - 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | 𝝂 ⁢ ( Π 𝝂 ⁢ ( x ) ) | 2 ,

because Π𝝂 and 𝝂∘Π𝝂 are constant along each direction 𝝂⁢(Π𝝂⁢(x)). Therefore, 𝒟⁢𝔡𝝂∈𝒞r-1, which entails 𝔡𝝂∈𝒞r and ends the proof of (a) implies (b).

The fact that part (b) implies part (c) is immediate. Next, we will prove that (b) implies (d) and (f). Suppose (b) and consider any outward vector field 𝝂∈𝒞r-1. Then 𝝂~:=𝝂/|𝝂|∈𝒞r-1. Let 𝒰, Π𝝂~ and 𝔡𝝂~ denote, respectively, the open set, the projection and the ‘regularized distance’  (1.8) provided by part (b). Then the function ψ𝝂:𝒰→ℝ defined by ψ𝝂:=-𝔡𝝂~ satisfies

∇ ⁡ ψ 𝝂 ⁢ ( x ) = 𝒟 ⁢ ψ 𝝂 ⁢ ( x ) = - 𝒟 ⁢ 𝔡 𝝂 ~ = 𝝂 ~ ⁢ ( Π 𝝂 ~ ⁢ ( x ) )

for all x∈𝒰. In particular, ∇⁡ψ𝝂⁢(x)=𝝂~⁢(x) for every x∈∂⁡Ω, and hence

∂ ⁡ ψ 𝝂 ∂ ⁡ 𝝂 ⁢ ( x ) = 〈 ∇ ⁡ ψ 𝝂 ⁢ ( x ) , 𝝂 ⁢ ( x ) 〉 = 〈 𝝂 ~ ⁢ ( x ) , 𝝂 ⁢ ( x ) 〉 = | 𝝂 ⁢ ( x ) | > 0 ,

which ends the proof of (b) implies (d). Actually, since |∇⁡ψ𝝂⁢(x)|=|𝝂~⁢(x)|=1 for all x∈∂⁡Ω, Ψ:=ψ𝝂 satisfies the requirements of part (f).

The fact that (d) implies (e) is trivial, and the proof of (c) implies (e) follows the same patterns as the proof of (b) implies (d). The fact that (f) implies (e) follows from the fact that 𝝂⁢(x):=∇⁡Ψ⁢(x) is an outward vector field of class 𝒞r-1 satisfying

∂ ⁡ Ψ ∂ ⁡ 𝝂 ⁢ ( x ) = | ∇ ⁡ Ψ ⁢ ( x ) | 2 = 1 > 0

for all x∈∂⁡Ω. Thus, part (e) holds by choosing ψ:=Ψ.

It remains to prove that (e) implies (a). By the properties of the function ψ guaranteed by part (e), it is apparent that ∂⁡Ω:=ψ-1⁢(0). Let us consider x0∈∂⁡Ω and 𝝂⁢(x0), and let {𝐞j}j=1N-1 be an orthonormal basis of span⁢[𝝂⁢(x0)]⊥ in ℝN. Subsequently, for every δ>0, we denote by Fδ:(-δ,δ)×(-δ,δ)N-1→ℝN the 𝒞∞ map defined through

F δ ⁢ ( z , 𝐲 ) := x 0 + z ⁢ 𝝂 ⁢ ( x 0 ) + ∑ j = 1 N - 1 y j ⁢ 𝐞 j , 𝐲 = ( y 1 , … , y N - 1 ) .

This map establishes a diffeomorphism onto its image, which is an open neighborhood of x0 denoted by 𝒲δ. Note that Fδ⁢(0,0)=x0. Choose δ>0 such that 𝒲δ⊂𝒰, where 𝒰 is the open neighborhood of ∂⁡Ω guaranteed by part (c). Lastly, consider the function

G δ := ψ ∘ F δ ∈ 𝒞 r ⁢ ( ( - δ , δ ) N ; ℝ ) .

Obviously, Gδ⁢(0,0)=0. Moreover,

∂ ⁡ G δ ∂ ⁡ z ⁢ ( 0 , 0 ) = [ 𝒟 ⁢ ψ ⁢ ( x 0 ) ] ⁢ ( ∂ ⁡ F δ ∂ ⁡ z ⁢ ( 0 , 0 ) ) = 𝒟 ⁢ ψ ⁢ ( x 0 ) ⁢ ( 𝝂 ⁢ ( x 0 ) ) = ∂ ⁡ ψ ∂ ⁡ 𝝂 ⁢ ( x 0 ) > 0 .

Thus, according to the implicit function theorem, there exists δ0>0 and ζ∈𝒞r⁢((-δ0,δ0)N-1;ℝ) such that

G δ 0 - 1 ⁢ ( 0 ) = { ( ζ ⁢ ( 𝐲 ) , 𝐲 ) ∈ ℝ N : 𝐲 ∈ ( - δ 0 , δ 0 ) N - 1 } .

In particular, the function (-δ0,δ0)N-1∋𝐲↦Fδ0⁢(ζ⁢(𝐲),𝐲) provides us with a class 𝒞r parametrization of ∂⁡Ω∩𝒲δ0. Since x0 was arbitrary, ∂⁡Ω is an (N-1)-manifold of class 𝒞r. This ends the proof of Theorem 1.3.

A further (deeper) analysis of the role played by the regularity of the outward vector field reveals the validity of the next result.

Corollary 2.1.

If ∂⁡Ω is an (N-1)-dimensional manifold of class Cr, r≥1, and 𝛎∈Ck⁢(∂⁡Ω;RN), k≥1, is an outward vector field, then there exist an open subset U of RN, with ∂⁡Ω⊂U, and a function Π𝛎∈Cmin⁡{r,k}⁢(U;∂⁡Ω) satisfying the requirements of Π𝛎 in the statement of Theorem 1.3  (b). In particular, the function d𝛎:U→R defined in (1.8) is of class Cmin⁡{r,k+1}.

3 A Canonical Transformation

As a byproduct of Theorem 1.3, the next result holds. It allows transforming the original problem into a problem with β≥0. So, without lost of generality, we can assume that β≥0 for the remaining of this paper.

Theorem 3.1.

Assume that ∂⁡Ω is of class C2. Then there exists E∈C2⁢(Ω¯), with E⁢(x)>0 for all x∈Ω¯, such that (1.1) can be equivalently expressed as

{ d ⁢ ℒ E ⁢ w = h E ⁢ ( w , x ) in  ⁢ Ω , ℬ E ⁢ w = 0 in  ⁢ ∂ ⁡ Ω ,

where

  1. h E ⁢ ( w , x ) = 1 E ⁢ ( x ) ⁢ h ⁢ ( E ⁢ ( x ) ⁢ w , x ) for all w ≥ 0 and x ∈ Ω ¯ ,

  2. ℒ E = - div ( A ∇ ⋅ ) + b E ∇ + c E , with

    b E := b - 2 ⁢ A ⁢ ∇ ⁡ E E ∈ ℳ 1 × N ⁢ ( 𝒞 ⁢ ( Ω ¯ ) ) , c E := ℒ ⁢ E E ∈ 𝒞 ⁢ ( Ω ¯ ) ,

  3. ℬ E = D on Γ D and ℬ E = ∂ ∂ ⁡ 𝝂 + β E on Γ R , with β E := ℬ ⁢ E E ≥ 0 .

Moreover, hE satisfies (H1), (H2) and (H4) if h does too.

Proof.

First, let us consider an arbitrary E∈𝒞2⁢(Ω¯) such that E⁢(x)>0 for all x∈Ω¯. Suppose that u is a non-negative solution of (1.1). Then w:=u/E satisfies

ℒ ⁢ u = ℒ ⁢ ( E ⁢ w ) = - div ⁡ ( A ⁢ ∇ ⁡ ( E ⁢ w ) ) + b ⁢ ∇ ⁡ ( E ⁢ w ) + c ⁢ E ⁢ w
= - div ⁡ ( E ⁢ A ⁢ ∇ ⁡ w ) - div ⁡ ( w ⁢ A ⁢ ∇ ⁡ E ) + E ⁢ b ⁢ ∇ ⁡ w + w ⁢ b ⁢ ∇ ⁡ E + w ⁢ c ⁢ E
= - ∇ ⁡ E ⁢ A ⁢ ∇ ⁡ w - E ⁢ div ⁡ ( A ⁢ ∇ ⁡ w ) - ∇ ⁡ w ⁢ A ⁢ ∇ ⁡ E - w ⁢ div ⁡ ( A ⁢ ∇ ⁡ E ) + E ⁢ b ⁢ ∇ ⁡ w + w ⁢ b ⁢ ∇ ⁡ E + w ⁢ c ⁢ E
= - E ⁢ div ⁡ ( A ⁢ ∇ ⁡ w ) + E ⁢ b ⁢ ∇ ⁡ w - ∇ ⁡ E ⁢ A ⁢ ∇ ⁡ w - ∇ ⁡ w ⁢ A ⁢ ∇ ⁡ E + w ⁢ ( - div ⁡ ( A ⁢ ∇ ⁡ E ) + b ⁢ ∇ ⁡ E + c ⁢ E ) .

By the symmetry of A, we have that ∇⁡w⁢A⁢∇⁡E=∇⁡E⁢A⁢∇⁡w, and thus

ℒ ⁢ u = E ⁢ ( - div ⁡ ( A ⁢ ∇ ⁡ w ) + ( b - 2 ⁢ A ⁢ ∇ ⁡ E E ) ⁢ ∇ ⁡ w + ℒ ⁢ E E ⁢ w ) = E ⁢ ℒ E ⁢ w   in  ⁢ Ω .

Hence,

d ⁢ ℒ E ⁢ w = 1 E ⁢ d ⁢ ℒ ⁢ u = 1 E ⁢ h ⁢ ( u , ⋅ ) = 1 E ⁢ h ⁢ ( E ⁢ w , ⋅ ) = h E ⁢ ( w , ⋅ )   in  ⁢ Ω .

As for the boundary, we find that ℬE⁢w⁢(x)=w⁢(x)=u⁢(x)/E⁢(x)=0 for all x∈ΓD, whereas

0 = ℬ ⁢ u ⁢ ( x ) = ℬ ⁢ ( E ⁢ w ) ⁢ ( x ) = ∂ ⁡ ( E ⁢ w ) ∂ ⁡ 𝝂 ⁢ ( x ) + β ⁢ ( x ) ⁢ E ⁢ ( x ) ⁢ w ⁢ ( x )
= E ⁢ ( x ) ⁢ ∂ ⁡ w ∂ ⁡ 𝝂 ⁢ ( x ) + ( ∂ ⁡ E ∂ ⁡ 𝝂 ⁢ ( x ) + β ⁢ ( x ) ⁢ E ⁢ ( x ) ) ⁢ w ⁢ ( x ) = E ⁢ ( x ) ⁢ ℬ E ⁢ w ⁢ ( x )

for all x∈ΓR. In order to choose E such that βE≥0, note that, according to Theorem 1.3 and the remarks after it, there exist an open set 𝒰, Ω¯⊂𝒰⊂ℝN, and a function ψ∈𝒞2⁢(𝒰) such that ψ⁢(x)<0 for all x∈Ω, ψ⁢(x)=0 for all x∈∂⁡Ω and minΓR⁡∂⁡ψ∂⁡𝝂>0. Consider

E := exp ⁡ ( μ ⁢ ψ ) ,

with μ>0 to be determined. Then, for each x∈ΓR, E⁢(x)=1, and hence

β E ⁢ ( x ) = ℬ ⁢ E ⁢ ( x ) E ⁢ ( x ) = β ⁢ ( x ) + 1 E ⁢ ( x ) ⁢ ∂ ⁡ E ∂ ⁡ 𝝂 ⁢ ( x ) = β ⁢ ( x ) + μ ⁢ ∂ ⁡ E ∂ ⁡ 𝝂 ⁢ ( x ) .

Thus, since minΓR⁡∂⁡ψ∂⁡𝝂>0, it becomes apparent that βE≥0 on ΓR for sufficiently large μ>0.

Now, let us analyze the properties of hE. The regularity required for (H1) is a byproduct of the regularity of both h and E. On the other hand, for every u>0 and x∈Ω¯, we have that

∂ ⁡ h E ∂ ⁡ w = ∂ ∂ ⁡ w ⁢ ( 1 E ⁢ ( x ) ⁢ h ⁢ ( E ⁢ ( x ) ⁢ w , x ) ) = 1 E ⁢ ( x ) ⁢ E ⁢ ( x ) ⁢ ∂ ⁡ h ∂ ⁡ u ⁢ ( E ⁢ ( x ) ⁢ w , x ) = ∂ ⁡ h ∂ ⁡ u ⁢ ( E ⁢ ( x ) ⁢ w , x ) < 0 .

Hence, hE satisfies (H2). To conclude, since h satisfies (H4), there exists M>0 such that maxΩ¯⁡h⁢(M,⋅)<0. Therefore, setting

M E := M min Ω ¯ ⁡ E > 0

and taking into account that h is decreasing in u by (H2), we conclude that, for every x∈Ω¯,

h E ⁢ ( M E , x ) = 1 E ⁢ ( x ) ⁢ h ⁢ ( M E ⁢ E ⁢ ( x ) , x ) = 1 E ⁢ ( x ) ⁢ h ⁢ ( E ⁢ ( x ) ⁢ M min Ω ¯ ⁡ E , x ) ≤ 1 E ⁢ ( x ) ⁢ h ⁢ ( M , x ) < 0 ,

which ends the proof. ∎

Remark 3.2.

It should be noted that one can achieve βE⁢(x)>0 for all x∈ΓR by choosing a sufficiently large μ>0 in the previous proof.

4 Monotonicity Properties of the Principal Eigenvalue

Throughout this section, for every d>0 and V∈𝒞⁢(Ω¯), we will denote by

Σ ⁢ ( d , V ) := σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ]

the principal eigenvalue of [d⁢ℒ+V;ℬ,Ω] in Wℬ2,∞⁢(Ω):=⋂p>N⁡Wℬ2,p⁢(Ω), The next result collects the main properties of Σ⁢(d,V). It extends [14, Theorem 2.1] to deal with general differential operators, ℒ, not necessarily self-adjoint. Part a provides us with the monotony of the principal eigenvalue with respect to the potential.

Theorem 4.1.

Σ ⁢ ( d , V ) has the following properties:

  1. For every d > 0 , the map Σ ⁢ ( d , ⋅ ) : C ⁢ ( Ω ¯ ) → ℝ is strictly increasing, i.e., Σ ⁢ ( d , V 1 ) < Σ ⁢ ( d , V 2 ) if V 1 , V 2 ∈ 𝒞 ⁢ ( Ω ¯ ) with V 1 ⪇ V 2 .

  2. For every V ∈ 𝒞 ⁢ ( Ω ¯ ) ,

    Σ ⁢ ( 0 , V ) := lim d → 0 ⁡ Σ ⁢ ( d , V ) = min Ω ¯ ⁡ V .

Proof.

Let φ1≫0 denote the (unique) principal eigenfunction associated to σ1⁢[d⁢ℒ+V1;ℬ,Ω] such that ∥φ1∥∞=1. Then

{ ( d ⁢ ℒ + V 2 - σ 1 ⁢ [ d ⁢ ℒ + V 1 ; ℬ , Ω ] ) ⁢ φ 1 ⪈ ( d ⁢ ℒ + V 1 - σ 1 ⁢ [ d ⁢ ℒ + V 1 ; ℬ , Ω ] ) ⁢ φ 1 = 0   in  ⁢ Ω , ℬ ⁢ φ 1 = 0   on  ⁢ ∂ ⁡ Ω .

Therefore, the function φ1 provides us with a positive strict supersolution of the differential operator d⁢ℒ+V2-σ1⁢[d⁢ℒ+V1;ℬ,Ω] subject to the boundary operator ℬ on ∂⁡Ω, and hence, thanks to the theorem of characterization provided by [31, Theorem 7.10], its principal eigenvalue must be positive. Thus,

σ 1 ⁢ [ d ⁢ ℒ + V 2 ; ℬ , Ω ] - σ 1 ⁢ [ d ⁢ ℒ + V 1 ; ℬ , Ω ] = σ 1 ⁢ [ d ⁢ ℒ + V 2 - σ 1 ⁢ [ d ⁢ ℒ + V 1 ; ℬ , Ω ] ; ℬ , Ω ] > 0 ,

which ends the proof of part (a).

For the convergence in part (b), we first note that, thanks to part (a),

σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ] ≥ d ⁢ σ 1 ⁢ [ ℒ ; ℬ , Ω ] + min Ω ¯ ⁡ V .

Thus,

lim inf d → 0 ⁡ σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ] ≥ min Ω ¯ ⁡ V .

Now, arguing by contradiction, suppose that

lim sup d → 0 ⁡ σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ] > min Ω ¯ ⁡ V .

Then there exist ε>0 and a sequence {dn}n≥1⊂(0,+∞), with limn→∞⁡dn=0, such that, for every n≥1,

σ 1 ⁢ [ d n ⁢ ℒ + V ; ℬ , Ω ] > min Ω ¯ ⁡ V + ε .

Equivalently,

σ 1 ⁢ [ d n ⁢ ℒ + V - min Ω ¯ ⁡ V - ε ; ℬ , Ω ] > 0 ,

and hence, by [31, Theorem 7.10], for every n≥1, the problem [dn⁢ℒ+V-minΩ¯⁡V-ε;ℬ,Ω] admits a strict supersolution φn≫0, i.e.,

{ ( d n ⁢ ℒ + V - min Ω ¯ ⁡ V - ε ) ⁢ φ n ≥ 0 in  ⁢ Ω , ℬ ⁢ φ n ≥ 0 on  ⁢ ∂ ⁡ Ω ,

with some of these inequalities strict. Let x0∈Ω¯ be such that V⁢(x0)=minΩ¯⁡V. By continuity, there exists ρ>0 such that

V ⁢ ( x ) < min Ω ¯ ⁡ V + ε 2

for all x∈Bρ⁢(x0)∩Ω¯. In particular, this estimate holds in an open ball B⊂Bρ⁢(x0)∩Ω. Thus, for every n≥1, we have that

{ ( d n ⁢ ℒ - ε 2 ) ⁢ φ n ⪈ 0 in  ⁢ B , φ n > 0 on  ⁢ ∂ ⁡ B .

Consequently, thanks again to [31, Theorem 7.10], we find that

σ 1 ⁢ [ d n ⁢ ℒ - ε 2 ; D , B ] > 0 ,

which contradicts the fact that

lim n → ∞ ⁡ σ 1 ⁢ [ d n ⁢ ℒ - ε 2 ; D , B ] = lim n → ∞ ⁡ d n ⁢ σ 1 ⁢ [ ℒ ; D , Ω ] - ε 2 = - ε 2 .

This contradiction ends the proof. ∎

For establishing the monotonicity of the principal eigenvalue with respect to the underlying domain, we need to introduce some notations.

Definition 4.2.

Let Ω0 be a subdomain of class 𝒞2 of Ω and ℬ0 a boundary operator on ∂⁡Ω0. We will say that (ℬ0,Ω0) is comparable with (ℬ,Ω), and write (ℬ0,Ω0)⪯(ℬ,Ω), when the following conditions are satisfied:

  1. Each component Γ of ∂⁡Ω0 is either a component of ∂⁡Ω, or Γ⊂Ω.

  2. The boundary operator ℬ0 satisfies

    ℬ 0 := { D on  ⁢ ∂ ⁡ Ω 0 ∩ Ω , ℬ ~ on  ⁢ ∂ ⁡ Ω 0 ∩ ∂ ⁡ Ω ,

    where for every component Γ of ∂⁡Ω0∩∂⁡Ω, either ℬ~=D on Γ, or Γ⊂ΓR and there is β0∈𝒞⁢(∂⁡Ω0) with β0≥β such that

    ℬ ~ = ∂ ∂ ⁡ 𝝂 + β 0   on  ⁢ Γ .

We will write (ℬ0,Ω0)≺(ℬ,Ω) if, in addition, (ℬ0,Ω0)≠(ℬ,Ω).

It should be noted that, according to [8, Theorem 9.1], the Dirichlet boundary operator on each component of ∂⁡Ω can be approximated by letting minΓ⁡β↑∞. Thus, the larger β0, the closer are ℬ0 and D. The next monotonicity result sharpens [14, Lemma 2.2].

Lemma 4.3.

Let Ω0 be a subdomain of class C2 of Ω and B0 a boundary operator on ∂⁡Ω0. If (B0,Ω0)≺(B,Ω), then

σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ] < σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ 0 , Ω 0 ]   for every  ⁢ d > 0 ⁢ and  ⁢ V ∈ 𝒞 ⁢ ( Ω ¯ ) .

Proof.

Let φ≫0 be the principal eigenfunction associated to σ1⁢[d⁢ℒ+V;ℬ,Ω], normalized so that ∥φ∥∞=1. Then, according to Definition 4.2, as long as (ℬ0,Ω0)≺(ℬ,Ω), there exist a component Γ≠∅ of ∂⁡Ω for which some of the following alternatives hold:

  1. Γ ⊂ Ω and ℬ0⁢φ=φ>0 on Γ. Actually, this occurs if Ω0 is a proper subdomain of Ω.

  2. Γ ⊂ Γ R and ℬ0⁢φ=φ on Γ. Then, since φ⁢(x)>0 for all x∈ΓR, we have that ℬ0⁢φ>0 on Γ.

  3. Γ ⊂ Γ R and ℬ0=∂∂⁡𝝂+β0, with β0⪈β on Γ. Then, since φ⁢(x)>0 for all x∈ΓR, we find that

    ℬ 0 ⁢ φ = ∂ ⁡ φ ∂ ⁡ 𝝂 + β 0 ⁢ φ ⪈ ∂ ⁡ φ ∂ ⁡ 𝝂 + β ⁢ φ = ℬ ⁢ φ = 0   on  ⁢ Γ .

Hence, φ satisfies

{ ( d ⁢ ℒ + V - σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ] ) ⁢ φ = 0 in  ⁢ Ω 0 , ℬ 0 ⁢ φ ⪈ 0 on  ⁢ ∂ ⁡ Ω 0 .

In particular, φ is a positive strict supersolution of [d⁢ℒ+V-σ1⁢[d⁢ℒ+V;ℬ,Ω];ℬ0,Ω0]. Therefore, we can conclude from [31, Theorem 7.10] that

σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ 0 , Ω 0 ] - σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ] = σ 1 ⁢ [ d ⁢ ℒ + V - σ 1 ⁢ [ d ⁢ ℒ + V ; ℬ , Ω ] ; ℬ 0 , Ω 0 ] > 0 ,

which ends the proof. ∎

5 The Generalized Diffusive Logistic Equation

We begin this section by proving Theorem 1.1, which characterizes the existence and establishes the uniqueness of the positive solution of (1.1) in terms of the linearized instability of u=0 as a steady-state solution of its parabolic counterpart. As pointed out in Section 3, without lost of generality, we can assume that β≥0. Moreover, h⁢(u,x) is supposed to satisfy (H1), (H2) and (H3) for some d>0.

Proof of Theorem 1.1.

As a consequence of (H2) and (H3), and since β can be assumed to be non-negative, u¯:=κ≥M>0 is a supersolution of (1.1). Now, suppose that σ1⁢[d⁢ℒ-h⁢(0,⋅);ℬ,Ω]<0 and let ϕ≫0 be any associated eigenfunction. We claim that u¯:=ε⁢ϕ is a subsolution of (1.1) for sufficiently small ε>0. Since ℬ⁢(ε⁢ϕ)=ε⁢ℬ⁢ϕ=0 on ∂⁡Ω, it suffices to show that

d ⁢ ℒ ⁢ ( ε ⁢ ϕ ) ≤ ε ⁢ ϕ ⁢ h ⁢ ( ε ⁢ ϕ , ⋅ )   in  ⁢ Ω .

By the choice of ϕ, we have that

d ⁢ ℒ ⁢ ( ε ⁢ ϕ ) = ε ⁢ ( σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] ⁢ ϕ + h ⁢ ( 0 , ⋅ ) ⁢ ϕ )   in  ⁢ Ω .

Hence, dividing by ε⁢ϕ, we should make sure that

(5.1) σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] ≤ h ⁢ ( ε ⁢ ϕ , ⋅ ) - h ⁢ ( 0 , ⋅ )   in  ⁢ Ω .

Since h is uniformly continuous on [0,1]×Ω¯ and ε⁢ϕ converges to 0 uniformly in Ω¯ as ε↓0, we find that

lim ε → 0 ∥ h ( ε ϕ , ⋅ ) - h ( 0 , ⋅ ) ∥ ∞ = 0 .

Thus, condition (5.1) holds for sufficiently small ε, and hence u¯:=ε⁢ϕ is a subsolution of (1.1). Since ε can be shortened up to get ε⁢ϕ≤κ, (1.1) possesses a (strong) positive solution u such that ε⁢ϕ≤u≤κ.

Next, we will show that σ1⁢[d⁢ℒ-h⁢(0,⋅);ℬ,Ω]<0 is necessary for the existence of a positive solutions. Indeed, if (1.1) admits a positive solution u, then σ1⁢[d⁢ℒ-h⁢(u,⋅);ℬ,Ω]=0, by the uniqueness of the principal eigenvalue. Thus, by (H2), it follows from Theorem 4.1 (a) that

σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] < σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( u , ⋅ ) ; ℬ , Ω ] = 0 .

As for establishing the uniqueness, assume that u1,u2∈⋂p>N⁡W2,p⁢(Ω) are two positive solutions of (1.1). In particular, u1,u2≫0. Thanks to the first part of the proof, we already know that (1.1) admits a subsolution u¯=ε⁢ϕ and a supersolution u¯=κ>M such that

u ¯ ≤ u 1 , u 2 ≤ u ¯ .

This can be easily obtained by shortening ε>0 and enlarging κ as much as necessary. For these choices, thanks to [1, Theorem 3], problem (1.1) admits two strong solutions, u*,u*∈⋂p>N⁡W2,p⁢(Ω), which are the minimal and maximal solutions of (1.1), respectively, in the order interval [u¯,u¯]. In particular, we have that

u ¯ ≤ u * ≤ u 1 , u 2 ≤ u * ≤ u ¯

and, since u1≠u2, necessarily u*<u*. Since they are solutions of (1.1), we already know that

(5.2) σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( u * , ⋅ ) ; ℬ , Ω ] = σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( u * , ⋅ ) ; ℬ , Ω ] = 0 ,

and, thanks to (H2),

h ⁢ ( u * , ⋅ ) ⪈ h ⁢ ( u * , ⋅ )   in  ⁢ Ω .

Thus, by Theorem 4.1 (a),

σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( u * , ⋅ ) ; ℬ , Ω ] < σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( u * , ⋅ ) ; ℬ , Ω ] ,

which contradicts (5.2). Therefore, u1=u2. This ends the proof. ∎

By linearizing (1.1) at u=0, it is easily seen that u=0 is linearly unstable if and only if

σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] < 0 ,

while it is linearly stable, or linearly neutrally stable, in any other case.

Throughout the rest of this paper, we will denote by θ{d,h}ℒ,ℬ,Ω the maximal non-negative solution of (1.1). By Theorem 1.1, θ{d,h}ℒ,ℬ,Ω=0 if σ1⁢[d⁢ℒ-h⁢(0,⋅);ℬ,Ω]≥0, while θ{d,h}ℒ,ℬ,Ω≫0 if σ1⁢[d⁢ℒ-h⁢(0,⋅);ℬ,Ω]<0. Should not exist any ambiguity, we will simply set

θ { d , h } := θ { d , h } ℒ , ℬ , Ω ,

or, alternatively, omit some of these indexes. As a byproduct of Theorems 4.1 (b) and 1.1, the positiveness of θ{d,h} can be characterized for small d>0 in terms of the sign of maxΩ¯⁡h⁢(0,⋅), as established by the next result.

Corollary 5.1.

Suppose that h⁢(u,x) satisfies (H3) for sufficiently small d>0. Then the following hold:

  1. If max Ω ¯ ⁡ h ⁢ ( 0 , ⋅ ) < 0 , then a maximal d 0 ∈ ( 0 , + ∞ ] exists such that θ { d , h } = 0 for d ∈ ( 0 , d 0 ) .

  2. If max Ω ¯ ⁡ h ⁢ ( 0 , ⋅ ) > 0 , then a maximal d 0 ∈ ( 0 , + ∞ ] exists such that θ { d , h } ≫ 0 for d ∈ ( 0 , d 0 ) .

In the intermediate case when maxΩ¯⁡h⁢(0,⋅)=0, Theorem 4.1 (b) implies that

lim d ↓ 0 ⁡ σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] = min Ω ¯ ⁡ ( - h ⁢ ( 0 , ⋅ ) ) = - max Ω ¯ ⁡ h ⁢ ( 0 , ⋅ ) = 0 .

Thus, the sign of the principal eigenvalue σ1⁢[d⁢ℒ-h⁢(0,⋅);ℬ,Ω] for sufficiently small d>0 might depend on the nature of the coefficients of ℒ as well as on the boundary operator ℬ, or even the geometry and the size of Ω. Indeed, if ℒ=-Δ is the Laplace operator and we assume that ΓR=∅, i.e., ℬ is the Dirichlet operator D and h⁢(0,⋅)=0, then

σ 1 ⁢ [ - d ⁢ Δ ; D , Ω ] = d ⁢ σ 1 ⁢ [ - Δ ; D , Ω ] > 0

for all d>0 and hence, by Theorem 1.1, θ{d,h}=0 for all d>0. But if we assume that ℒ=-Δ-1, h⁢(0,⋅)=0, ΓD=∅ and β≡0 on ΓR=∂⁡Ω, i.e., ℬ is the Neumann operator R0, then

σ 1 ⁢ [ d ⁢ ( - Δ - 1 ) ; R 0 , Ω ] = d ⁢ σ 1 ⁢ [ - Δ ; R 0 , Ω ] - d = - d < 0

for all d>0. Therefore, due to Theorem 1.1, θ{d,h}≫0 for all d>0. Finally, note that, according to a celebrated variational inequality of Faber and Krahn (see, e.g., [31, Proposition 8.6]), the sign of

σ 1 ⁢ [ d ⁢ ( - Δ - 1 ) ; D , Ω ] = d ⁢ ( σ 1 ⁢ [ - Δ ; D , Ω ] - 1 )

depends on the Lebesgue measure of Ω. Indeed, for sufficiently small |Ω|, σ1⁢[-Δ;D,Ω]>1, and hence θ{d,h}=0 for all d>0, while, for sufficiently large |Ω|, σ1⁢[-Δ;D,Ω]<1, and therefore θ{d,h}≫0 for all d>0.

The following result provides us with a substantial sharpening of [14, Lemma 2.5].

Lemma 5.2.

Suppose that h⁢(u,x) satisfies (H3) for some d>0. Let Ω0 be a subdomain of class C2 of Ω, B0 a boundary operator on ∂⁡Ω0 such that, according to Definition 4.2, (B0,Ω0)⪯(B,Ω), and suppose h0∈C⁢(R×Ω¯0) satisfies (H1), (H2), (H3) and h0≤h in [0,+∞)×Ω¯0. Then

θ { d , h 0 } ℒ , ℬ 0 , Ω 0 ≤ θ { d , h } ℒ , ℬ , Ω   in  ⁢ Ω 0 .

If, in addition, (B0,Ω0)≺(B,Ω), or h0⁢(u,⋅)≠h⁢(u,⋅) in Ω0 for all u≥0, then

θ { d , h 0 } ℒ , ℬ 0 , Ω 0 ≪ θ { d , h } ℒ , ℬ , Ω   in  ⁢ Ω 0

provided θ{d,h}L,B,Ω>0.

Proof.

For the sake of simplicity, throughout this proof we will denote

θ := θ { ν , h } ℒ , ℬ , Ω , θ 0 := θ { ν , h 0 } ℒ , ℬ 0 , Ω 0 .

By Theorem 4.1 (a) and Lemma 4.3, we have that

σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] ≤ σ 1 ⁢ [ d ⁢ ℒ - h 0 ⁢ ( 0 , ⋅ ) ; ℬ 0 , Ω 0 ] .

Thus, due to Theorem 1.1,

θ = θ 0 = 0 if  ⁢ σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] ≥ 0 ,
θ ≫ θ 0 = 0 if  ⁢ σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] < 0 ≤ σ 1 ⁢ [ d ⁢ ℒ - h 0 ⁢ ( 0 , ⋅ ) ; ℬ 0 , Ω 0 ] .

Hence, it remains to study the case when

σ 1 ⁢ [ d ⁢ ℒ - h 0 ⁢ ( 0 , ⋅ ) ; ℬ 0 , Ω 0 ] ≤ σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( 0 , ⋅ ) ; ℬ , Ω ] < 0 .

Then, by Theorem 1.1, θ,θ0≫0. Subsequently, we will consider the function f∈𝒞⁢(Ω¯0) defined, for each x∈Ω¯0, by

f ⁢ ( x ) := { θ ⁢ ( x ) ⁢ h 0 ⁢ ( θ ⁢ ( x ) , x ) - θ 0 ⁢ ( x ) ⁢ h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) θ ⁢ ( x ) - θ 0 ⁢ ( x ) if  ⁢ θ ⁢ ( x ) ≠ θ 0 ⁢ ( x ) , h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) + θ 0 ⁢ ( x ) ⁢ ∂ ∂ ⁡ u ⁢ h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) if  ⁢ θ ⁢ ( x ) = θ 0 ⁢ ( x ) .

By definition, θ-θ0 satisfies

d ⁢ ℒ ⁢ ( θ - θ 0 ) = θ ⁢ h ⁢ ( θ , ⋅ ) - θ 0 ⁢ h 0 ⁢ ( θ 0 , ⋅ ) ≥ θ ⁢ h 0 ⁢ ( θ , ⋅ ) - θ 0 ⁢ h 0 ⁢ ( θ 0 , ⋅ ) = ( θ - θ 0 ) ⁢ f   in  ⁢ Ω 0 ,

with strict inequality if h⁢(u,⋅)⪈h0⁢(u,⋅) in Ω0 for every u>0. Moreover, since (ℬ0,Ω0)⪯(ℬ,Ω), we have that

ℬ 0 ⁢ ( θ - θ 0 ) = ℬ 0 ⁢ θ ≥ 0   on  ⁢ ∂ ⁡ Ω 0 ,

with strict inequality if (ℬ0,Ω0)≺(ℬ,Ω). Thus, θ-θ0 is a supersolution of

{ ( d ⁢ ℒ - f ) ⁢ u = 0 in  ⁢ Ω 0 , ℬ 0 ⁢ u = 0 on  ⁢ ∂ ⁡ Ω 0 ,

and, actually, it is a strict supersolution if (ℬ0,Ω0)≺(ℬ,Ω), or h0⁢(u,⋅)≠h⁢(u,⋅) in Ω0 for all u≥0. We claim that σ1⁢[d⁢ℒ-f;ℬ0,Ω0]>0. Thanks to [31, Theorem 7.10], this entails that θ-θ0≥0 in Ω0 and that, actually, θ≫θ0 if it is strict, and so concluding the proof.

To prove σ1⁢[d⁢ℒ-f;ℬ0,Ω0]>0, we can argue as follows. Let x∈Ω¯0 be such that θ⁢(x)=θ0⁢(x). Then, by definition, and thanks to (H2),

f ⁢ ( x ) = h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) + θ 0 ⁢ ( x ) ⁢ ∂ ⁡ h 0 ∂ ⁡ u ⁢ ( θ 0 ⁢ ( x ) , x ) ≤ h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) ,

with strict inequality if θ0⁢(x)>0, while if x∈Ω¯0, with θ⁢(x)≠θ0⁢(x), then

f ⁢ ( x ) = θ ⁢ ( x ) ⁢ h 0 ⁢ ( θ ⁢ ( x ) , x ) - θ 0 ⁢ ( x ) ⁢ h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) θ ⁢ ( x ) - θ 0 ⁢ ( x )
= h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) + θ ⁢ ( x ) ⁢ h 0 ⁢ ( θ ⁢ ( x ) , x ) - h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) θ ⁢ ( x ) - θ 0 ⁢ ( x )
≤ h 0 ⁢ ( θ 0 ⁢ ( x ) , x ) ,

with strict inequality if θ⁢(x)>0. Note that θ⁢(x)>0 and θ0⁢(x)>0 for all x∈Ω0, and hence both inequalities are strict for all x∈Ω0. Therefore,

f ⪇ h 0 ⁢ ( θ 0 , ⋅ )   in  ⁢ Ω ¯ 0 ,

and hence, owing to Theorem 4.1 (a),

σ 1 ⁢ [ d ⁢ ℒ - f ; ℬ 0 , Ω 0 ] > σ 1 ⁢ [ d ⁢ ℒ - h 0 ⁢ ( θ 0 , ⋅ ) ; ℬ 0 , Ω 0 ] = 0 ,

which ends the proof. ∎

6 Proof of Theorem 1.2

Throughout this section, we assume that h satisfies (H1), (H2) and (H4). Hence, (H3) holds for sufficiently small d>0. The precise range of d where this occurs is unimportant for the proof, and so it is not specified. It should be remembered that the function

(6.1) Θ h ⁢ ( x ) := { 0 if  ⁢ h ⁢ ( ξ , x ) ⁢ < 0 ⁢  for all  ⁢ ξ > ⁢ 0 , ξ if there exists  ⁢ ξ > 0 ⁢  such that  ⁢ h ⁢ ( ξ , x ) = 0 ,

is well defined for all x∈Ω¯ and it is continuous in Ω¯. Let ΓR+ denote the union of the components of ΓR where Θh is everywhere positive. This section gives the proof of Theorem 1.2.

Remark 6.1.

For every x∈Ω¯, Θh⁢(x) provides us with the unique non-negative linearly stable, or linearly neutrally stable, steady-state solution of the associated kinetic model

{ u ′ ⁢ ( t ) = u ⁢ ( t ) ⁢ h ⁢ ( u ⁢ ( t ) , x ) , t ∈ [ 0 , + ∞ ) , u ⁢ ( 0 ) = u 0 ≥ 0 .

Note that

Θ h ⁢ ( x ) = 0 if  ⁢ h - 1 ⁢ ( ⋅ , x ) ⁢ ( 0 ) = ∅ ,
Θ h ⁢ ( x ) = max ⁡ { 0 , h - 1 ⁢ ( ⋅ , x ) ⁢ ( 0 ) } if  ⁢ h - 1 ⁢ ( ⋅ , x ) ⁢ ( 0 ) ≠ ∅ .

Remark 6.2.

The condition (H4) is necessary for the continuity of Θh on Ω¯, as the following simple example shows:

{ d ⁢ ( - Δ ⁢ u + u ) = u ⁢ ( - x 2 + e - u ) in  ⁢ Ω = ( - 1 , 1 ) , ℬ ⁢ u = 0 on  ⁢ ∂ ⁡ Ω = { - 1 , 1 } ,

where h⁢(u,x)=-x2+e-u for all x∈(-1,1) and u∈ℝ. According to (6.1), it becomes apparent that

Θ h ⁢ ( x ) = - log ⁡ x 2 , x ∈ [ - 1 , 1 ] ∖ { 0 } ,

which is discontinuous, and unbounded, at x=0. It turns out that in this example the function h⁢(u,x) satisfies (H1), (H2) and (H3) for sufficiently small d>0, however, it does not satisfies (H4). Therefore, condition (H4) is the minimal necessary condition required to guarantee the continuity of Θh⁢(x).

The proof of Theorem 1.2 follows after a series of results of a technical nature, some of them of great interest on their own. The first one is a consequence of Theorem 1.3 in the special case r=2.

Lemma 6.3.

Let ξ1,ξ2∈C⁢(Ω¯) be such that ξ1⁢(x)<ξ2⁢(x) for all x∈Ω¯. Then the following hold:

  1. There exists Φ ∈ 𝒞 2 ⁢ ( Ω ¯ ) such that ξ 1 ≤ Φ ≤ ξ 2 in Ω ¯ and R ⁢ Φ ⁢ ( x ) > 0 for all x ∈ Γ R .

  2. There exists Φ ∈ 𝒞 2 ⁢ ( Ω ¯ ) such that ξ 1 ≤ Φ ≤ ξ 2 in Ω ¯ and R ⁢ Φ ⁢ ( x ) < 0 for all x ∈ Γ R .

Proof.

By Theorem 1.3 applied to the conormal vector field, there exist an open neighborhood 𝒰⊂ℝN of ∂⁡Ω, a function ψ∈𝒞2⁢(𝒰;ℝ) and a constant τ>0 such that ψ⁢(x)<0 for all x∈𝒰∩Ω, ψ⁢(x)=0 for each x∈∂⁡Ω and

(6.2) ∂ ⁡ ψ ∂ ⁡ 𝝂 ⁢ ( x ) ≥ τ   for all  ⁢ x ∈ ∂ ⁡ Ω .

Let ε>0 be such that

ε < min Ω ¯ ⁡ ( ξ 2 - ξ 1 ) .

Then

ξ 1 ⁢ ( x ) + ε 2 < ξ 2 ⁢ ( x ) - ε 2   for all  ⁢ x ∈ Ω ¯ ,

and hence there exists ϕ∈𝒞∞⁢(Ω¯) such that

ξ 1 ⁢ ( x ) + ε 2 < ϕ ⁢ ( x ) < ξ 2 ⁢ ( x ) - ε 2   for all  ⁢ x ∈ Ω ¯ .

Consider, for each M∈ℝ, the map ϕM∈𝒞2⁢(𝒰∩Ω¯) defined by

ϕ M ⁢ ( x ) := ϕ ⁢ ( x ) - 1 + e M ⁢ ψ ⁢ ( x ) , x ∈ 𝒰 ∩ Ω ¯ .

By the continuity of ϕM, and the fact that ϕM⁢(x)=ϕ⁢(x) for all x∈∂⁡Ω, we can reduce 𝒰 to some open set 𝒰M, with ∂⁡Ω⊂𝒰M⊂𝒰, so that

ξ 1 ⁢ ( x ) + ε 2 < ϕ M ⁢ ( x ) < ξ 2 ⁢ ( x ) - ε 2   for all  ⁢ x ∈ 𝒰 M ∩ Ω ¯ .

On the other hand, since ψ⁢(x)=0 for all x∈∂⁡Ω, it becomes apparent that, for every x∈ΓR,

R ⁢ ϕ M ⁢ ( x ) = R ⁢ ϕ ⁢ ( x ) + R ⁢ ( e M ⁢ ψ ⁢ ( x ) - 1 )
= R ⁢ ϕ ⁢ ( x ) + M ⁢ e M ⁢ ψ ⁢ ( x ) ⁢ ∂ ⁡ ψ ∂ ⁡ 𝝂 ⁢ ( x ) + β ⁢ ( x ) ⁢ ( e M ⁢ ψ ⁢ ( x ) - 1 )
= R ⁢ ϕ ⁢ ( x ) + M ⁢ ∂ ⁡ ψ ∂ ⁡ 𝝂 ⁢ ( x ) .

According to (6.2), for sufficiently large M>0, one can get R⁢ϕM⁢(x)>0 for all x∈ΓR. So, in order to get part (a), it suffices to choose Φ equal to ϕM in a neighborhood of ∂⁡Ω. Similarly, by choosing M<0 sufficiently large, part (b) can be easily accomplished.

In each of these cases, once we have fixed the appropriate M, it remains to take Φ as any smooth extension of ϕM from a neighborhood 𝒱 of ∂⁡Ω, with 𝒱⊂𝒰M, to Ω¯ in such a way that ξ1⁢(x)<Φ⁢(x)<ξ2⁢(x) for all x∈Ω¯. This can be accomplished through an appropriate cutoff function of class 𝒞∞. ∎

Remark 6.4.

Note that if ξ1≥0 in Ω¯, then the function Φ provided by Lemma 6.3 (a) satisfies ℬ⁢Φ≥0 on ∂⁡Ω, whereas if ξ2≤0 in Ω¯, then the function Φ provided by Lemma 6.3 (b) verifies ℬ⁢Φ≤0 on ∂⁡Ω.

The next result provides us with a global uniform estimate in Ω¯, when d∼0, for the non-negative solutions of (1.1).

Lemma 6.5.

For every ε>0, there exists d0=d0⁢(ε)>0 such that θ{d,h}≤Θh+ε in Ω¯ for all d∈(0,d0).

Proof.

Subsequently, we suppose that d has been chosen sufficiently small so that (H3) holds. For a given ε>0, set

ξ 1 := Θ h + ε 2 > 0 , ξ 2 := Θ h + ε .

By Lemma 6.3 (a) and Remark 6.4, there exists Φ∈C2⁢(Ω¯) such that

0 < Θ h + ε 2 ≤ Φ ≤ Θ h + ε   in  ⁢ Ω ¯    and    ℬ ⁢ Φ ≥ 0   on  ⁢ ∂ ⁡ Ω .

In particular, Φ⁢(x)>Θh⁢(x) for all x∈Ω¯. Thus, since h⁢(Θh⁢(x),x)≤0 for all x∈Ω¯ and, owing to (H2), it is strictly decreasing in the first variable, we find that

h ⁢ ( Φ ⁢ ( x ) , x ) < 0   for all  ⁢ x ∈ Ω ¯ .

Hence, setting

d 0 := max x ∈ Ω ¯ ⁡ ( Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) ) min ⁡ { 0 , min x ∈ Ω ¯ ⁡ ℒ ⁢ Φ ⁢ ( x ) } ∈ ( 0 , + ∞ ] ,

it becomes apparent that, for every d<d0,

Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) ≤ max x ∈ Ω ¯ ⁡ ( Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) ) ≤ d ⁢ min x ∈ Ω ¯ ⁡ ℒ ⁢ Φ ⁢ ( x ) ≤ d ⁢ ℒ ⁢ Φ ⁢ ( x )   in  ⁢ Ω ¯ .

Note that this estimate holds true for all d>0 if minx∈Ω¯⁡ℒ⁢Φ⁢(x)≥0, because, by construction,

Φ ⁢ h ⁢ ( Φ , ⋅ ) < 0   in  ⁢ Ω ¯ .

This explains why we are setting d0=+∞ when minx∈Ω¯⁡ℒ⁢Φ⁢(x)≥0. On the other hand, when we have minx∈Ω¯⁡ℒ⁢Φ⁢(x)<0, the value of d0 becomes

d 0 := max x ∈ Ω ¯ ⁡ ( Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) ) min x ∈ Ω ¯ ⁡ ℒ ⁢ Φ ⁢ ( x ) = - max x ∈ Ω ¯ ⁡ ( Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) ) - min x ∈ Ω ¯ ⁡ ℒ ⁢ Φ ⁢ ( x ) > 0 .

Thus,

- d ⁢ min x ∈ Ω ¯ ⁡ ℒ ⁢ Φ ⁢ ( x ) < - max x ∈ Ω ¯ ⁡ ( Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) )

for all d<d0, or, equivalently,

max x ∈ Ω ¯ ⁡ ( Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) ) < d ⁢ min x ∈ Ω ¯ ⁡ ℒ ⁢ Φ ⁢ ( x ) ,

which also shows the previous estimate in this case.

Consequently, Φ provides us with a positive supersolution of (1.1). Consider the function f∈𝒞⁢(Ω¯) defined, for each x∈Ω¯, by

f ⁢ ( x ) := { Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) - θ { d , h } ⁢ ( x ) ⁢ h ⁢ ( θ { d , h } ⁢ ( x ) , x ) Φ ⁢ ( x ) - θ { d , h } ⁢ ( x ) if  ⁢ Φ ⁢ ( x ) ≠ θ { d , h } ⁢ ( x ) , h ⁢ ( Φ ⁢ ( x ) , x ) + Φ ⁢ ( x ) ⁢ ∂ ⁡ h ∂ ⁡ u ⁢ ( Φ ⁢ ( x ) , x ) if  ⁢ Φ ⁢ ( x ) = θ { d , h } ⁢ ( x ) .

Therefore, the function Φ-θ{d,h} is a supersolution of

{ ( d ⁢ ℒ - f ) ⁢ u = 0 in  ⁢ Ω , ℬ ⁢ u = 0 on  ⁢ ∂ ⁡ Ω .

Now, either θ{d,h}≡0, which ends the proof, or

θ { d , h } ≫ 0 , σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( θ { d , h } , ⋅ ) ; ℬ , Ω ] = 0 .

In the latter case, it is easily seen that (H2) implies f⪇h⁢(θ{d,h},⋅) in Ω¯. Thus, for every d∈(0,d0), it follows from Theorem 4.1 (a) that

σ 1 ⁢ [ d ⁢ ℒ - f ; ℬ , Ω ] > σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( θ { d , h } , ⋅ ) ; ℬ , Ω ] = 0 .

By [31, Theorem 7.10], we may infer that, for every 0<d<d0,

θ { d , h } ⁢ ( x ) ≤ Φ ⁢ ( x ) ≤ Θ h ⁢ ( x ) + ε   for all  ⁢ x ∈ Ω ¯ .

The proof is complete. ∎

The following result provides us with Theorem 1.2 in the special case when ΓR=∅.

Proposition 6.6.

For any compact subset K of Ω∪Θh-1⁢(0), we have that

lim d ↓ 0 ⁡ θ { d , h } Ω = Θ h   uniformly in  ⁢ K .

Proof.

Fix ε>0. By Lemma 6.5, there exists d0=d0⁢(ε)>0 such that

θ { d , h } ≤ Θ h + ε   for all  ⁢ x ∈ K ⊂ Ω ¯ , d ∈ ( 0 , d 0 ) .

In order to get a lower estimate, we will first assume h⁢(u,x) to be autonomous, i.e., h⁢(u,x)=h⁢(u) for all (u,x)∈ℝ×Ω¯. In such a case, Θh is a non-negative constant. Since θ{d,h} is non negative, it is obvious that θ{d,h}>Θh-ε in Ω¯ for all d>0 if Θh=0. Thus, the following estimate holds:

Θ h - ε ≤ θ { d , h } ≤ Θ h + ε   for all  ⁢ x ∈ K ⊂ Ω ¯ , d ∈ ( 0 , d 0 ) .

In order to get the lower estimate when Θh is a positive constant (necessarily, h⁢(0)>0, h⁢(Θh)=0 and K⊂Ω, because Θh-1⁢(0)=∅), we consider ε~∈(0,min⁡{2⁢Θh,ε}), x0∈K, and ρ>0 be such that ρ<dist⁢(K,∂⁡Ω). For these choices, B¯ρ⁢(x0)⊂Ω. Let φ≫0 be the principal eigenfunction associated to σ1⁢[ℒ;D,Bρ⁢(x0)] normalized so that ∥φ∥∞=1/2, and define the function ϕ∈⋂p>N⁡W2,p⁢(Bρ⁢(x0)) through

ϕ := { φ in  ⁢ B ¯ ρ ⁢ ( x 0 ) ∖ B ¯ ρ / 2 ⁢ ( x 0 ) , φ ~ in  ⁢ B ¯ ρ / 2 ⁢ ( x 0 ) ,

where φ~ is any sufficiently smooth function chosen so that ϕ⁢(x)>0 for all x∈Bρ⁢(x0), ϕ⁢(x0)=1 and ∥ϕ∥∞=1. Set

Φ := ( Θ h - ε ~ 2 ) ⁢ ϕ   in  ⁢ B ¯ ρ ⁢ ( x 0 ) .

Then, by construction, D⁢Φ=0 on ∂⁡Bρ⁢(x0) and

0 < Φ ⁢ ( x ) ≤ Θ h - ε ~ 2   for all  ⁢ x ∈ B ρ ⁢ ( x 0 ) ,

since Θh is a constant greater than ε~/2 and ∥ϕ∥∞=1. Thus, taking into account that, owing to (H2), h⁢(u) is strictly decreasing in u>0, it is apparent that

h ⁢ ( Φ ⁢ ( x ) ) ≥ h ⁢ ( Θ h - ε ~ 2 ) > h ⁢ ( Θ h ) = 0   for all  ⁢ x ∈ B ¯ ρ ⁢ ( x 0 ) ,

and hence

min x ∈ B ¯ ρ ⁢ ( x 0 ) ⁡ h ⁢ ( Φ ⁢ ( x ) ) ≥ h ⁢ ( Θ h - ε ~ 2 ) > 0 .

On the other hand, the function ℒ⁢Φ/Φ is continuous in B¯ρ⁢(x0) because Φ⁢(x)>0 for all x∈Bρ⁢(x0) and

ℒ ⁢ Φ ⁢ ( x ) / Φ ⁢ ( x ) = σ 1 ⁢ [ ℒ ; D , B ρ ⁢ ( x 0 ) ] ∈ ℝ   for all  ⁢ x ∈ ∂ ⁡ B ρ ⁢ ( x 0 ) .

Although unnecessary, ρ⁢(x0) can be shortened so that σ1⁢[ℒ;D,Bρ⁢(x0)]>0, because, due to the Faber–Krahn inequality, limρ→0⁡σ1⁢[ℒ;D,Bρ⁢(x0)]=+∞ (see, e.g., [31, Proposition 8.6]). Thus, setting

0 < d x 0 < min B ¯ ρ ⁢ ( x 0 ) ⁡ h ⁢ ( Φ ) max B ¯ ρ ⁢ ( x 0 ) | ℒ Φ / Φ | ,

we have that, for every d∈(0,dx0),

d max B ¯ ρ ⁢ ( x 0 ) | ℒ Φ / Φ | ⪇ h ( Φ )   in  B ¯ ρ ( x 0 ) ,

and so

d ⁢ ℒ ⁢ Φ = d ⁢ Φ ⁢ ℒ ⁢ Φ / Φ ≤ d ⁢ Φ ⁢ max B ¯ ρ ⁢ ( x 0 ) ⁡ | ℒ ⁢ Φ / Φ | ⪇ Φ ⁢ h ⁢ ( Φ )   in  ⁢ B ¯ ρ ⁢ ( x 0 ) .

Therefore, Φ provides us with a strict subsolution of

{ d ⁢ ℒ ⁢ u = u ⁢ h ⁢ ( u ) in  ⁢ B ρ ⁢ ( x 0 ) , u = 0 on  ⁢ ∂ ⁡ B ρ ⁢ ( x 0 ) .

Equivalently, θ{d,h}D,Bρ⁢(x0)-Φ is a strict supersolution of

{ ( d ⁢ ℒ - f ) ⁢ u = 0 in  ⁢ B ρ ⁢ ( x 0 ) , u = 0 on  ⁢ ∂ ⁡ B ρ ⁢ ( x 0 ) ,

where f∈𝒞⁢(B¯ρ⁢(x0)) stands for the function defined, for every x∈B¯ρ⁢(x0), by

f ( x ) := { θ { d , h } D , B ρ ⁢ ( x 0 ) ⁢ ( x ) ⁢ h ⁢ ( θ { d , h } D , B ρ ⁢ ( x 0 ) ⁢ ( x ) ) - Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) ) θ { d , h } D , B ρ ⁢ ( x 0 ) ⁢ ( x ) - Φ ⁢ ( x ) if  ⁢ Φ ⁢ ( x ) ≠ θ { d , h } D , B ρ ⁢ ( x 0 ) ⁢ ( x ) , h ⁢ ( Φ ⁢ ( x ) ) + Φ ⁢ ( x ) ⁢ h ′ ⁢ ( Φ ⁢ ( x ) ) if  ⁢ Φ ⁢ ( x ) = θ { d , h } D , B ρ ⁢ ( x 0 ) ⁢ ( x ) .

Moreover, thanks to (H2), f⪇h⁢(θ{d,h}D,Bρ⁢(x0)) in B¯ρ⁢(x0) and thus, by the monotonicity of the principal eigenvalue with respect to the potential established by Theorem 4.1 (a), it becomes apparent that

σ 1 ⁢ [ d ⁢ ℒ - f ; D , B ρ ⁢ ( x 0 ) ] > σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( θ { d , h } D , B ρ ⁢ ( x 0 ) ) ; D , B ρ ⁢ ( x 0 ) ] = 0 .

Note that h⁢(0)>0, and hence, owing to Corollary 5.1 (b), θ{d,h}D,Bρ⁢(x0)≫0 for sufficiently small d>0. Consequently, by [31, Theorem 7.10], we find that, for every d∈(0,dx0),

(6.3) Φ ⁢ ( x ) < θ { d , h } D , B ρ ⁢ ( x 0 ) ⁢ ( x )   for all  ⁢ x ∈ B ρ ⁢ ( x 0 ) .

Moreover, by Lemma 5.2,

(6.4) θ { d , h } D , B ρ ⁢ ( x 0 ) ≤ θ { d , h } ℬ , Ω   in  ⁢ B ρ ⁢ ( x 0 ) .

On the other hand, since Φ∈𝒞⁢(B¯ρ⁢(x0)) and Φ⁢(x0)=Θh-ε~/2, there exist ρx0∈(0,ρ) such that

(6.5) Φ ⁢ ( x ) ≥ Θ h - ε ~ > Θ h - ε   for all  ⁢ x ∈ B ρ x 0 ⁢ ( x 0 ) .

According to (6.3)–(6.5), we find that

θ { d , h } ℬ , Ω > Θ h - ε   in  ⁢ B ρ x 0 ⁢ ( x 0 ) ⁢  for all  ⁢ d ∈ ( 0 , d x 0 ) .

As K is compact, we can extract x1,…,xn∈K such that K⊂⋃i=1nBρxi⁢(xi), and hence for every d<mini⁡dxi, d>0, the estimate θ{d,h}ℬ,Ω≥Θh-ε holds in K. This ends the proof when h is independent of x.

Subsequently, we will assume that h⁢(u,x) is a general function satisfying (H1), (H2) and (H4). Then, for sufficiently small d>0, it is obvious that

θ { d , h } ℬ , Ω ⁢ ( x ) ≥ 0 ≥ Θ h ⁢ ( x ) - ε   for all  ⁢ x ∈ Θ h - 1 ⁢ ( [ 0 , ε ] ) .

As this provides us with a satisfactory lower estimate in K∩Θh-1⁢([0,ε]), in order to extend it to K, it remains to show that there exists d1>0 such that, for every d∈(0,d1),

θ { d , h } ℬ , Ω ⁢ ( x ) ≥ Θ h ⁢ ( x ) - ε   for all  ⁢ x ∈ K 0 := K ∩ Θ h - 1 ⁢ ( [ ε , max Ω ¯ ⁡ Θ h ] ) ⊂ Ω .

Should K0 be empty, the proof is complete. So, suppose that K0 is nonempty and pick x0∈K0 and ρ>0 such that

B ¯ ρ ⁢ ( x 0 ) ⊂ Ω ∩ Θ h - 1 ⁢ ( ε 2 , + ∞ ) .

By construction, Θh⁢(x)>ε/2>0 for all x∈B¯ρ⁢(x0) and hence, owing to (H2),

min B ¯ ρ ⁢ ( x 0 ) ⁡ h ⁢ ( 0 , ⋅ ) > 0   and   min B ¯ ρ ⁢ ( x 0 ) ⁡ Θ h > ε 2 > 0 .

Actually, by continuity, ρ>0 can be shortened, if necessary, so that

(6.6) min B ¯ ρ ⁢ ( x 0 ) ⁡ Θ h ≥ Θ h ⁢ ( x ) - ε 2   for all  ⁢ x ∈ B ¯ ρ ⁢ ( x 0 ) .

The rest of the proof consists in reducing ourselves to the previous case, by establishing the existence of an autonomous function H⁢(u) satisfying (H1), (H2), (H4), and such that

(6.7) H ⁢ ( u ) ≤ h ⁢ ( u , x )   for all  ⁢ u ≥ 0 , x ∈ B ¯ ρ ⁢ ( x 0 ) ,

and

(6.8) min x ∈ B ¯ ρ ⁢ ( x 0 ) ⁡ Θ h ⁢ ( x ) - ε 4 ≤ Θ H ≤ min x ∈ B ¯ ρ ⁢ ( x 0 ) ⁡ Θ h ⁢ ( x ) .

The most natural candidate function for a (globally defined in ℝ) H⁢(u) is

h min ⁢ ( u ) := { min x ∈ B ¯ ρ ⁢ ( x 0 ) ⁡ h ⁢ ( u , x ) if  ⁢ u ≥ 0 , min x ∈ B ¯ ρ ⁢ ( x 0 ) ⁡ h ⁢ ( 0 , x ) - u if  ⁢ u < 0 .

Obviously, hmin∈𝒞⁢(ℝ) and it is strictly decreasing, though, in general, it is not of class 𝒞1⁢(ℝ). Thus, in order to construct H⁢(u) satisfying (6.7), (6.8), (H1), (H2) and (H4), we begin by considering the function

G ⁢ ( u ) := min ⁡ { - δ , 4 ε ⁢ h min ⁢ ( u + ε 4 ) } < 0 , u ∈ ℝ ,

with sufficiently small δ>0, to be chosen later, and then we take, for every u∈ℝ,

H ⁢ ( u ) := ∫ Θ min - ε 4 u G ⁢ ( s ) ⁢ 𝑑 s , where  ⁢ Θ min ≡ min x ∈ B ¯ ρ ⁢ ( x 0 ) ⁡ Θ h ⁢ ( x ) .

Since G is a continuous function, H is a function of class 𝒞1⁢(ℝ), and hence (H1) holds. Moreover, by definition, H′⁢(u)=G⁢(u)<0 for all u∈ℝ. Thus, (H2) holds. Furthermore, since H⁢(Θmin-ε4)=0, (H4) also holds, because H⁢(u)<0 for all u>Θmin-ε4. Actually, (6.8) holds too, since ΘH=Θmin-ε4, by definition (see (6.1) if necessary). It remains to shorten δ, if necessary, to get (6.7). Suppose u≤Θmin-ε4. Then u+ε4≤Θmin, and hence hmin⁢(u+ε4)≥0 and G⁢(u)=-δ, which implies

H ⁢ ( u ) = - δ ⁢ ( u - Θ min + ε 4 ) .

Thus, for sufficiently small δ,

H ⁢ ( 0 ) = δ ⁢ ( Θ min - ε 4 ) < h min ⁢ ( Θ min - ε 4 ) ,

and therefore

H ⁢ ( u ) ≤ H ⁢ ( 0 ) < h min ⁢ ( Θ min - ε 4 ) ≤ h min ⁢ ( u )

for all u∈[0,Θmin-ε4]. So, (6.7) holds in this interval. When u∈(Θmin-ε4,Θmin), by construction,

H ⁢ ( u ) = ∫ Θ min - ε 4 u G ⁢ ( s ) ⁢ 𝑑 s < 0 < h min ⁢ ( u ) ,

and hence (6.7) holds in [0,Θmin). Finally, when u≥Θmin, we find that

G ⁢ ( u ) = min ⁡ { - δ , 4 ε ⁢ h min ⁢ ( u + ε 4 ) } ≤ 4 ε ⁢ h min ⁢ ( u + ε 4 ) < 0

and, consequently,

H ⁢ ( u ) = ∫ Θ min - ε 4 u G ⁢ ( s ) ⁢ 𝑑 s ≤ 4 ε ⁢ ∫ Θ min - ε 4 u h min ⁢ ( s + ε 4 ) ⁢ 𝑑 s = 4 ε ⁢ ∫ Θ min u + ε 4 h min ⁢ ( t ) ⁢ 𝑑 t
≤ 4 ε ⁢ ∫ u u + ε 4 h min ⁢ ( t ) ⁢ 𝑑 t < 4 ε ⁢ ∫ u u + ε 4 h min ⁢ ( u ) ⁢ 𝑑 t = h min ⁢ ( u ) ,

which shows (6.7).

By Lemma 5.2, for sufficiently small d>0, the following estimate holds:

(6.9) θ { d , H } D , B ρ ⁢ ( x 0 ) ≤ θ { d , h } ℬ , Ω   in  ⁢ B ρ ⁢ ( x 0 ) .

By the first part of the proof, since H⁢(u) does not depend on x∈Ω, there exists dx0,ε>0 such that

(6.10) Θ H - ε 4 ≤ θ { d , H } D , B ρ ⁢ ( x 0 )   in  ⁢ B ¯ ρ / 2 ⁢ ( x 0 ) ⁢  for all ⁢ d ∈ ( 0 , d x 0 , ε ) .

Combining (6.6), (6.8), (6.9) and (6.10) yields

Θ h - ε ≤ min B ¯ ρ ⁢ ( x 0 ) ⁡ Θ h - ε 2 ≤ Θ H - ε 4 ≤ θ { d , H } D , B ρ ⁢ ( x 0 ) ≤ θ { d , h } ℬ , Ω   in  ⁢ B ρ / 2 ⁢ ( x 0 )

for all d∈(0,dx0,ε). Lastly, since K0 is compact, there exist x1,…,xn∈K0 such that K0⊂⋃i=1nBρi/2⁢(xi). Therefore,

Θ h - ε ≤ θ { d , h } ℬ , Ω   in  ⁢ K 0 ⁢ for all  ⁢ d < d 0 := min 1 ≤ i ≤ n ⁡ d x i , ε ,

which ends the proof. ∎

We already have all the necessary tools to complete the proof of Theorem 1.2.

Proof of Theorem 1.2.

Since h satisfies (H2), Θh≡0 if maxΩ¯⁡h⁢(0,⋅)≤0. Should it be the case, the result is a direct consequence from Proposition 6.6. So, subsequently, we assume that

max Ω ¯ ⁡ h ⁢ ( 0 , ⋅ ) > 0 .

Then, by Corollary 5.1 (b), θ{d,h}≫0 for sufficiently small d>0.

Thanks to Proposition 6.6, Theorem 1.2 holds on any compact subset of Ω∪Θh-1⁢(0). Hence, it remains to prove the theorem on a neighborhood of ΓR+. Let γ be a component of ΓR+. By the definition of ΓR+, we have that Θh⁢(x)>0 for all x∈γ. By the continuity of Θh, there exists ρ>0 such that

ε 0 := min Ω ¯ γ , ρ ⁡ Θ h > 0 , where  ⁢ Ω γ , ρ ≡ { x ∈ Ω : dist ⁡ ( x , γ ) < ρ } .

Pick ε∈(0,ε0). By the proof of Theorem 1.3, we can shorten ρ, if necessary, so that

{ x ∈ Ω : dist ⁡ ( x , γ ) = ρ } = ∂ ⁡ Ω γ , ρ ∩ Ω

is diffeomorphic to γ, and so of class 𝒞2. Hence, Ωγ,ρ is an open subdomain of Ω with boundary of class 𝒞2, consisting of two components ∂⁡Ωγ,ρ∩Ω and γ for sufficiently small ρ>0.

Subsequently, we consider the compact subset of Ω

K γ , ρ := { x ∈ Ω : ρ / 2 ≤ dist ⁡ ( x , γ ) ≤ ρ } .

By Proposition 6.6, there exists dρ>0 such that

(6.11) Θ h - ε 2 ≤ θ { d , h }   in  ⁢ K γ , ρ ⁢  for all  ⁢ d < d ρ .

By applying Lemma 6.3 (a) and Remark 6.4 with the choices

ξ 1 ( x ) := Θ h ( x ) - ε ( ≥ ε 0 - ε > 0 )

and

ξ 2 ⁢ ( x ) := Θ h ⁢ ( x ) - 3 ⁢ ε 4 < Θ h ⁢ ( x ) , x ∈ Ω ¯ γ , ρ / 2 ,

there exists Φ∈𝒞2⁢(Ω¯γ,ρ/2) such that

(6.12) Θ h - ε ≤ Φ ≤ Θ h - 3 ⁢ ε 4   in  ⁢ Ω γ , ρ / 2 ⁢  and  ⁢ R ⁢ Φ ≤ 0 ⁢  on  ⁢ γ .

In particular, since ∂⁡Ωγ,ρ/2∩Ω⊂Kγ,ρ, we may infer from (6.11) and (6.12) that

(6.13) θ { d , h } ≥ Θ h - ε 2 = Θ h - 3 ⁢ ε 4 + ε 4 ≥ Φ + ε 4   on  ⁢ ∂ ⁡ Ω γ , ρ / 2 ∩ Ω ⁢  for all  ⁢ d < d ρ .

Moreover, by (H2), since Φ⁢(x)<Θh⁢(x) for all x∈Ω¯γ,ρ/2, we have that

min x ∈ Ω ¯ γ , ρ / 2 ⁡ h ⁢ ( Φ ⁢ ( x ) , x ) > min x ∈ Ω ¯ γ , ρ / 2 ⁡ h ⁢ ( Θ h ⁢ ( x ) , x ) = 0 .

Thus, shortening dρ, if necessary, so that

d ρ < min x ∈ Ω ¯ γ , ρ / 2 ⁡ Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) max ⁡ { 0 , max Ω ¯ γ , ρ / 2 ⁡ ℒ ⁢ Φ } ,

we are driven to

d ⁢ ℒ ⁢ Φ ≤ Φ ⁢ h ⁢ ( Φ , ⋅ )   in  ⁢ Ω γ , ρ / 2 ⁢  for all  ⁢ d < d ρ .

Let us denote by f∈𝒞⁢(Ω¯γ,ρ/2) the function defined, for every x∈Ω¯γ,ρ/2, through

f ( x ) := { θ { d , h } ⁢ ( x ) ⁢ h ⁢ ( θ { d , h } ⁢ ( x ) , x ) - Φ ⁢ ( x ) ⁢ h ⁢ ( Φ ⁢ ( x ) , x ) θ { d , h } ⁢ ( x ) - Φ ⁢ ( x ) if  ⁢ θ { d , h } ⁢ ( x ) ≠ Φ ⁢ ( x ) , h ⁢ ( Φ ⁢ ( x ) , x ) + Φ ⁢ ( x ) ⁢ ∂ ⁡ h ∂ ⁡ u ⁢ ( Φ ⁢ ( x ) , x ) if  ⁢ θ { d , h } ⁢ ( x ) ≠ Φ ⁢ ( x ) .

Then, for every d<dρ, taking into account (6.13), the function w:=θ{d,h}-Φ satisfies

{ ( d ⁢ ℒ - f ) ⁢ w ≥ 0 in  ⁢ Ω γ , ρ / 2 , ℬ ⁢ w = R ⁢ w > 0 on  ⁢ γ , w ≥ ε 4 > 0 on  ⁢ ∂ ⁡ Ω γ , ρ / 2 ∩ Ω .

Therefore, w provides us with a strict supersolution of [d⁢ℒ-f;ℬ0,Ωγ,ρ/2], where

ℬ 0 := { ℬ on  ⁢ γ , D on  ⁢ ∂ ⁡ Ω γ , ρ / 2 ∖ γ .

Since, owing to (H2), h is strictly decreasing in the first variable, f⪇h⁢(θ{d,h},⋅) in Ωγ,ρ/2. Moreover, we already know that θ{d,h}≫0 for sufficiently small d>0. Thus, it follows from Theorem 4.1 (a) and Lemma 4.3 that

σ 1 ⁢ [ d ⁢ ℒ - f ; ℬ 0 , Ω γ , ρ / 2 ] > σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( θ { d , h } , ⋅ ) ; ℬ 0 , Ω γ , ρ / 2 ]
> σ 1 ⁢ [ d ⁢ ℒ - h ⁢ ( θ { d , h } , ⋅ ) ; ℬ , Ω ] = 0

for sufficiently small d>0. Therefore, due to [31, Theorem 7.10], and taking into account (6.12), we conclude that

θ { d , h } ≫ Φ ≥ Θ h - ε   in  ⁢ Ω γ , ρ / 2 ⁢  for sufficiently small  ⁢ d > 0 .

The proof is complete. ∎

Communicated by Laurent Veron

Funding source: Ministry of Economy and Competitiveness of Spain

Award Identifier / Grant number: MT2015-65899-P

Funding source: Ministry of Education and Culture of Spain

Award Identifier / Grant number: FPU15/04755

Funding statement: This work has been partially supported by the Ministry of Economy and Competitiveness of Spain under Research Grant MT2015-65899-P, by the Institute of Interdisciplinary Mathematics (IMI) of Complutense University, and by the Ministry of Education and Culture of Spain under Fellowship Grant FPU15/04755.

Acknowledgements

The authors thank the reviewer for an extremely careful reading of the paper and for bringing to our attention the role played by the tubular neighborhood theorem on some classical results involving the normal distance.

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Received: 2018-04-05
Revised: 2018-09-09
Accepted: 2018-10-02
Published Online: 2018-10-31
Published in Print: 2019-02-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2018-2034
Keywords for this article
Singular Perturbation; Positive Solution; Generalized Logistic Equation; Non-classical Mixed Boundary Conditions; Conormal Vector Field; Conormal Projection; Conormal Distance; Regularity
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