Quasilinear Elliptic Equations with Singular Nonlinearity

João Marcos do Ó; Esteban da Silva

doi:10.1515/ans-2015-5031

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Quasilinear Elliptic Equations with Singular Nonlinearity

João Marcos do Ó and Esteban da Silva

Published/Copyright: April 6, 2016

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From the journal Advanced Nonlinear Studies Volume 16 Issue 2

Abstract

In this paper, motivated by recent works on the study of the equations which model electrostatic MEMS devices, we study the quasilinear elliptic equation

(Pλ) { - ( r α ⁢ | u ′ | β ⁢ u ′ ) ′ = λ ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ⁢ ( r ) < 1 , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 .

According to the choice of the parameters α, β, and γ, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the p-Laplacian, and the k-Hessian. We prove the existence of an extremal parameter λ∗>0 such that, for λ∈(0,λ∗), there exists a minimal solution u¯λ and, for λ>λ∗, there exists no solution of any kind. We also study the behavior of the minimal branch of solutions and we prove uniqueness of solutions of (Pλ∗) for β>-1.

Keywords: Quasilinear Elliptic Equation; MEMS Capacitor; Extremal Solutions; Minimal Solutions; Regularity; Singular Nonlinearities

MSC 2010: 35J62; 35J70; 34B15; 35B35

1 Introduction

The main purpose of this paper is to investigate a class of radial quasilinear elliptic differential equations involving a singular nonlinearity of an inverse square type, which has recently received much attention due to its applicability on the modeling of electrostatic MEMS (micro electro-mechanical systems).

1.1 Motivation and Related Problems

MEMS are microdevices consisting of electrical and mechanical components combined together on a chip to produce a system of miniature dimensions (between 1 and 100 micrometers, that is, 0.001 and 0.1 millimeters, respectively – less than the thickness of a human hair). They are essential components of the modern technology that is currently driving telecommunications, commercial systems, biomedical engineering, and space exploration. For more information on the applications, development, etc. of the fundamental partial differential equations that model these devices, see Bernstein and Pelesko [20] and Esposito, Ghoussoub, and Guo [10].

We are motivated by recent works on the study of the equations that model a special MEMS component – an electrostatically controlled tunable deformable capacitor – that can be understood as consisting of a thin and deformable microplate whose shape we are representing by Ω (a bounded domain of ℝN), fixed along its boundary, coated with a negligibly thin metallic conducting film, and lying above one unit of a parallel rigid grounded plate as shown in Figure 1.

Figure 1

A deformable capacitor.

The membrane deflects towards the conducting plate when a voltage (represented here by λ) is applied. It may occur that the membrane touches the plate or gets ripped due the loss of stability between the forces acting on the system, thus, creating a singularity. The modeling is then based on the equilibrium between these forces. On the one hand, we have tension due to the stretching (given by the Laplacian of the deformation function) and rigidity (given by the bi-Laplacian of the deformation function). On the other hand, we have the electrostatic force that according to Coulomb’s law is inversely proportional to the square of the distance between the two charged plates. We can also consider the elastic and the electric potential energies associated with the deformation. In the stationary case, a very general model for the deformation u of the membrane is

(1.1) { α ⁢ Δ 2 ⁢ u = S ⁢ ( u ) ⁢ Δ ⁢ u + λ ⁢ f ⁢ ( x ) D ⁢ ( u ) ⁢ ( 1 - u ) 2 in ⁢ Ω , 0 ≤ u < 1 in ⁢ Ω , u = ∂ ⁡ u ∂ ⁡ η = 0 on ⁢ ∂ ⁡ Ω ,

where f is a varying dielectric permittivity profile, η is the unit outward normal to ∂⁡Ω, the constant α≥0 represents the membrane width, and the terms

S ⁢ ( u ) = β ⁢ ∫ 0 1 | ∇ ⁡ u | 2 ⁢ d x + γ and D ⁢ ( u ) = 1 + χ ⁢ ∫ 0 1 d ⁢ x ( 1 - u ) 2 with β , γ , χ ≥ 0

represent nonlocal parameters that affect the membrane’s deformation (the elastic and the electric potentials, respectively).

In the limit case of zero plate thickness, hence, for a thin membrane with zero rigidity, and neglecting inertial effects as well as nonlocal effects, that is, in dimensionless constants, we set α=β=χ=0. Then, (1.1) reduces to the semilinear elliptic problem

{ Δ ⁢ u = λ ⁢ f ⁢ ( x ) ( 1 - u ) 2 in ⁢ Ω , 0 < u < 1 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω ,

where, for simplicity, we have set γ=1. This problem has been studied in the past in a very general context as we can see in [7, 8, 9, 11, 12, 14, 13, 18].

Recently Castorina, Esposito, and Sciunzi [3] studied, for 1<p≤2, the p-MEMS equation (the MEMS equation for the p-Laplacian operator), that is,

{ Δ p ⁢ u = λ ⁢ h ⁢ ( x ) ⁢ f ⁢ ( u ) in ⁢ Ω , 0 < u < 1 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω ,

where f is a nondecreasing positive function, defined on [0,1) with a singularity at u=1. They established uniqueness results for semistable solutions and stability (in a strict sense) of minimal solutions for 1<p≤2. In [2, 3, 4], the authors also proved radial symmetry of the first eigenfunction for the p-MEMS problem on a ball. The radial form of the p-MEMS equation on a ball B∈ℝn, n≥2, is

(1.2) { - ( r n - 1 ⁢ | u ′ | p - 2 ⁢ u ′ ) ′ = λ ⁢ r n - 1 ⁢ f ⁢ ( r ) ( 1 - u ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ⁢ ( r ) < 1 , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 .

1.2 Problem Formulation

Motivated by the works cited above, and in the spirit of the pioneering work [6], we study here the following more general class of quasilinear elliptic equations, which includes (1.2) as a particular case, since we consider a continuous variation of the exponents in (1.2), including the radial case for non-integer-dimensional spaces, that is,

(Pλ) { - ( r α ⁢ | u ′ | β ⁢ u ′ ) ′ = λ ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ⁢ ( r ) < 1 , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 .

We prove the existence of a constant λ∗ such that, for λ∈(0,λ*), there exists a minimal classical solution uλ and, for λ>λ*, there are no solutions of any kind. In addition, we provide some estimates for λ∗. We also prove that the function λ↦uλ is increasing in the pointwise sense. Furthermore, we provide a way to compute the minimal solutions by approximation. The main result of this work concerns the uniqueness of the solution of (Pλ*) (the problem (Pλ) for λ=λ∗) for β>-1, which improves the results in [2] since our arguments can be also applied to the p-Laplace operator even for p>2 and when f⁢(r) is a general function under suitable assumptions.

Problems involving the quasilinear operator of the form L⁢u:=-(rα⁢|u′|β⁢u′)′ have been studied in various contexts and applications. It should be mentioned that the operator Lu corresponds to the radial form of various operators for a convenient choice of the parameters α, β, and γ as shown in Table 1.

Table 1

Radial form of various operators for choices of the parameters α, β, and γ.

Operator	α	β	γ
Laplacian	n - 1	0	n - 1
p-Laplacian (p>1)	n - 1	p - 2	n - 1
k-Hessian	n - k	k - 1	n - 1

We recall that the k-Hessian operator, usually represented by Sk⁢(D2⁢u), is defined as the sum of all principal k×k minors of the Hessian matrix D2⁢u. For instance, S1⁢(D2⁢u)=Δ⁢u and SN⁢(D2⁢u)=det⁡D2⁢u, the Monge–Ampère operator. Recently, Jacobsen and Schmitt [15] studied singular problems involving this class of operators. They determined precise existence and multiplicity results for radial solutions of the Liouville–Bratu–Gelfand problem associated with this class of quasilinear radial operators. In 1996, Clément, de Figueiredo, and Mitidieri [5] studied Brezis–Nirenberg-type problems for this class of quasilinear elliptic operators. For related problems on this subject, we also refer to [7, 16] and the references therein.

1.3 Outline

The paper is organized as follows. In the next subsection, we introduce some notation and terminologies used throughout the text. In Section 1.5, we present the main results of our work. In Section 2.1, we present some preliminary results, starting with some properties of eventual solutions, following with the existence of λ∗, and finishing with the presentation of a sub- and supersolution result suitable for our study. The proofs of the main results are given in Section 3.

1.4 Notation

Before stating our main results, we need to introduce some basic notation, terminologies, and the function space appropriate for studying problems involving the operator L⁢u=-(rα⁢|u′|β⁢u′)′.

Let X~ be the set of the Lloc1 real functions defined on (0,1) with distributional derivative in Lloc1 satisfying

(1.3) ∥ u ∥ β + 2 := ∫ 0 1 r α ⁢ | u ⁢ ( r ) | β + 2 ⁢ d r + ∫ 0 1 r α ⁢ | u ′ ⁢ ( r ) | β + 2 ⁢ d r < ∞ .

If β≥-1, equipping X~ with the usual operations makes it a real vector space and (1.3) defines a norm on this space. Every u∈X~ is, in particular, absolutely continuous on (0,1). Now, consider the subspace X^ of the elements of X~ such that

lim r → 1 ⁡ u ⁢ ( r ) = 0

and X, the completion of X^ with respect to the equivalent norm

(1.4) ∥ u ∥ X := ( ∫ 0 1 r α ⁢ | u ′ ⁢ ( r ) | β + 2 ⁢ d r ) 1 / ( β + 2 ) .

It is known that α<β+1 is a sufficient condition in order to have X=X^. See Kufner and Opic [19, 17] for details concerning the properties remarked in this paragraph.

Now, let us establish the notions of solution used in this work. For that, consider the general problem

(1.5) { - ( r α ⁢ | u ′ ⁢ ( r ) | β ⁢ u ′ ⁢ ( r ) ) ′ = h ⁢ ( r , u ⁢ ( r ) ) , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 ,

where h is a real-valued continuous function acting over a set H⊂ℝ2 whose projection over the first variable is (0,1).

Definition 1.1

Definition 1.1 (Kinds of Solutions of Problem (1.5))

(Integral Solution) A function u∈X is an integral solution of (1.5) provided that (r,u⁢(r))∈H for every r∈(0,1), h⁢(⋅,u)∈L1⁢((0,1)), and

(1.6) - r α ⁢ | u ′ ⁢ ( r ) | β ⁢ u ′ ⁢ ( r ) = ∫ 0 r h ⁢ ( s , u ⁢ ( s ) ) ⁢ d s a.e. in ⁢ ( 0 , 1 ) .
(Integral Supersolution) We say that u¯∈X is an integral supersolution of (1.5) when (r,u¯⁢(r))∈H for every r∈(0,1), h⁢(⋅,u¯)∈L1⁢((0,1)), and

- r α ⁢ | u ¯ ′ ⁢ ( r ) | β ⁢ u ¯ ′ ⁢ ( r ) ≥ ∫ 0 r h ⁢ ( s , u ¯ ⁢ ( s ) ) ⁢ d s a.e. in ⁢ ( 0 , 1 ) .
(Integral Subsolution) In the same way, a function u¯∈X is said to be an integral subsolution of (1.5) if (r,u¯⁢(r))∈H for every r∈(0,1), h⁢(⋅,u¯)∈L1⁢((0,1)), and

- r α ⁢ | u ¯ ′ ⁢ ( r ) | β ⁢ u ¯ ′ ⁢ ( r ) ≤ ∫ 0 r h ⁢ ( s , u ¯ ⁢ ( s ) ) ⁢ d s a.e. in ⁢ ( 0 , 1 ) .
(Minimal Solution) We call minimal solution a positive (in any sense defined above) solution u∈X of (1.5) such that, for any other positive solution v∈X of (1.5), it satisfies 0≤u⁢(r)≤v⁢(r) for all r∈(0,1).

1.5 Statements of the Main Results

Now, we are able to state the main results of this work. In the following, we are always assuming β>-1, f:(0,1)→(0,+∞) continuous, and the pairwise condition over γ and f

(1.7) 0 < ∫ 0 1 r γ ⁢ f ⁢ ( r ) ⁢ d r < + ∞ for ⁢ r ∈ ( 0 , 1 ) .

First, we assert the existence of a finite pull-in voltage. In MEMS modeling, the pull-in voltage is the value of λ (which represents the voltage) that leads the system to the loss of equilibrium between the acting forces. We also establish estimates for the pull-in voltage, which is important in applications.

Theorem 1.2

There exists a finite pull-in voltage λ∗>0 such that

if 0 ≤ λ < λ ∗ , there exists at least one regular solution of ( (Pλ) );
if λ = λ ∗ , there exists at least one integral solution of ( (Pλ) );
if λ > λ ∗ , there is no solution of any kind of ( (Pλ) ).

Moreover, we have the lower bound

(1.8) 0 < ( β + 1 β + 3 ) β + 1 ⁢ ( 1 - β + 1 β + 3 ) 2 ⁢ C ≤ λ ∗ ≤ C ,

where C=C⁢(α,β,γ,f) is the well-defined constant

C := ( φ β - 1 ⁢ ( ∫ 0 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s t γ ⁢ f ⁢ ( t ) ⁢ d t ) ⁢ d s ) ) - 1 ,

where

φ β ⁢ ( s ) := { s ⁢ | s | - β / ( β + 1 ) if ⁢ s ≠ 0 , 0 if ⁢ s = 0 .

We also prove the existence of a branch of minimal solutions. Each minimal solution can be obtained as the limit of a recursive sequence, which enables one to make numerical approximations.

Theorem 1.3

Theorem 1.3 (Existence of the Minimal Branch)

For each λ∈(0,λ∗), problem (Pλ) admits a unique positive minimal solution uλ which is obtained as the limit of the sequence (un), where u0=0 and un is the solution of the problem

(Pλ(n)) { - r - γ ⁢ ( r α ⁢ | u n + 1 ′ | β ⁢ u n + 1 ′ ) ′ = λ ⁢ f ( 1 - u n ) 2 , r ∈ ( 0 , 1 ) , u n + 1 ′ ⁢ ( 0 ) = u n + 1 ⁢ ( 1 ) = 0 .

Moreover, each minimal solution is regular and the function λ↦uλ is increasing on (0,λ∗), that is, if λ1<λ2, then uλ1≤uλ2 and uλ1≠uλ2.

On the critical problem with respect to the variable λ, we prove that (Pλ*) has a unique integral solution that can be considered as the pointwise limit of the family of solutions (uλ)λ, namely,

u ∗ ⁢ ( r ) := lim λ ↗ λ ∗ ⁡ u λ ⁢ ( r ) , r ∈ ( 0 , 1 ) .

Theorem 1.4

The function u∗ is an integral solution of (Pλ) with λ=λ∗. Moreover, if λ=λ∗, then (Pλ) has a unique integral solution and, consequently, u∗ is a minimal solution of (Pλ*).

2 Preliminary Results

Before we present the proofs of the results stated in the last section, let us introduce some convenient notation. Observe that (Pλ) is equivalent to the integral equation

(2.1) u ⁢ ( r ) = φ β ⁢ ( λ ) ⁢ ∫ r 1 φ β ⁢ ( s - α ) ⁢ G ⁢ ( u , s ) ⁢ d s ,

where

(2.2) G ⁢ ( u , r ) := φ β ⁢ ( ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u ⁢ ( s ) ) 2 ⁢ d s ) .

Observe that the term G is an increasing function with respect to the first variable. Observe also that if the right-hand side of (2.1) is well-defined for some u∈C⁢([0,1]) and the equality holds, then u∈X and it is an integral solution of (Pλ) (see Definition 1.1).

2.1 Early Comments on the Solutions of (Pλ)

Here, we will discuss some properties of eventual solutions of (Pλ). The following proposition, among other results, asserts that the solutions of (Pλ) cannot cross the value 1.

Proposition 2.1

Any integral solution of (Pλ) is decreasing, that is,

(2.3) u ′ ⁢ ( r ) < 0 for ⁢ r ∈ ( 0 , 1 ) .

Moreover,

(2.4) u ∈ C ⁢ ( [ 0 , 1 ] ) , u ⁢ ( 0 ) = ∥ u ∥ ∞ ≤ 1 , and u ⁢ ( r ) < 1 ⁢ for ⁢ r ∈ ( 0 , 1 ) ,

and

if ∥ u ∥ ∞ < 1 , then u ∈ C 2 ⁢ ( ( 0 , 1 ] ) ∩ C ⁢ ( [ 0 , 1 ] ) ;
if γ > α - 1 and ∥ u ∥ ∞ < 1 , then u ∈ C 2 ⁢ ( ( 0 , 1 ] ) ∩ C 1 , μ ⁢ ( [ 0 , 1 ] ) and u ′ ⁢ ( 0 ) = 0 ;
if γ>α+β and ∥u∥∞<1, then we can consider u∈C2⁢([0,1])∩C1,μ⁢([0,1]). In addition, u′⁢(0)=u′′⁢(0)=0.

Proof.

Looking at (2.1), we can easily see that u∈C⁢((0,1)). Since G⁢(u,r) is positive for r∈(0,1), it follows that u′⁢(r)<0 for r∈(0,1), that is, u is decreasing. Now, assume the existence of r0∈(0,1) with u⁢(r0)=1. Let δ>0 be such that [r0-δ,r0+δ]⊂(0,1). Since r-α and G⁢(u,r) are bounded for u fixed and r varying in [r0-δ,r0+δ], we have

| 1 - u ⁢ ( r ) | = | u ⁢ ( r 0 ) - u ⁢ ( r ) | = | ∫ r r 0 φ β ⁢ ( λ ⁢ s - α ) ⁢ G ⁢ ( u , s ) ⁢ d s | ≤ C ⁢ | r 0 - r | .

Note that

C := inf ⁡ { r γ ⁢ f ⁢ ( r ) : r ∈ [ r 0 - δ , r 0 + δ ] } > 0 .

It follows that

∫ r 0 - δ r 0 + δ r γ ⁢ f ⁢ ( r ) ( 1 - u ⁢ ( r ) ) 2 ⁢ d r ≥ C ⁢ ∫ r 0 - δ r 0 + δ 1 ( r - r 0 ) 2 ⁢ d r = ∞ ,

which is in contradiction with the definition of an integral solution once we require

∫ 0 1 r γ ⁢ f ⁢ ( r ) ( 1 - u ⁢ ( r ) ) 2 ⁢ d r < ∞ .

Thus, we conclude that (2.4) holds.

To verify (i), we just have to use (1.6) and the fact that rγ⁢f⁢(r)⁢(1-u⁢(r))-2 is continuous for r∈(0,1] and φβ is a diffeomorphism on (0,∞).

In order to prove (ii), suppose ∥u∥∞<1 and denote

g ⁢ ( r ) := ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u ⁢ ( s ) ) 2 ⁢ d s .

By L’Hôpital’s rule, we have

(2.5) lim r → 0 ⁡ g ⁢ ( r ) = lim r → 0 ⁡ r γ ⁢ f ⁢ ( r ) α ⁢ r α - 1 ⁢ ( 1 - u ⁢ ( r ) ) 2 .

We can see that g is a continuous function over [0,1] with

g ⁢ ( 0 ) = f ⁢ ( 0 ) α ⁢ ( 1 - u ⁢ ( 0 ) ) 2 if ⁢ γ = α - 1

g ⁢ ( 0 ) = 0 if ⁢ γ > α - 1 .

In both cases, we conclude that u∈C1⁢([0,1]). We emphasize that if γ>α-1, we have u′⁢(0)=0.

To prove the Hölder’s continuity of u′, since φβ is Hölder continuous, it is sufficient to prove that g is Hölder continuous. Since g is differentiable on (0,1], we have

g ⁢ ( r ) - g ⁢ ( s ) = ∫ s r g ′ ⁢ ( t ) ⁢ d t for every ⁢ r , s ∈ ( 0 , 1 ] .

Using Hölder’s inequality with p>1 and 1/p+1/q=1, we have

(2.6) | g ⁢ ( r ) - g ⁢ ( s ) | ≤ | r - s | 1 / p ⁢ ( ∫ s r | g ′ ⁢ ( t ) | q ⁢ d t ) 1 / q for every ⁢ r , s ∈ ( 0 , 1 ] .

Since

g ′ ⁢ ( r ) = α r α + 1 ⁢ ∫ 0 r λ ⁢ s γ ⁢ f ⁢ ( s ) ( 1 - u ⁢ ( s ) ) 2 ⁢ d s - r γ - α ⁢ λ ⁢ f ⁢ ( r ) ( 1 - u ⁢ ( r ) ) 2 ,

using L’Hôpital’s rule, we get

lim r → 0 ⁡ g ′ ⁢ ( r ) = lim r → 0 ⁡ ( α α + 1 - 1 ) ⁢ r γ - α ⁢ λ ⁢ f ⁢ ( r ) ( 1 - u ⁢ ( r ) ) 2 ,

that is, g′⁢(r)=O⁢(rγ-α) as r→0 and the integral in (2.6) is bounded if (γ-α)⁢q>-1 or, equivalently,

(2.7) γ - α + 1 > 1 - 1 q = 1 p .

Thus, if γ>α-1, we can find p>1 satisfying (2.7) and we conclude that u′ is Hölder continuous on [0,1] with exponent

μ < { γ - α + 1 if - 1 < β < 0 , γ - α + 1 β + 1 if ⁢ β ≥ 0 .

Now we are going to verify that u′ is differentiable at 0. Note that

lim r → 0 ⁡ φ β ⁢ ( λ ⁢ g ⁢ ( r ) ) r = lim r → 0 ⁡ φ β ⁢ ( λ ⁢ g ⁢ ( r ) r β + 1 ) = φ β ⁢ ( λ ⁢ lim r → 0 ⁡ g ⁢ ( r ) r β + 1 )

and, again by L’Hôpital’s rule, limr→0⁡g⁢(r)/rβ+1 exists when γ≥α+β and vanishes when γ>α+β. This completes the proof. ∎

Remark 2.2

Back to the p-Laplacian case, that is, for α=γ=n-1 and β=p-2, we see from Proposition 2.1 (iii) that solutions of (1.1) are classical for 1<p<2, Ω=B, where B is the unidimensional ball of ℝn, h⁢(x)=f⁢(|x|), and f⁢(u)=(1-u)-2, provided that ∥u∥∞<1.

2.2 Existence of the Pull-In Voltage

Here, with the help of Proposition 2.1, we prove the existence of an extremal parameter λ∗>0 so that (Pλ) admits solutions for 0<λ<λ∗, while no solutions exist for λ>λ∗. In the following, we obtain an upper bound for the set of parameters λ for which (Pλ) may have a solution.

Proposition 2.3

Suppose that (Pλ) has an integral solution u≥0. Then,

∫ 0 1 φ β ⁢ ( r - α ) ⁢ G ⁢ ( r , 0 ) ⁢ d r < ∞

and

λ ≤ ( φ β - 1 ( ∫ 0 1 φ β ( r - α ) G ( r , 0 ) d r ) ) - 1 .

Proof.

Assumption (1.7) implies that G⁢(r,0) (defined in (2.2)) is a well-determined function on (0,1). Supposing that (Pλ) admits an integral solution u≥0, we have

1 ≥ u ⁢ ( 0 ) = φ β ⁢ ( λ ) ⁢ ∫ 0 1 φ β ⁢ ( r - α ) ⁢ G ⁢ ( r , u ) ⁢ d r ≥ φ β ⁢ ( λ ) ⁢ ∫ 0 1 φ β ⁢ ( r - α ) ⁢ G ⁢ ( r , 0 ) ⁢ d r .

In the estimate above, we used the fact that G is an increasing function with respect to its second variable. ∎

Remark 2.4

We emphasize that in Proposition 2.3 we have proved that (Pλ) has no solution, even in the integral sense, if

λ > ( φ β - 1 ⁢ ( ∫ 0 1 φ β ⁢ ( r - α ) ⁢ G ⁢ ( r , 0 ) ⁢ d r ) ) - 1 .

Then, we can define

(2.8) λ ∗ := sup ⁡ { λ > 0 : (P λ ) has a classical solution } ,

where classical means that u∈C2⁢(0,1]∩C⁢([0,1]) and solves (Pλ) for any point in (0,1). We can also define

(2.9) λ ∗ ∗ := sup ⁡ { λ > 0 : (P λ ) has an integral solution } .

If follows immediately that

(2.10) λ ∗ ≤ λ ∗ ∗ < ( φ β - 1 ⁢ ( ∫ 0 1 φ β ⁢ ( r - α ) ⁢ G ⁢ ( r , 0 ) ⁢ d r ) ) - 1 .

Indeed, we will prove in Section 3.3 that

λ ∗ = λ ∗ ∗ .

2.3 A Sub- and Supersolution Argument

Here, we construct a suitable sub- and supersolution procedure to approach problem (Pλ) (Proposition 2.5). Using such a technique, we can prove that (Pλ) admits a classical solution whenever λ∈(0,λ∗) and no solution, even in the integral sense, for λ>λ∗, where λ∗ is defined in (2.8). In other words, writing

Λ ∗ := { λ > 0 : (P λ ) has a classical solution } ,

we prove that Λ is a nondegenerate and bounded interval.

Before we state the sub- and supersolution method suitable for our study, we point out some facts about the particular case when h is constant with respect to the second variable, namely,

(2.11) { - ( r α ⁢ | u ′ ⁢ ( r ) | β ⁢ u ′ ⁢ ( r ) ) ′ = g ⁢ ( r ) , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 ,

where α, β are real constants and g is a continuous and integrable real function over the interval (0,1) satisfying the following conditions. Consider

g ~ ⁢ ( r ) := 1 r α ⁢ ∫ 0 r g ⁢ ( s ) ⁢ d s ,

suppose β>-1, and let

(2.12) lim s → 0 ⁡ g ~ ⁢ ( s ) = 0 ,

which are hypotheses on α and g. In view of this, one can define a function u:[0,1]→ℝ given by

u ⁢ ( r ) := ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s g ⁢ ( t ) ⁢ d t ) ⁢ d s for ⁢ r ∈ ( 0 , 1 ) .

In fact, due to (2.12), g~ is continuous on [0,1]. Since β>-1, then φβ defines a homeomorphism from ℝ into itself. Thus, φβ∘g~ is continuous on [0,1] and, consequently, u is a well-defined element of C1⁢[0,1]. Observe that u satisfies

- r α ⁢ | u ′ ⁢ ( r ) | β ⁢ u ′ ⁢ ( r ) = ∫ 0 r g ⁢ ( s ) ⁢ d s pointwise in ⁢ ( 0 , 1 ) .

Observe, in particular, that u is an integral solution of (Pλ) for h⁢(r,z)=g⁢(r).

Now, we are ready to present the method of sub- and supersolution.

Proposition 2.5

Proposition 2.5 (Method of Sub- and Supersolution)

Suppose that β>-1, that h is continuous on H and nondecreasing on the second variable, and assume that there exist u¯≤u¯, an integral subsolution and an integral supersolution, respectively, of (1.5) with

lim s → 0 ⁡ 1 s α ⁢ ∫ 0 s h ⁢ ( t , u ¯ ⁢ ( t ) ) ⁢ d t = lim s → 0 ⁡ 1 s α ⁢ ∫ 0 s h ⁢ ( t , u ¯ ⁢ ( t ) ) ⁢ d t = 0 .

Then, there exists an integral solution u∈X of (1.5) satisfying

u ¯ ≤ u ≤ u ¯ .

Moreover, u is given by

u ⁢ ( r ) := lim k → ∞ ⁡ u k ⁢ ( r ) ,

where (uk)⊂X is a sequence given recursively as follows. We set u0=u¯ and we define uk+1 as the unique solution of the problem

(2.13) { - ( r α ⁢ | u k + 1 ′ ⁢ ( r ) | β ⁢ u k + 1 ′ ⁢ ( r ) ) ′ = h ⁢ ( r , u k ⁢ ( r ) ) in ⁢ ( 0 , 1 ) , u k + 1 ′ ⁢ ( 0 ) = u k + 1 ⁢ ( 1 ) = 0 ,

which can be explicitly written as

(2.14) u k + 1 ⁢ ( r ) = ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s h ⁢ ( t , u k ⁢ ( t ) ) ⁢ d t ) ⁢ d s .

In particular, uk→u in X.

Proof.

We divide the proof into three parts. Claim. The sequence (uk) is well-defined and

u ¯ ≤ u k ≤ u ¯ for all ⁢ k ∈ ℕ .

Suppose that the claim is true for some k∈ℕ. Since h⁢(r,u¯⁢(r))∈L1⁢(0,1) and

h ⁢ ( r , u ¯ ⁢ ( r ) ) ≤ h ⁢ ( r , u k ⁢ ( r ) ) ≤ h ⁢ ( r , u ¯ ⁢ ( r ) ) ,

it follows that h⁢(r,uk⁢(r))∈L1⁢(0,1). Observe that

lim s → 0 ⁡ 1 s α ⁢ ∫ 0 s h ⁢ ( r , u ¯ ⁢ ( r ) ) ⁢ d t ≤ lim s → 0 ⁡ 1 s α ⁢ ∫ 0 s h ⁢ ( r , u k ⁢ ( r ) ) ⁢ d t ≤ lim s → 0 ⁡ 1 s α ⁢ ∫ 0 s h ⁢ ( r , u ¯ ⁢ ( r ) ) ⁢ d t = 0 .

Then, taking g:=h⁢(⋅,uk) in (2.11), since such a g is continuous on (0,1), the conditions on α, β, and g in the beginning of this section are satisfied. So, we conclude that uk+1 given by (2.14) is a well-defined element of C1⁢([0,1]).

Since φβ is increasing for β>-1, we have

u k + 1 ⁢ ( r ) = ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s h ⁢ ( t , u k ⁢ ( t ) ) ⁢ d t ) ⁢ d s ≤ ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s h ⁢ ( t , u ¯ ⁢ ( t ) ) ⁢ d t ) ⁢ d s ≤ u ¯ ⁢ ( r )

and, analogously,

u k + 1 ⁢ ( r ) = ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s h ⁢ ( t , u k ⁢ ( t ) ) ⁢ d t ) ⁢ d s ≥ ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s h ⁢ ( t , u ¯ ⁢ ( t ) ) ⁢ d t ) ⁢ d s ≥ u ¯ ⁢ ( r ) .

Whence, we conclude that

u ¯ ≤ u ≤ u ¯ .

Observe also that

u k + 1 ′ ⁢ ( r ) = - φ β ⁢ ( 1 r α ⁢ ∫ 0 r h ⁢ ( s , u k ⁢ ( s ) ) ⁢ d s ) ≤ - φ β ⁢ ( 1 r α ⁢ ∫ 0 r h ⁢ ( s , u ¯ ⁢ ( s ) ) ⁢ d s ) = u ¯ ⁢ ( r )

and

u k + 1 ′ ⁢ ( r ) = - φ β ⁢ ( 1 r α ⁢ ∫ 0 r h ⁢ ( s , u k ⁢ ( s ) ) ⁢ d s ) ≥ - φ β ⁢ ( 1 r α ⁢ ∫ 0 r h ⁢ ( s , u ¯ ⁢ ( s ) ) ⁢ d s ) ≥ u ¯ ′ ⁢ ( r ) ,

that is,

u ¯ ′ ≤ u k + 1 ′ ≤ u ¯ ′

pointwise on [0,1]. Then,

r α ⁢ | u ¯ ′ ⁢ ( r ) | β + 2 ≤ r α ⁢ | u k + 1 ′ ⁢ ( r ) | β + 2 ≤ r α ⁢ | u ¯ ′ ⁢ ( r ) | β + 2

pointwise on (0,1]. Thus, we have that uk+1∈X. Since the claim is automatically true for k=0, we conclude by an induction argument that (uk) is well-defined and u¯≤uk≤u¯ for all k∈ℕ.

Claim. The sequence (uk) is nondecreasing.We proved above that u1⁢(r)≥u¯⁢(r)=u0⁢(r). Taking k∈ℕ such that our claim is true up to k, we have

u k + 1 ⁢ ( r ) = ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s h ⁢ ( t , u k ⁢ ( t ) ) ⁢ d t ) ⁢ d s ≥ ∫ r 1 φ β ⁢ ( 1 s α ⁢ ∫ 0 s h ⁢ ( t , u k - 1 ⁢ ( t ) ) ⁢ d t ) ⁢ d s = u k ⁢ ( r ) .

Then, our claim is valid inductively for every k∈ℕ.

Claim. The sequence (uk) converges in X to an integral solution u of (1.5). Moreover, for any r∈(0,1], we have

(2.15) u ⁢ ( r ) := lim k → ∞ ⁡ u k ⁢ ( r ) .

Due to the arguments above, the function u:[0,1]→ℝ given by (2.15) is well-defined. By the monotone convergence theorem, rα/(β+2)⁢u⁢(r)∈Lβ+2⁢((0,1)) and

∫ 0 1 r α ⁢ | u k ⁢ ( r ) | β + 2 ⁢ d r → ∫ 0 1 r α ⁢ | u ⁢ ( r ) | β + 2 ⁢ d r .

Now, using the Brezis–Lieb lemma (see [1]), we have

∫ 0 1 r α ⁢ | u k ⁢ ( r ) - u ⁢ ( r ) | β + 2 ⁢ d r → 0 .

The monotonicity of (uk) with respect to k implies a monotonicity property for the sequence of the corresponding derivatives (uk′). In fact, for every k≥1, we have

u k + 1 ′ ⁢ ( r ) = - φ β ⁢ ( 1 r α ⁢ ∫ 0 r h ⁢ ( s , u k ⁢ ( s ) ) ⁢ d s ) ≤ - φ β ⁢ ( 1 r α ⁢ ∫ 0 r h ⁢ ( s , u k - 1 ⁢ ( s ) ) ⁢ d s ) = u k ′ ⁢ ( r ) .

Thus, for almost every r∈[0,1], we can define

v ⁢ ( r ) := lim k → ∞ ⁡ u k ′ ⁢ ( r ) .

Observe that (rα⁢|uk′|β+2⁢(r)) is a nondecreasing sequence and, as a consequence of the monotone convergence theorem,

∫ 0 1 r α ⁢ | u k ′ ⁢ ( r ) | β + 2 ⁢ d r → ∫ 0 1 r α ⁢ | v ⁢ ( r ) | β + 2 ⁢ d r as ⁢ k → ∞ .

Furthermore, the sequence (rα/(β+2)⁢uk′⁢(r)) is bounded in Lβ+2⁢((0,1)) since

∫ 0 1 r α ⁢ | u k ′ ⁢ ( r ) | β + 2 ⁢ d r ≤ ∫ 0 1 r α ⁢ | u ¯ ′ ⁢ ( r ) | β + 2 ⁢ d r < ∞ .

Then, by the Brezis–Lieb lemma, we have

∫ 0 1 r α ⁢ | u k ′ ⁢ ( r ) - v ⁢ ( r ) | β + 2 ⁢ d r → 0 .

Thus, (uk) is a Cauchy sequence in the complete space X. Since the norms defined in (1.4) and (1.3) are equivalent and v⁢(0)=limk→∞⁡uk′⁢(0)=0, we conclude that

∫ 1 r v ⁢ ( s ) ⁢ d s = u ⁢ ( r ) ∈ X .

For almost every r∈[0,1], we have

- r α ⁢ | u ′ ⁢ ( r ) | β ⁢ u ′ ⁢ ( r ) = - lim k → ∞ ⁡ r α ⁢ | u k + 1 ′ ⁢ ( r ) | β ⁢ u k + 1 ′ ⁢ ( r ) = lim k → ∞ ⁡ ∫ 0 r h ⁢ ( s , u k ⁢ ( s ) ) ⁢ d s = ∫ 0 r h ⁢ ( s , u ⁢ ( s ) ) ⁢ d s .

Due to the monotonicity property of h⁢(r,z) for z∈I, it follows that

∫ 0 1 h ⁢ ( r , u ⁢ ( r ) ) ⁢ d r ≤ ∫ 0 1 h ⁢ ( r , u ¯ ⁢ ( r ) ) ⁢ d r < ∞ .

Then, we conclude that u is an integral solution of (1.5). ∎

3 Proofs of the Main Results

First, we study the values of λ for which (Pλ) admits solutions in the classical sense. We prove that there exists λ∗>0 so that (Pλ) admits a classical solution whenever 0<λ<λ∗ and no solution, even in the integral sense, for λ>λ∗. Let

Λ := { λ > 0 : (P λ ) has a classical solution } .

Our task is to prove that Λ is a nondegenerate and bounded interval.

3.1 Proof of Theorem 1.2 (i)

First of all, we are going to prove the existence of a solution of (Pλ) for λ near 0. Let v∈C2⁢([0,1]) be the unique solution of (2.11) for g⁢(r)=rγ⁢f and take δ>0 small enough to ensure that u¯:=δ⁢v satisfies ∥u¯∥∞<1, that is, 0<δ<∥v∥∞-1. Note that

- ( r α ⁢ | u ¯ ′ | β ⁢ u ¯ ′ ) ′ = δ β + 1 ⁢ ( r α ⁢ | v ′ | β ⁢ v ′ ) ′ = δ β + 1 ⁢ r γ ⁢ f ⁢ ( r ) = δ β + 1 ⁢ ( 1 - ∥ u ¯ ∥ ∞ ) 2 ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - ∥ u ¯ ∥ ∞ ) 2 .

Since (1-z)-2 is increasing on (-∞,0), it follows that

- ( r α ⁢ | u ¯ ′ | β ⁢ u ¯ ′ ) ′ ≥ δ β + 1 ⁢ ( 1 - ∥ u ¯ ∥ ∞ ) 2 ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ¯ ) 2 ≥ λ ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ¯ ) 2

whenever

0 < λ ≤ δ β + 1 ⁢ ( 1 - ∥ u ¯ ∥ ∞ ) 2 = δ β + 1 ⁢ ( 1 - δ ⁢ ∥ v ∥ ∞ ) 2 .

So, we have found a supersolution of (Pλ). Since zero is a subsolution of (Pλ), we can conclude that (Pλ) admits an integral solution u∈X (cf. Proposition 2.5) satisfying

0 ≤ u ≤ u ¯ .

This implies that ∥u∥∞≤∥u¯∥∞<1. Then, u is a classical solution for (Pλ) by Proposition 2.1. Finally, observing that

( β + 1 ( β + 3 ) ⁢ ∥ v ∥ ∞ ) β + 1 ⁢ ( 1 - β + 1 β + 3 ) 2 = max 0 < δ < ∥ v ∥ ∞ - 1 ⁡ δ β + 1 ⁢ ( 1 - δ ⁢ | v | ∞ ) 2 ,

we get (1.8).

Given λ¯∈Λ, a classical solution u¯ of (Pλ¯) is a supersolution of (Pλ) for every λ∈(0,λ¯). By Proposition 2.5 we assert the existence of a classical solution of (Pλ) for λ∈(0,λ¯). Then, we conclude that Λ is in fact an interval.

3.2 Proof of Theorem 1.2 (iii) and Proof of Theorem 1.3

First, we need to establish a lemma which will be used many times in our upcoming arguments. In the following, we denote

G ⁢ ( u , r ) := φ β ⁢ ( ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u ⁢ ( s ) ) 2 ⁢ d s ) ,

where φβ⁢(z):=|z|-β/(β+1)⁢z.

Lemma 3.1

Given two continuous functions u1,u2:[0,1]→ℝ, we consider

u t := ( 1 - t ) ⁢ u 1 + t ⁢ u 2 .

Suppose

u t ⁢ ( r ) < 1 for every ⁢ ( t , r ) ∈ [ 0 , 1 ] × ( 0 , 1 )

and G⁢(ut,r) is well-defined for the same range of values. Then, we have

G ⁢ ( u t , r ) ≤ ( 1 - t ) ⁢ G ⁢ ( u 1 , r ) + t ⁢ G ⁢ ( u 2 , r ) for ⁢ ( t , r ) ∈ [ 0 , 1 ] × [ 0 , 1 ] .

Moreover, if u1⁢(r)<u2⁢(r) strictly on (δ1,δ2)⊂[0,1], there exists, for each t∈(0,1), a positive constant c=c⁢(t) such that

G ⁢ ( u t , r ) + c ≤ ( 1 - t ) ⁢ G ⁢ ( u 1 , r ) + t ⁢ G ⁢ ( u 2 , r ) for every ⁢ r ∈ ( δ 1 , 1 ] .

Proof.

For t∈[0,1], consider

ξ ⁢ ( t , r ) := G ⁢ ( u t , r ) - t ⁢ G ⁢ ( u 2 , r ) - ( 1 - t ) ⁢ G ⁢ ( u 1 , r ) .

Observe that ξ⁢(0,r)≡ξ⁢(1,r)≡0. Moreover, since φβ is differentiable for β>-1 and the integrands in the definition of ξ are continuous for every (t,s)∈[0,1]×[0,r] and r∈(0,1) as well as its derivative with respect to t, it follows by Leibniz’s rule that ξ is derivable with respect to t with

d d ⁢ t ⁢ ξ ⁢ ( t , r ) = G ⁢ ( u 1 , r ) - G ⁢ ( u 2 , r ) + 2 β + 1 ⁢ ∫ 0 r s γ ⁢ f ⁢ ( s ) ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) ( 1 - u t ⁢ ( s ) ) 3 ⁢ d s ⁢ | ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u t ⁢ ( s ) ) 2 ⁢ d s | - β / ( β + 1 ) .

Applying again the same argument, we calculate

d 2 d ⁢ t 2 ⁢ ξ ⁢ ( t , r ) = - 4 ⁢ β ( β + 1 ) 2 ⁢ | ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u t ⁢ ( s ) ) 2 ⁢ d s | - ( 2 ⁢ β + 1 ) / ( β + 1 ) ⁢ ( ∫ 0 r s γ ⁢ f ⁢ ( s ) ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) ( 1 - u t ⁢ ( s ) ) 3 ⁢ d s ) 2

+ 6 β + 1 ⁢ | ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u t ⁢ ( s ) ) 2 ⁢ d s | - β / ( β + 1 ) ⁢ ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u t ⁢ ( s ) ) 4 ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) 2 ⁢ d s

or, equivalently,

(3.1) d 2 d ⁢ t 2 ⁢ ξ ⁢ ( t , r ) = ζ ⁢ ( t , r ) ⁢ ( 3 ⁢ η 1 ⁢ ( t , r ) - 2 ⁢ β β + 1 ⁢ η 2 ⁢ ( t , r ) ) ,

where

{ ζ ⁢ ( t , r ) : = 2 β + 1 | ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u t ⁢ ( s ) ) 2 d s | - ( 2 ⁢ β + 1 ) / ( β + 1 ) , η 1 ⁢ ( t , r ) : = ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u t ⁢ ( s ) ) 2 d s ∫ 0 r s γ ⁢ f ⁢ ( s ) ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) 2 ( 1 - u t ⁢ ( s ) ) 4 d s , η 2 ⁢ ( t , r ) : = ( ∫ 0 r s γ ⁢ f ⁢ ( s ) ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) ( 1 - u t ⁢ ( s ) ) 3 d s ) 2 .

Observe that ζ⁢(t,r)≥0 for all (t,r)∈[0,1]×[0,1]. In fact, the term inside the parenthesis in (3.1) is also nonnegative for all β>-1 since

2 ⁢ β β + 1 > - 2

and by the Cauchy–Schwarz inequality we have

( ∫ 0 r s γ ⁢ f ⁢ ( s ) ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) ( 1 - u t ⁢ ( s ) ) 3 ⁢ d s ) 2 = ( ∫ 0 r ( s γ ⁢ f ⁢ ( s ) ) 1 / 2 1 - u t ⁢ ( s ) ⁢ ( s γ ⁢ f ⁢ ( s ) ) 1 / 2 ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) ( 1 - u t ⁢ ( s ) ) 2 ⁢ d s ) 2

(3.2) ≤ ∫ 0 r s γ ⁢ f ⁢ ( s ) ( 1 - u t ⁢ ( s ) ) 2 ⁢ d s ⁢ ∫ 0 r s γ ⁢ f ⁢ ( s ) ⁢ ( u 2 ⁢ ( s ) - u 1 ⁢ ( s ) ) 2 ( 1 - u t ⁢ ( s ) ) 4 ⁢ d s .

That is, ξ′′ is nonnegative and we conclude that ξ is nonpositive.

If u1⁢(r)<u2⁢(r) strictly on (δ1,δ2)⊂[0,1], then the terms η1 and η2 are strictly positive on [0,1]×(δ1,1], consequently, ξ′′ is strictly positive for (t,r)∈[0,1]×(δ1,1], and we finally conclude that ξ is strictly negative for (t,r)∈(0,1)×(δ1,1]. ∎

3.3 Proof of Theorem 1.3 Completed

The first part of Theorem 1.3 is a consequence of the sub- and supersolution method (cf. Proposition 2.5). For the second part, consider λ1,λ2∈(0,λ∗∗) with λ1<λ2 and the respective minimal integral solutions u1 and u2 of (Pλ1) and (Pλ2). Observe that u2≥0, u2≠0, and that u2 is a supersolution of (Pλ1). Then, (Pλ1) admits a nonnegative solution u≠0 satisfying u≤u2. By the definition of a minimal solution we must have u1≤u2. We cannot have u1=u2 because this implies λ1=λ2.

Now, consider

λ 0 = sup ⁡ { λ ∈ Λ : the associated minimal solution ⁢ u λ ⁢ is regular } .

Suppose λ0<λ∗∗ and take λ1∈(0,λ0) and λ2∈(λ0,λ∗∗). Consider also the minimal solutions u1 and u2 associated with (Pλ1) and (Pλ2), respectively. Choose t∈(0,1) such that

λ 1 1 / ( β + 1 ) < λ 0 1 / ( β + 1 ) < t ⁢ λ 1 1 / ( β + 1 ) + ( 1 - t ) ⁢ λ 2 1 / ( β + 1 ) < λ 2 1 / ( β + 1 )

and set

λ = ( t ⁢ λ 1 1 / ( β + 1 ) + ( 1 - t ) ⁢ λ 2 1 / ( β + 1 ) ) β + 1 > λ 0

and v=t⁢u1+(1+t)⁢u2. We see from Lemma 3.1 that

- r α / ( β + 1 ) ⁢ v ′ = t ⁢ λ 1 1 / ( β + 1 ) ⁢ G ⁢ ( u 1 ) + ( 1 - t ) ⁢ λ 2 ( β + 1 ) ⁢ G ⁢ ( u 2 ) ≥ λ 1 / ( β + 1 ) ⁢ G ⁢ ( v ) .

Thus, v is a supersolution of (Pλ). Let u be a integral solution of (Pλ), ensured by the sub- and supersolution method. Since ∥v∥∞≤t⁢∥u1∥∞+(1-t)⁢∥u2∥∞<1 and 0≤u≤v, we deduce that u is a regular solution. This is in contradiction with the definition of λ0. Therefore, we have proved that λ0=λ∗=λ∗∗ and, consequently, we have also proved Theorem 1.2 (iii).

3.4 Proof of Theorem 1.2 (ii) and Proof of Theorem 1.4

Here, we study the critical problem (Pλ) with λ=λ∗, that is,

(Pλ*) { - ( r α ⁢ | u ′ | β ⁢ u ′ ) ′ = λ ∗ ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ⁢ ( r ) ≤ 1 , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 .

We prove that (Pλ*) has a unique solution for -1<β, as stated in Theorem 1.4. Since, for each fixed r∈[0,1], the function λ↦uλ⁢(r) is nondecreasing (see Theorem 1.3) and bounded by 1 (see Proposition 2.1), we have that the function u∗:[0,1]→ℝ given by

u ∗ ⁢ ( r ) := lim λ ↗ λ ∗ ⁡ u λ ⁢ ( r ) , r ∈ [ 0 , 1 ] ,

is well-defined. Then, by using an argument similar to the one in the last part of the proof of Proposition 2.5, we can conclude that u∗ is an integral solution of (Pλ*) and, consequently, we have proved Theorem 1.2 (ii). Now, since we have proved the existence of a solution for (Pλ*), consider an arbitrary solution v. Due to the monotonicity property of the term G, we have that v is an integral supersolution of (Pλ) for every 0<λ<λ∗. Then, uλ⁢(r)≤v⁢(r) for every r∈[0,1], where uλ is the minimal solution of (Pλ). In the limit case, it follows that

u ∗ ≤ v

in the pointwise sense. Therefore, u∗ is minimal.

For the uniqueness part, suppose that there exists a solution v of (Pλ*) with v≢u∗. Since u∗ is minimal and no solution blows over the value 1, we know that

u ∗ ⁢ ( r ) ≤ v ⁢ ( r ) < 1 for ⁢ r ∈ ( 0 , 1 ) .

Denote by (δ1,δ2)⊂[0,1] the open interval on which

u ∗ ⁢ ( r ) < v ⁢ ( r ) for ⁢ r ∈ ( δ 1 , δ 2 ) .

Consider

u ¯ := u ∗ + v 2

and observe that

u ¯ ⁢ ( r ) < 1 for ⁢ r ∈ ( 0 , 1 ) .

Applying Lemma 3.1 with t=1/2, we guarantee the existence of a positive constant c such that

{ - u ¯ ′ ⁢ ( r ) ≥ φ β ⁢ ( λ ∗ ⁢ r - α ) ⁢ G ⁢ ( u ¯ , r ) , r ∈ ( 0 , 1 ) , - u ¯ ′ ⁢ ( r ) ≥ φ β ⁢ ( λ ∗ ⁢ r - α ) ⁢ G ⁢ ( u ¯ , r ) + c , r ∈ [ δ 1 , 1 ] .

Thus,

- u ¯ ′ ⁢ ( r ) ≥ φ β ⁢ ( λ ∗ ⁢ r - α ) ⁢ G ⁢ ( u ¯ , r ) + ξ ⁢ ( r ) for ⁢ r ∈ ( 0 , 1 ) ,

where ξ is a nonzero continuous function supported on [δ1,1] and satisfying

ξ ⁢ ( r ) ≤ c and ξ ⁢ ( 1 ) > 0 .

So, u¯ is a supersolution of the problem

(3.3) { - u ′ ⁢ ( r ) = φ β ⁢ ( λ ∗ ⁢ r - α ) ⁢ G ⁢ ( u , r ) + ξ ⁢ ( r ) , r ∈ ( 0 , 1 ) , u ⁢ ( 1 ) = 0 .

Since the zero function is a subsolution of (3.3), we conclude via Proposition 2.5 that (3.3) has an integral solution u0 satisfying

0 ≤ u 0 ≤ u ¯ ≤ 1 .

Considering

ψ ⁢ ( r ) := ∫ r 1 ξ ⁢ ( r ) ⁢ d r ,

it is easy to see, in particular, that u is a solution of (3.3) if and only if u-ψ is a solution of (Pλ).

Denote

η ⁢ ( r ) := u 0 ⁢ ( r ) ψ ⁢ ( r ) .

Since

lim r → 1 ⁡ u 0 ⁢ ( r ) = lim r → 1 ⁡ ψ ⁢ ( r ) = 0 ,

we can apply L’Hôpital’s rule to obtain

lim r → 1 ⁡ η ⁢ ( r ) = lim r → 1 ⁡ u 0 ′ ⁢ ( r ) ψ ′ ⁢ ( r ) = φ β ⁢ ( λ ∗ ) ⁢ G ⁢ ( u 0 , 1 ) + ξ ⁢ ( 1 ) ξ ⁢ ( 1 ) .

Thus, η defines a continuous function over [0,1]. Then, we can take δ>0 sufficiently small such that

(3.4) δ ⁢ u 0 ≤ ψ .

Choose ε>0 sufficiently small such that

φ β ⁢ ( λ ∗ + ε ) φ β ⁢ ( λ ∗ ) - 1 < δ

and define

w ε = φ β ⁢ ( λ ∗ + ε ) φ β ⁢ ( λ ∗ ) ⁢ u 0 - ψ .

Due to (3.4) we see that

u 0 - w ε = ψ + ( 1 - φ β ⁢ ( λ ∗ + ε ) φ β ⁢ ( λ ∗ ) ) ⁢ u 0 > ψ - δ ⁢ u 0 ≥ 0 ,

that is,

w ε ≤ u 0 ≤ 1 .

Since G is monotone on the variable u for u≤1, it follows that

- w ε ′ ⁢ ( r ) = φ β ⁢ ( λ ∗ + ε ) φ β ⁢ ( λ ∗ ) ⁢ φ β ⁢ ( λ ∗ ⁢ r - α ) ⁢ G ⁢ ( u 0 , r ) + ( φ β ⁢ ( λ ∗ + ε ) φ β ⁢ ( λ ∗ ) - 1 ) ⁢ G 0 ⁢ ( u 0 , r ) ≥ φ β ⁢ ( ( λ ∗ + ε ) ⁢ r - α ) ⁢ G ⁢ ( w ε , r ) .

Thus, wε is a supersolution of (Pλ) for λ=λ∗+ε. Moreover, wε≥0. Then, via Proposition 2.5, there exists a solution of (Pλ) for λ>λ∗, which contradicts the definition of λ∗.

Funding statement: Research supported in part by INCTmat/MCT/Brazil, CNPq, and CAPES/Brazil.

References

[1] Brézis H. and Lieb E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. 10.1007/978-3-642-55925-9_42Search in Google Scholar

[2] Castorina D., Esposito P. and Sciunzi B., p-MEMS equation on a ball, Methods Appl. Anal. 15 (2008), no. 3, 277–283. 10.4310/MAA.2008.v15.n3.a2Search in Google Scholar

[3] Castorina D., Esposito P. and Sciunzi B., Degenerate elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations 34 (2009), no. 3, 279–306. 10.1007/s00526-008-0184-3Search in Google Scholar

[4] Castorina D., Esposito P. and Sciunzi B., Spectral theory for linearized p-Laplace equations, Nonlinear Anal. 74 (2011), no. 11, 3606–3613. 10.1016/j.na.2011.03.009Search in Google Scholar

[5] Clément P., de Figueiredo D. G. and Mitidieri E., Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 133–170. 10.12775/TMNA.1996.006Search in Google Scholar

[6] Crandall M. G. and Rabinowitz P. H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal. 58 (1975), no. 3, 207–218. 10.1007/BF00280741Search in Google Scholar

[7] Dávila J., Singular solutions of semi-linear elliptic problems, Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. VI, Elsevier/North-Holland, Amsterdam (2008), 83–176. 10.1016/S1874-5733(08)80019-8Search in Google Scholar

[8] Esposito P., Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Commun. Contemp. Math. 10 (2008), no. 1, 17–45. 10.1142/S0219199708002697Search in Google Scholar

[9] Esposito P., Ghoussoub N. and Guo Y., Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math. 60 (2007), no. 12, 1731–1768. 10.1002/cpa.20189Search in Google Scholar

[10] Esposito P., Ghoussoub N. and Guo Y., Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lect. Notes Math. 20, American Mathematical Society, Providence, 2010. 10.1090/cln/020Search in Google Scholar

[11] Ghoussoub N. and Guo Y., On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1423–1449. 10.1137/050647803Search in Google Scholar

[12] Guo Z. and Wei J., Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math. 120 (2006), no. 2, 193–209. 10.1007/s00229-006-0001-2Search in Google Scholar

[13] Guo Z. and Wei J., Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity, Commun. Pure Appl. Anal. 7 (2008), no. 4, 765–786. 10.3934/cpaa.2008.7.765Search in Google Scholar

[14] Guo Z. and Wei J., Infinitely many turning points for an elliptic problem with a singular non-linearity, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 21–35. 10.1112/jlms/jdm121Search in Google Scholar

[15] Jacobsen J. and Schmitt K., The Liouville–Bratu–Gelfand problem for radial operators, J. Differential Equations 184 (2002), no. 1, 283–298. 10.1006/jdeq.2001.4151Search in Google Scholar

[16] Jacobsen J. and Schmitt K., Radial solutions of quasilinear elliptic differential equations, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam (2004), 359–435. 10.1016/S1874-5725(00)80006-1Search in Google Scholar

[17] Kufner A. and Opic B., How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin. 25 (1984), no. 3, 537–554. Search in Google Scholar

[18] Mignot F. and Puel J.-P., Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations 5 (1980), no. 8, 791–836. 10.1080/03605308008820155Search in Google Scholar

[19] Opic B. and Kufner A., Hardy-Type Inequalities, Pitman Res. Notes Math. Ser. 219, Longman Scientific & Technical, Harlow, 1990. Search in Google Scholar

[20] Pelesko J. A. and Bernstein D. H., Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, 2003. 10.1201/9781420035292Search in Google Scholar

Received: 2015-06-27

Accepted: 2015-11-28

Published Online: 2016-04-06

Published in Print: 2016-05-01

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

Articles in the same Issue

https://doi.org/10.1515/ans-2015-5031

Keywords for this article

Quasilinear Elliptic Equation; MEMS Capacitor; Extremal Solutions; Minimal Solutions; Regularity; Singular Nonlinearities

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