Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls

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Ann. I. H. Poincaré – AN 29 (2012) 783–812

www.elsevier.com/locate/anihpc

Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls

Norihisa Ikoma

^a,^∗

, Hitoshi Ishii

^b,c

aMathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai-Shi, Miyagi 980-8578, Japan

bDepartment of Mathematics, Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan

cDepartment of Mathematics, Faculty of Science, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia Received 6 August 2011; accepted 19 April 2012

Available online 23 May 2012

Abstract

We study the eigenvalue problem for positively homogeneous, of degree one, elliptic ODE on finite intervals and PDE on balls. We establish the existence and completeness results for principal and higher eigenpairs, i.e., pairs of an eigenvalue and its corresponding eigenfunction.

1. Introduction

We consider the eigenvalue problem for fully nonlinear elliptic PDE F

D²u, Du, u, x

+μu=0 inΩ,

u=0 on∂Ω, (1.1)

whereΩ is a bounded domain inR^N,u:Ω¯ →Randμ∈Rrepresent the unknown function (eigenfunction) and constant (eigenvalue), respectively, andF:S^N×R^N×R×Ω→Ris a given function, whereS^N denotes the space of real symmetricN×Nmatrices.

Recently there has been much interest in eigenvalue problems for fully nonlinear PDE since the work of P.-L. Li- ons[15]. See[3,13,4,18,1,17]for these developments. See also[2,8,12]for some earlier related works. In this regard, most of work has been devoted to the questions concerning principal eigenvalues, while recent work by Esteban, Felmer and Quaas[10](see also[4]) has established the existence of other eigenvalues beyond the principal eigenvalues and of the corresponding eigenfunctions in the one-dimensional or the radially symmetric problem. In this paper we extend the scope of the work of Esteban, Felmer and Quaas[10]to the eigenvalue problem set in theL^q framework.

* Corresponding author.

E-mail addresses:[email protected](N. Ikoma),[email protected](H. Ishii).

http://dx.doi.org/10.1016/j.anihpc.2012.04.004

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We thus study (1.1) in the one-dimensional or radially symmetric domains. That is, in what follows, we are con- cerned with the case whereΩis an open interval(a, b), with−∞< a < b <∞, or an open ballB_R=B_R(0)inR^N of radiusR∈(0,∞)with center at the origin.

We now introduce our basic assumptions (F1)–(F3) on the functionF. Given constantsλ∈(0,∞)andΛ∈ [λ,∞], P^± denote the Pucci operators defined as the functions on S^N given, respectively, byP⁺(M)≡P⁺(M;λ, Λ)= sup{trAM: A∈S^N, λINAΛIN}andP⁻(M)= −P⁺(−M), whereIN denotes the N×N identity matrix and the relation, XY, is the standard order relation betweenX, Y ∈S^N. Note that if N=1 andΛ= ∞, then P⁺(m)=λmform0 andP⁺(m)= ∞form >0.

(F1) The functionF:S^N×R^N×R×Ω→Ris a Carathéodory function, i.e., the functionx→F (M, p, u, x)is measurable for any(M, p, u)∈S^N×R^N⁺¹and the function(M, p, u)→F (M, p, u, x)is continuous for a.a.

x∈Ω.

(F2) There exist constantsλ∈(0,∞),Λ∈ [λ,∞],q∈ [1,∞]and functionsβ, γ∈L^q(Ω)such that F (M₁, p₁, u₁, x)−F (M₂, p₂, u₂, x)P⁺(M₁−M₂)+β(x)|p₁−p₂| +γ (x)|u₁−u₂| for all(M₁, p₁, u₁), (M₂, p₂, u₂)∈S^N×R^N⁺¹and a.a.x∈Ω.

(F3) F (t M, tp, t u, x)=t F (M, p, u, x)for allt0, all(M, p, u)∈S^N×R^N⁺¹and a.a.x∈Ω.

Of course, ifΛ= ∞andM1M2, then the inequality in condition (F2) is trivially satisfied.

We make an additional assumption in the multi-dimensional case.

(F4) The functionF is radially symmetric in the sense that for any(m, l, q, u)∈R⁴and a.a.r∈(0, R), the function ω→F

mω⊗ω+l(I_N−ω⊗ω), qω, u, rω

is constant on the unit sphereS^N⁻¹⊂R^N. Here and henceforthx⊗xdenotes the matrix inS^N with the(i, j ) entry given byx_ix_j ifx∈R^N.

We study the eigenvalue problem (1.1) in the Sobolev spaceW^2,q(Ω). For any pair(μ, ϕ)∈R×W^2,1(Ω)which satisfies the PDE in the almost everywhere sense and the boundary condition of (1.1) in the pointwise sense, we call μandϕ aneigenvalueandeigenfunctionof (1.1), respectively, providedϕ(x)≡0. We call such a pair aneigenpair of (1.1).

We state our main results in this paper.

Theorem 1.1.LetN=1andΩ=(a, b), and assume that(F1), (F2), withΛ= ∞, and(F3)hold. Then:

(i) For any n∈N, there exist eigenpairs (μ⁺_n, ϕ_n⁺), (μ⁻_n, ϕ_n⁻)∈R×W^2,q(a, b) of (1.1) and finite sequences (x_n,j⁺ )ⁿ_j₌₀, (x_n,j⁻ )ⁿ_j₌₀⊂ [a, b]such that

⎧⎪

⎨

⎪⎩

a=x_n,0⁺ < x_n,1⁺ <· · ·< x_n,n⁺ =b, a=x_n,0⁻ < x_n,1⁻ <· · ·< x_n,n⁻ =b, (−1)^j⁻¹ϕ_n⁺(x) >0 in

x_n,j⁺ ₋₁, x_n,j⁺

forj=1, . . . , n, (−1)^jϕ⁻_n(x) >0 in

x⁻_n,j₋₁, x_n,j⁻

forj=1, . . . , n.

(ii) The eigenpairs(μ⁺_n, ϕ⁺_n)and(μ⁻_n, ϕ_n⁻)are complete in the sense that for any eigenpair(μ, ϕ)∈R×W^2,q(a, b) of (1.1), there existn∈Nandθ >0such that either(μ, ϕ)=(μ⁺_n, θ ϕ_n⁺)or(μ, ϕ)=(μ⁻_n, θ ϕ⁻_n)holds.

Forq∈ [1,∞], letWr^2,q(BR)denote the space of those functionsϕ∈W^2,q(BR)which are radially symmetric. We may identify any functionf inW_r^2,q(B_R)with a functiongon[0, R]such thatf (x)=g(|x|)for a.a.x∈B_Rand we employ the standard abuse of notation:f (x)=f (|x|)forx∈BR. We setλ_∗=λ/Λandq_∗=N/(λ_∗N+1−λ_∗)if Λ <∞. Note that 0< λ_∗1 andq_∗∈ [1, N ).

Theorem 1.2.Let N2 andΩ=B_R, and assume that(F1), (F2)withΛ <∞,(F3)and(F4)hold. Assume that q >max{N/2, q_∗}and thatβ∈L^N(B_R)ifq < N. Then:

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(i) For each n∈N, there exist eigenpairs (μ⁺_n, ϕ_n⁺), (μ⁻_n, ϕ_n⁻)∈R×W_r^2,q(B_R) of (1.1) and finite sequences (r_n,j^± )ⁿ_j₌₀⊂ [0, R]such that

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

0=r_0,n⁺ < r_n,1⁺ <· · ·< r_n,n⁺ =R, 0=r_0,n⁻ < r_n,1⁻ <· · ·< r_n,n⁻ =R, (−1)^j⁻¹ϕ⁺_n(r) >0 in

r_n,j⁺ ₋₁, r_n,j⁺

forj=1, . . . , n, (−1)^jϕ_n⁻(r) >0 in

r_n,j⁻ ₋₁, r_n,j⁻

forj=1, . . . , n, ϕ_n⁺(0) >0> ϕ⁻_n(0).

(ii) The eigenpairs(μ⁺_n, ϕ⁺_n)and(μ⁻_n, ϕ⁺_n)are complete in the sense that for any eigenpair(μ, ϕ)∈R×W_r^2,q(B_R) of(1.1), there existn∈Nandθ >0such that either(μ, ϕ)=(μ⁺_n, θ ϕ⁺_n)or(μ, ϕ)=(μ⁻_n, θ ϕ_n⁻)is valid.

A comparison of these results with those of[10]might be in order. The results above treat the same eigenvalue problems as in[10]. The main differences are twofold: one is our weaker regularity assumptions onF and the other is in the method of proof. In the above results the regularity ofF is imposed through (F1) and (F2), where the functions β andγ are assumed to be in someL^qspace. We use here fairly elementary arguments to prove the existence of the principal eigenvalues and the higher eigenvalues based, respectively, on the so-called inverse power method and on the monotonicity on the domains of the eigenvalues.

Another feature of this article is this. Regarding the regularity hypotheses (F1) and (F2) onF in caseN2, our requirement onβin Theorem1.2is only thatβ∈L^q(B_R)∩L^N(B_R). From the viewpoint of the existence of a solution, this requirement seems relatively sharp in comparison with the known results[11,14,19,9,5,6]. See Theorem7.5in this connection. We refer also to[16]for some recent results concerning regularity of axially symmetric solutions of uniformly elliptic Hessian equations.

In general, condition (F4) onF is different from Eq. (1.1) being Hessian. LetN2. For simplicity of the argument, we assume thatF depends only onM∈S^N. According to[20], the uniformly elliptic equation (1.1) is called Hessian (cf.[7]) if the functionF:S^N→Ris invariant under conjugation of the orthogonal matrices, i.e.,F (Q⁻¹MQ)= F (M) for all M∈S^N and Q∈O(N ), where O(N ) denotes the group of orthogonal matrices of order N. This condition is stated asF (M)being a symmetric function of the eigenvalues ofM.

Note that the eigenvalues of the matrixM=mω⊗ω+l(IN−ω⊗ω), withω∈S^N⁻¹andm, l∈R, aremandl.

IfF (M)is a symmetric function of the eigenvalues ofM, then F

mω⊗ω+l(I_N−ω⊗ω)

, withω∈S^N⁻¹andm, l∈R, is a function ofm, l. That is, if (1.1) is Hessian, thenF satisfies (F4).

IfN =2, then any symmetric matrixM∈S^N can be represented asM=λ₁ω⊗ω+λ₂(I₂−ω⊗ω), whereλ₁ andλ₂are the eigenvalues ofMandω∈S¹is an eigenvector corresponding toλ₁, and therefore, we see that (1.1) is Hessian if and only ifF satisfies (F4).

However, ifN3, then there are functionsF which satisfy (F4) but are not invariant under conjugation of the matricesQ∈O(N ). For such an example see Appendix A.

The rest of this article is organized as follows. Section 2 is devoted to the study of the solvability of the Dirichlet problem for fully nonlinear ODE on a finite interval as well as some estimates of solutions of fully nonlinear ODE. In Section 3 we establish the existence of principal eigenpairs of fully nonlinear (homogeneous) ODE, and in Section 4 we present basic properties of eigenpairs of fully nonlinear ODE. Section 5 is devoted to completing the proof of one of the main results, Theorem1.1. In Section 6, we turn the multi-dimensional radially symmetric problem (1.1) into one-dimensional problem. Section 7 collects several estimates on radial functions including the W^2,q estimates of radial solutions of fully nonlinear PDE. Section 8 is devoted to the proof of Theorem1.2. In Appendix A, we give an example ofF which satisfies (F1)–(F4) but is not invariant under conjugation of the orthogonal matrices whenN3 (i.e., (1.1) is not a Hessian equation).

2. Solvability of the Dirichlet problem in one dimension

In this section we deal with the one-dimensional case and study the solvability of the Dirichlet problem

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F

u, u, u, x

=0 in(a, b), (2.1)

u(a)=u(b)=0, (2.2)

whereu=du/dxandu=d²u/dx².

We assume throughout this section that (F1) and (F2), withq=1 andΛ= ∞, hold. We thus useP^±(m)to denote P^±(m;λ,∞)in this section.

In what follows, we use the following notation. For any function u∈W^2,1(a, b), F[u](x):=F (u(x), u(x), u(x), x)andP^±[u](x)=P^±(u(x)). In particular, we haveF[0](x)=F (0,0,0, x). A functionu∈W^2,1(a, b)is said to be a subsolution (resp., supersolution) of (2.1) ifF[u](x)0 (resp.,F[u](x)0) a.e. in(a, b).

The following lemma is an adaptation of[10, Lemma 2.1].

Lemma 2.1.There is a functiong_F:R²×(a, b)→Rsuch that for a.a.x∈(a, b)and all(m, p, u)∈R³, we havem= g_F(p, u, x)(resp.,m < g_F(p, u, x)orm > g_F(p, u, x))if and only ifF (m, p, u, x)=0 (resp.,F (m, p, u, x) <0or F (m, p, u, x) >0). The functiong_F satisfies

gF(p1, u1, x)−gF(p2, u2, x) λ⁻¹

β(x)|p1−p2| +γ (x)|u1−u2| for all(p₁, u₁), (p₂, u₂)∈R²and a.a.x∈(a, b). Moreover, we have

g_F(0,0, x) λ⁻¹ F (0,0,0, x) for a.a.x∈(a, b).

Proof. Observe by (F1) and (F2) that for a.a.x ∈(a, b) and any(p, u)∈R², the functionm→F (m, p, u, x)is continuous onRand, ifm1, m2∈Randm1< m2, then we have

F (m₁, p, u, x)−F (m₂, p, u, x)λ(m₁−m₂),

which implies that the functionm→F (m, p, u, x)is (strictly) increasing on R and has the rangeR. Hence, for a.a.x ∈(a, b)and any(p, u)∈R², there exists a uniquegF =gF(p, u, x)such thatm=gF(p, u, x)(resp., m >

gF(p, u, x)orm < gF(p, u, x)) if and only ifF (m, p, u, x)=0 (resp.,F (m, p, u, x) >0 orF (m, p, u, x) <0).

Next we check the Lipschitz property of the functiongF:R²×(a, b)→R. Let(p1, u1), (p2, u2)∈R² and set g_i=g_F(p_i, u_i, x), withi=1,2. Ifg₁< g₂, then, by (F2), we have

0=F (g₁, p₁, u₁, x)−F (g₂, p₂, u₂, x)

λ(g1−g2)+β(x)|p1−p2| +γ (x)|u1−u2| for a.a.x∈(a, b),

which ensures the required Lipschitz property ofg_F. Moreover, for a.a.x∈(a, b), we get similarly to the above, F (0,0,0, x)−λgF(0,0, x) ifgF(0,0, x) >0,

and

−F (0,0,0, x)λg_F(0,0, x) otherwise,

and we have|g_F(0,0, x)|λ⁻¹|F (0,0,0, x)|for a.a.x∈(a, b). 2

Let g_F be the function from Lemma2.1. It is clear that (2.1) is equivalent to the ordinary differential equation (ODE for short) of the normal form

u(x)=g_F

u(x), u(x), x

in(a, b). (2.3)

Together with this observation and Lemma2.1, the standard theory of ODE guarantees the existence of a solution to the Cauchy problem for (2.1) as stated in the following.

Theorem 2.2. Letα1, α2∈Rand c∈ [a, b]. Assume that the functionF[0] ∈L¹(a, b). Then there exists a unique solutionu∈W^2,1(a, b)of (2.1)satisfyingu(c)=α₁andu(c)=α₂.

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We remark that the mapping(α₁, α₂)→u fromR²toC([a, b])is continuous, whereu is the solution of (2.1) given by the above theorem. We omit here giving the proof of the above theorem and this remark on the continuous dependence of the solution of (2.1).

In what follows, given a functionf on[a, b], we denote by f₊ andf₋ the functions x→max{f (x),0}and x→max{−f (x),0}, respectively.

Lemma 2.3.Letc∈ [a, b],f ∈L¹(a, b)andu∈W^2,1(a, b). Assume that λu(x)+β(x) u(x) +f (x)0 a.a.x∈(a, b).

Then we have u

−(x) u

−(c)exp x

c

λ⁻¹β(r)dr

+ x

c

λ⁻¹f₊(r)exp x

r

λ⁻¹β(t)dt

dr for allx∈ [c, b], (2.4)

u

+(x) u

+(c)exp c

x

λ⁻¹β(r)dr

+ c x

λ⁻¹f₊(r)exp r

x

λ⁻¹β(t)dt

dr for allx∈ [a, c], (2.5)

and, ifu(a)0andu(b)0, max[a,b]u(b−a)expλ⁻¹β

L¹(a,b)λ⁻¹f₊

L¹(a,b). (2.6)

To see the role of the above lemma in the context of (2.1), it is worth noting that, iff (x)0, the inequality λu(x)+β(x)|u(x)| +f (x)0 is equivalent to the inequalityP⁺[u](x)+β|u(x)| +f (x)0.

The assertion (2.6) can be regarded as a weak version of the Aleksandrov–Bakelman–Pucci maximum principle.

In the following arguments, we use the fact that if f is absolutely continuous on [a, b], then f₊ and f₋ are absolutely continuous on[a, b]and, for a.a.x∈(a, b),

⎧⎨

⎩

(f₊)(x)=f(x) and (f₋)(x)=0 iff (x) >0, (f₊)(x)=0 and (f₋)(x)= −f(x) iff (x) <0, (f₊)(x)=(f₋)(x)=0 iff (x)=0.

Proof of Lemma 2.3. We write βˆ andfˆforλ⁻¹β andλ⁻¹f, respectively. Settingv=(u)₋ andw=(u)₊, we observe thatvβvˆ + ˆf₊andw− ˆβw− ˆf₊a.e. in(a, b). Hence, (2.4) and (2.5) are consequences of Gronwall’s inequality.

For the proof of (2.6), we may assume that max_[a,b]u >0. We may moreover assume by replacing the interval [a, b]by a smaller interval thatu(x) >0 for allx∈(a, b). We choose a pointcin(a, b)so thatu(c)=max_[a,b]u, and apply (2.5), to obtain

max[a,c]

u

+exp

ˆβ_L¹_(a,c)

ˆf₊_L¹_(a,c),

and moreover

u(c)u(c)−u(a)= c a

u(r)dr c a

u

+(r)dr(b−a)exp

ˆβL¹(a,b)

ˆf₊L¹(a,b), which completes the proof. 2

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Letu, v∈W^2,1(a, b), and observe that for a.a.x∈(a, b),

F[u](x)−F[v](x)P⁺[u−v](x)+β(x) u(x)−v(x) +γ (x) u(x)−v(x) . (2.7) Henceforth we fix anyκ0, and define the functionFκonR³×(a, b)by

Fκ(m, p, u, x)=F (m, p, u, x)−κu.

As above, for anyu, v∈W^2,1(a, b)and a.a.x∈(a, b), we have Fκ[u](x)−Fκ[v](x)P⁺[u−v](x)+β(x) u(x)−v(x)

+

γ (x)−κ

+

u(x)−v(x)

ifu(x)v(x). (2.8)

We set

σ=σ_κ:=(b−a)exp

λ⁻¹βL¹(a,b)λ⁻¹(γ−κ)₊

L¹(a,b), (2.9)

and note that limκ→∞σκ=0.

The following comparison principle holds for (2.1).

Theorem 2.4.Letf, g∈L¹(a, b)andu, v∈W^2,1(a, b). Assume thatσ_κ<1,u(x)v(x)forx=a, b, and F_κ[v](x)+g(x)F_κ[u](x)+f (x) for a.a.x∈(a, b).

Then

max[a,b](u−v) b−a

(1−σκ)expλ⁻¹β

L¹(a,b)λ⁻¹(f −g)₊

L¹(a,b).

Proof. Setw=u−v. As in the proof of Lemma2.3, we may assume that max_[a,b]w >0 andw(x) >0 in(a, b).

By (2.8), we get for a.a.x∈(a, b), P⁺[w](x)+β(x) w(x) +

γ (x)−κ

+w(x)+(f−g)₊(x)0.

Applying Lemma2.3yields max[a,b]w(b−a)exp

ˆβL¹(a,b)λ⁻¹

(γ −κ)₊w+(f−g)₊

L¹(a,b), whereβˆ=λ⁻¹β. Hence, we get

max[a,b]wσ_κmax

[a,b]w+(b−a)exp

ˆβL¹(a,b)λ⁻¹(f−g)₊

L¹(a,b), from which we easily obtain the desired bound on max_[a,b]w. 2

A simple consequence of the above theorem is the following.

Corollary 2.5. Letu∈W^2,1(a, b)andv∈W^2,1(a, b)be, respectively, a subsolution and a supersolution of (2.1), withF replaced byFκ. Assumeσκ<1. Ifu(x)v(x)forx=a, b, thenu(x)v(x)for allx∈ [a, b].

Next, we state and prove a strong comparison principle for (2.1).

Theorem 2.6.Letu, v∈W^2,1(a, b)satisfy F[v](x)F[u](x) for a.a.x∈(a, b)

and u(x)v(x)in[a, b]. Then eitheru(x)≡v(x) oru(x) < v(x)holds in(a, b). Furthermore if u(x) < v(x)in (a, b), then

max

(v−u)(a), (v−u)(a)

>0 and max

(v−u)(b),−(v−u)(b)

>0.

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Proof. Setw=v−uand observe that

P⁻[w] −β w −γ w0 a.e. in(a, b).

It is enough to show that if either max{w(a), w(a)}0, or max{w(b),−w(b)}0, or w(c)=0 for some c∈ (a, b), then w(x) ≡0 in [a, b]. Moreover, it is enough to show that if either max{w(a), w(a)}0 or max{w(b),−w(b)}0, then w(x)≡0 in [a, b]. Indeed, observing that if w(c)=0 for some c∈(a, b), then w(c)=w(c)=0 and applying the above assertion in the intervals [a, c] and[c, b], we deduce thatw(x)≡0 in both of two intervals[a, c]and[c, b].

We consider the case wherew(a)0 andw(a)0. Sincew0 in[a, b], we have indeedw(a)=w(a)=0.

Sincez:= −wsatisfiesP⁺[z] +β|z| +γ w0 a.e. in[a, b], we deduce from Lemma2.3that for allr∈ [a, b], w

+(r)exp

ˆβL¹(a,b)

^r

a

ˆ

γ (t )w(t )dt,

whereβˆ=λ⁻¹β andγˆ=λ⁻¹γ. Integrating this over[a, x], we get forx∈ [a, b], w(x)exp

ˆβL¹(a,b)

^x

a

dr r a

ˆ

γ (t )w(t )dt(b−a)exp

ˆβL¹(a,b)

^x

a

ˆ

γ (t )w(t )dt.

From this, using Gronwall’s inequality, we see thatw(x)≡0 in[a, b].

An argument parallel to the above ensures that if max{w(b),−w(b)} =0, thenw(x)≡0 in[a, b]. 2

Theorem 2.7.Letκ∈ [0,∞). Assume thatF[0] ∈L¹(a, b)andσκ<1, whereσκis the constant defined by(2.9). Then there is a unique solutionu∈W^2,1(a, b)of the Dirichlet problem(2.1)and(2.2), withFκin place ofF. Moreover, if β, γ , F[0] ∈L^q(a, b)for someq∈(1,∞], thenu∈W^2,q(a, b).

Proof. The uniqueness assertion is a direct consequence of Corollary2.5. It is thus enough to show the existence of a solution inW^2,1(a, b)of (2.1) and (2.2), withF_κin place ofF.

For anyd∈R, we denote byu(x;a, d)the unique solution inW^2,1(a, b)of the Cauchy problem for (2.1), withFκ

in place ofF, satisfying the initial condition(u(a;a, d), u(a;a, d))=(0, d), whereu(x;a, d):=∂u(x;a, d)/∂x.

As we have remarked after Theorem2.2, we know that the functiond→u(b;a, d)is continuous fromRtoR.

Letd₁, d₂∈Rbe such thatd₁> d₂. Setw(x)=u(x;a, d₁)−u(x;a, d₂)forx∈ [a, b]. Sincew∈C¹([a, b])and w(a)=d₁−d₂>0, there is a pointc∈(a, b]such thatw(x) >0 for all(a, c].

Fix such a pointc∈(a, b]. Noting thatw(x) >0 andw(x) >0 for allx∈(a, c] andP⁺[w] +β|w| +(γ − κ)₊w0 a.e. in(a, c), we find by Lemma2.3that for allx∈ [a, c],

d₁−d₂= w

+(a)e^B^ˆ

w(x)+w(x) x a

λ⁻¹

γ (t)−κ

+dt

, (2.10)

whereBˆ:= λ⁻¹βL¹(a,b).

We show that w(x) >0 for all x ∈ [a, b]. Indeed, if this is not the case, there is a point e∈(a, b] such that w(e)=0 andw(x) >0 for allx∈ [a, e). Using Lemma2.3again, we get for allx∈ [a, e],

w(x)e^B^ˆw(e) e x

λ⁻¹

γ (t)−κ

+dt=e^B^ˆw(e)λ⁻¹(γ −κ)₊

L¹(a,b). (2.11)

Integrating (2.11) over(a, e), we getw(e)σ_κw(e), which yieldsw(e)0. This is a contradiction, and we conclude thatw(x) >0 for allx∈ [a, b], which shows that (2.10) holds withc=b. Integrating (2.10) over(a, b), we get

(b−a)(d₁−d₂)e^B^ˆw(b)

1+(b−a)λ⁻¹(γ−κ)₊

L¹(a,b)

. That is,

u(b;a, d₁)−u(b;a, d₂) (b−a)(d1−d2)

e^B^ˆ(1+(b−a)λ⁻¹(γ −κ)₊L¹(a,b)) .

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This strict monotonicity and the continuity of the functiond→u(b;a, d)guarantee that there is a uniqued_∗∈Rsuch thatu(b;a, d_∗)=0. The functionu(x;a, d_∗)ofxis a solution of (2.1) and (2.2), withF_κin place ofF.

Now, we assume thatβ, γ , F[0] ∈L^q(a, b) for someq∈(1,∞]. Observe by (F2) that bothϕ=uandϕ= −u satisfy

λϕ(x)+β(x) ϕ(x) +

γ (x)+κ ϕ(x) + F[0](x) 0 for a.a.x∈(a, b).

Hence,

u(x) λ⁻¹β(x) u(x) +

γ (x)+κ u(x) + F[0](x) for a.a.x∈(a, b).

Noting thatu∈C¹([a, b]), we conclude thatu∈L^q(a, b)and, accordingly,u∈W^2,q(a, b). 2

Remark 2.8.The same assertion as Theorem2.7concerning the existence, uniqueness and regularity of solutionsu∈ W^2,1(a, b)of the Dirichlet problem for (2.1) is valid under the general boundary conditionu(a)=d1andu(b)=d2, whered1, d2∈Rare any given constants. Indeed, one can prove this assertion in the same fashion as in the proof above.

3. Principal eigenvalues in one dimension

This section is devoted to the existence of principal eigenpairs of (1.1) in one dimension under hypotheses (F1)–

(F3).

Throughout this section we assume thatN=1,Ω=(a, b), where−∞< a < b <∞, and (F1), (F2) withΛ= ∞ and (F3) hold. We remark that, by assumption (F3), we haveF[0] =0.

We fix a constantκ0 so that σ=σ_κ:=(b−a)expλ⁻¹β

L¹(a,b)λ⁻¹(γ−κ)₊

L¹(a,b)<1, (3.1)

and, as before, setF_κ(m, p, u, x):=F (m, p, u, x)−κu. We consider the eigenvalue problem F_κ

u, u, u, x

+νu=0 in(a, b),

u(a)=u(b)=0. (3.2)

We prove here the following proposition, which is obviously a special case (i.e., the casen=1) of Theorem1.1.

Theorem 3.1. There exist eigenpairs (ν⁺, ϕ⁺), (ν⁻, ϕ⁻)∈R×W^2,q(a, b) of (3.2) such that ϕ⁺(x) > 0 and ϕ⁻(x) <0in(a, b).

The constantsν⁺andν⁻in the above theorem are called, respectively, the positive and negative principal eigenvalues of (3.2). The functionsϕ⁺andϕ⁻are called, respectively, positive and negative principal eigenfunctions of (3.2).

Similarly, the pairs(ν⁺, ϕ⁺)and(ν⁻, ϕ⁻)are called, respectively, positive and negative principal eigenpairs of (3.2).

Letf ∈L^q(a, b), and we consider the Dirichlet problem F_κ

u, u, u, x

+f=0 in(a, b),

u(a)=u(b)=0. (3.3)

SetF (m, p, u, x)˜ :=F_κ(m, p, u, x)−f (x). Then it is easily seen thatF˜ satisfies (F1), (F2) andF˜[0] ∈L^q(a, b).

Hence, according to Theorem2.7, there is a unique solutionu∈W^2,q(a, b)of (3.3). We introduce the solution map- pingT :L^q(a, b)→W^2,q(a, b)byTf=u.

Basic properties of the mapT are stated in the following lemma.

Lemma 3.2.

(i) The mapT is positively homogeneous of degree one, i.e.,T (sf )=sTf for alls0andf ∈L^q(a, b).

(ii) If f ∈L^q(a, b) and f (x)0 for a.a. x ∈(a, b), then (Tf )(x)0 in [a, b]. Furthermore, if f ≡0, then (Tf )(x) >0in(a, b),(Tf )(a) >0and(Tf )(b) <0.

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(iii) There is a constantC >0, depending only onb−a,κ,λ,βL^q(a,b)andγL^q(a,b), such that

Tf −T gW^2,q(a,b)Cf−gL^q(a,b) for allf, g∈L^q(a, b). (3.4) Proof. Letf ∈L^q(a, b). By assumption (F3), we see thatsTf, withs0, is a solution of (3.3) withf replaced bysf, which tells us thatsTf =T (sf ), proving the homogeneity ofT.

Suppose thatf is a nonnegative function. We observe by (F3) thatv≡0 is a subsolution ofF_κ[v] +f =0 in (a, b). Theorem2.4tells us thatTf (x)0 in[a, b]. In the case wheref (x)≡0, we have(Tf )(x)≡0. Hence, we find by Theorem2.6(or the uniqueness assertion of Theorem2.2) thatu(x) >0 in(a, b),u(a) >0 andu(b) <0.

Letf, g∈L^q(a, b)and setu=Tf−T g. By Theorem2.4we have uL^∞(a,b)(b−a)e^B^ˆ

1−σ λ⁻¹(f−g)

L¹(a,b)(b−a)²⁻^q¹e^B^ˆ

1−σ λ⁻¹(f−g)

L^q(a,b), whereBˆ= λ⁻¹βL¹(a,b). Both of the functionsϕ=uandϕ= −usatisfy

λϕ+β ϕ +(γ +κ)|ϕ| + |f−g|0 a.e. in(a, b). (3.5) Hence, noting thatu(c)=0 for somec∈(a, b)and applying (2.4) and (2.5) of Lemma2.3, we get

u

L^∞(a,b)e^B^ˆλ⁻¹(γ+κ)

L¹(a,b)uL^∞(a,b)+λ⁻¹(f−g)

L¹(a,b)

. Finally, we observe by (3.5) that

u

L^q(a,b)λ⁻¹

βL^q(a,b)u

L^∞(a,b)+ γ+κL^q(a,b)uL^∞(a,b)+ f−gL^q(a,b)

,

proving (3.4). 2 Next we define

X:=

f∈C¹ [a, b]

:f (a)=f (b)=0, f(a) >0, f(b) <0, f (x) >0 in(a, b) ,

and observe by Lemma3.2thatTf ∈Xiff ∈X. We introduce the mappingR fromXto the functions on[a, b]as follows:

Rf (x):=

⎧⎨

⎩

Tf (x)

f (x) ifx∈(a, b),

(Tf )(x)

f(x) ifx=a, b.

It follows from the homogeneity ofT that for eacht >0 andf∈X,

R(tf )(x)=Rf (x) for allx∈ [a, b]. (3.6)

Lemma 3.3.

(i) For anyf∈X, we haveRf∈C([a, b])and 0< min

x∈[a,b]Rf (x)= inf

x∈(a,b)

Tf (x) f (x) sup

x∈(a,b)

Tf (x)

f (x) = max

x∈[a,b]Rf (x) <∞.

(ii) The mapR:X→C([a, b])is continuous, provided thatXis equipped with theC¹([a, b])topology.

Proof. Sincef, Tf ∈X, l’Hôpital’s rule tells us thatRf is continuous ataandb, and thusRf∈C([a, b]). It is then clear that the other assertions of (i) hold.

Next we prove the continuity ofR. Letψ denote the function on (a, b)given byψ (x)=(x−a)⁻¹(b−x)⁻¹. Note that 0<infa<x<bψ (x)f (x) <∞ for any f ∈X. Note also that for any function f ∈C¹([a, b])satisfying f (a)=f (b)=0,

ψ (x)f (x)

ψ (x)x

a |f(t)|dt_(b₋²_a)fL^∞(a,b) fora < x(a+b)/2, ψ (x)b

x|f(t)|dt_(b₋²_a)fL^∞(a,b) for(a+b)/2x < b.

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Using these observations, we compute that for anyf, g∈Xandx∈(a, b), Rf (x)−Rg(x) =|g(x)(Tf (x)−T g(x))+(g(x)−f (x))T g(x)|

f (x)g(x)

4gL^∞(a,b)(Tf−T g)L^∞(a,b)+ (f−g)L^∞(a,b)(T g)L^∞(a,b)

(b−a)²inf(a,b)ψ²f g .

From this we see thatR:X→C([a, b])is continuous. 2 Lemma 3.4.Letf ∈Xandu=Tf. Then

min[a,b]Rfmin

[a,b]Rumax

[a,b]Rf.

Moreover, ifmin_[a,b]Rf=min_[a,b]Ru, then T u(x)=

min[a,b]Rf

u(x) for everyx∈ [a, b].

Proof. Set v=T u and θ=min_[a,b]Rf. Since θf (x)u(x)for all x∈ [a, b], the functionv is a supersolution of (3.3), withf replaced byθf. By the homogeneity ofF_κ, the functionθ uis a solution of (3.3), withf replaced byθf. By Theorem2.4, we getθ uvin[a, b], which yields min_[a,b]Rf=θmin_[a,b]Ru. In a similar fashion one can prove that max_[a,b]Rumax_[a,b]Rf.

Now, we assume that min_[a,b]Rf =min_[a,b]Ru. Setting θ =min_[a,b]Rf, we note that θf u in [a, b] and Fκ[v] = −u−θf =Fκ[θ u]a.e. in(a, b). By Theorem2.6, we have eitherθ u(x)≡v(x)in[a, b], or elseθ u(x) <

v(x) in(a, b),v(a) > θ u(a) >0 and v(b) < θ u(b) <0. If the latter is the case, then we have θ <min_[a,b]Ru, which is a contradiction. Thus we must haveθ u=vin[a, b]. 2

Proof of Theorem 3.1. Fix f₀∈X so that f₀C([a,b])=1, and define the sequences(u_k)_k_∈N, (f_k)_k_∈N⊂X and (M_k)_k_∈Nby setting inductivelyu_k:=Tf_k₋₁,M_k:=max_[a,b]u_k andf_k(x):=u_k(x)/M_k fork∈N. Then setθ_k:=

min_[a,b]Ru_k andΘ_k :=max_[a,b]Ru_k. From (3.6) and Lemma3.4, we obtainθ_kθ_k₊1Θ_k₊1Θ_k. Hence, the sequence(θk)k∈Nis convergent. We setθ:=limk→∞θk.

Sincef_kC([a,b])=1, the sequence(u_k)is bounded inW^2,q(a, b)thanks to (3.4). Hence, by the Ascoli–Arzela theorem, (u_k) has a subsequence (u_k_j) converging to a nonnegative function u in C¹([a, b]). Since Rf_k(x)= Ru_k(x)=u_k₊₁(x)/f_k(x)for allx∈(a, b), we have

θ_kf_k(x)u_k₊₁(x)Θ_kf_k(x) for allx∈ [a, b]. (3.7)

SincefkC([a,b])=1, we therefore getθkmax_[a,b]uk+1=Mk+1Θk. Noting thatfk_j(x)=M_k⁻¹

j uk_j(x), we see that, asj → ∞,f_k_j→f:=(max_[a,b]u)⁻¹uinC¹([a, b]). By Lemma3.2, we see that the sequence(Tf_k_j)converges toTf inC¹([a, b]), which reads that(u_k_j₊₁)converges toTf inC¹([a, b]). Settingv:=Tf, by Lemma3.3, we thus obtain

min[a,b]Rv= lim

j→∞min

[a,b]Ru_k_j₊₁= lim

j→∞θ_k_j₊₁=θ. (3.8)

SinceRT u_k_j₊1=RTf_k_j₊1=Ru_k_j₊2, we obtain as above min[a,b]RT v= lim

j→∞min

[a,b]Ruk_j+2=θ. (3.9)

Consequently, by Lemma3.4, we getT v(x)≡θ v(x)in[a, b], which implies thatvis a solution of (3.2) withν=θ⁻¹. The pair(ν⁺, ϕ⁺)=(θ⁻¹, v)is an eigenpair of (3.2) satisfyingϕ⁺(x) >0 for allx∈(a, b).

Note that the functionG(m, p, u, x):= −F (−m,−p,−u, x)satisfies (F1)–(F3), with the same constantsλ,Λ=

∞ and functionsβ, γ. If we define the functionG_κ by the formulaG_κ(m, p, u, x)=G(m, p, u, x)−κu, then we haveGκ(m, p, u, x)= −Fκ(−m,−p,−u, x). Observe also thatu∈W^2,q(a, b)satisfiesFκ[u] +νu=0 a.e. in(a, b) if and only ifv:= −usatisfiesGκ[v] +νv=0 a.e. in(a, b). We apply the previous observation on the existence of an eigenpair of (3.2) to the eigenvalue problem (3.2), withG_κin place ofF_κ, to find an eigenpair(ν⁻, ψ⁻)of (3.2), with G_κin place ofF_κ, such thatψ⁻(x) >0 for allx∈(a, b). If we putϕ⁻(x)= −ψ⁻(x), then(μ⁻, ϕ⁻)is an eigenpair of (3.2) such thatϕ⁻(x) <0 for allx∈(a, b). 2