Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Electronic Journal of Qualitative Theory of Differential Equations

2018, No.65, 1–3; https://doi.org/10.14232/ejqtde.2018.1.65 www.math.u-szeged.hu/ejqtde/

Corrigendum to

Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Tomas Godoy

and Alfredo Guerin

FaMAF, Universidad Nacional de Cordoba, Ciudad Universitaria, Cordoba, 5000, Argentina

Received 19 June 2018, appeared 27 July 2018 Communicated by Maria Alessandra Ragusa

Abstract. This paper serves as a corrigendum to the paper “Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems”, published inElectron J. Qual. Theory Differ. Equ. 2017, No. 100, 1–30. We modify one of the assumptions of that paper and we present a correct proof of the Lemma 2.11 of that paper.

Keywords: singular elliptic problems, positive solutions, sub- and supersolutions, bi- furcation problems.

2010 Mathematics Subject Classification: Primary 35J75; Secondary 35D30, 35J20.

1 Introduction

Lemma 2.11 in [1], under the assumptions stated there, is false. In order to correct this situation, the assumption H2) of [1], Theorem 1.1 (assumed, jointly with H1) and H3)–H5), in the quoted lemma and throughout the whole article [1]) must be replaced (throughout the whole article [1]) by the (slightly stronger) following new version of it:

H2) a∈ L^∞(_Ω),a≥0a.e.inΩ,and there exists δ>0such thatinf_Aδa>0.

Here and below, forρ>0,

A_ρ:={x∈ _Ω:d_Ω(x)≤ρ},

where d_Ω := dist(·,∂Ω); and, for a measurable subset Eof Ω, inf_E means the essential infi- mum on E. In the next section we give (assuming the stated new version of H2)) a correct proof of [1, Lemma 2.11]. With these changes, all the results contained in [1] hold.

BCorresponding author. Email: [email protected]

2 T. Godoy and A. Guerin

2 Correct proof of [1, Lemma 2.11]

Below, “problem (2.4)” refers to the problem labeled (2.4) in [1]; i.e., refers to the problem











−_∆u=χ_{u>0}a(x)u^−α+ζ inΩ, u=_{0 on}∂Ω,

u≥0 inΩ, u>0 a.e. in {a>0},

whereζ ∈ L^∞(_Ω). Recall that the new version ofH2)is assumed in the following lemma.

Lemma 2.1([1, Lemma 2.11]). Assume1<α<3,and letζ ∈ L^∞(_Ω)be such thatζ ≥0. Let u be the solution to problem(2.4)given by [1, Lemma 2.5] (in the sense stated there). Then there exists a positive constant c,independent ofζ,such that u≥cd

2 1+α

Ω inΩ.

Proof. From [1, Lemma 2.5], there exists a positive constant c⁰, independent of ζ, such that u ≥ c⁰d_Ω a.e. in Ω. Then (since inf_Ω\A_δ

4

d_Ω > 0), there exists a positive constant c⁰⁰ (that depends onδ, but not onζ) such that

u≥c⁰⁰d

2 1+α

Ω a.e. inΩ\Aδ

4. (2.1)

LetU be aC^1,1 domain such that A3δ

4 ⊂ U ⊂ A_δ. Note that ∂U\_∂Ω ⊂ _Ω\Aδ

2. Indeed, let z ∈ ∂U\_{∂Ω. Since}U ⊂ A_δ∪∂Ω, we havez ∈ _{Ω. If}z ∈ Aδ

2, then, for some open setVz such thatz ∈ Vz ⊂ Ω, we would haved_Ω ≤ ³₄δ onVz, and soVz ⊂ A_δ ⊂U, which contradicts that z∈∂U. Then∂U\_∂Ω⊂_Ω\Aδ

2. We claim that

d_U =d_Ω in Aδ

8, (2.2)

where d_U := dist(·,∂U). Indeed, let x ∈ Aδ

8, let y_x ∈ ∂Ω be such that d_Ω(x) = |x−y_x|, and let w ∈ ∂U\∂Ω. Since ∂U\∂Ω ⊂ _Ω\Aδ

2, we have |w−yx| ≥ d_Ω(z) > ^δ₂. Also,

|x−yx| = d_Ω(x) ≤ ₈^δ. Therefore, by the triangle inequality, |w−x| ≥ |w−yx| − |x−yx| >

δ

2 −₈^δ = ^3δ₈. Then dist(x,∂U\_∂Ω)≥ ^3δ₈ for any x ∈ Aδ

8, and so (sinced_Ω(x)≤ ^δ₈),d_U(x) = min{dist(x,∂U\_∂Ω),d_Ω(x)}=d_Ω(x)for allx ∈ Aδ

8

Since U ⊂ A_δ we have that a := infUa > 0. Let σ₁ be the principal eigenvalue for −_∆ in U with homogeneous Dirichlet boundary condition and weight function a, and let ψ₁ be the corresponding positive principal eigenfunction, normalized bykψ₁k_∞ = 1. Observe that ψ

2 1+α

1 ∈ H₀¹(U)∩L^∞(U)(because 1<α<3), and that a computation gives

−_∆

ψ

1+2α

1

= ²

1+ασ₁aψ₁^1+2α+ ² 1+α

α−1 1+α

ψ

1+2α

1

−α

|∇ψ₁|²

≤βa

ψ

2 1+α

1

−α

a.e. inU,

whereβ:= ₁₊²

ασ₁+ ₁₊²

α α−1 1+α 1

ak∇ψ₁k²_∞. Then

−_∆

β⁻

1 1+αψ

2 1+α

1

≤a

β⁻

1 1+αψ

2 1+α

1

−α

inU

Corrigendum to the article EJQTDE 2017, No. 100 3

in the weak sense of [1, Lemma 2.5] (i.e., with test functions in H¹₀(U)∩L^∞(U)). Moreover, again in the weak sense of [1, Lemma 2.5], −_∆u ≥ au^−α in U. Also u ≥ β⁻

1 1+αψ

2 1+α

1 in ∂U.

Then, by the weak maximum principle in [2, Theorem 8.1],u≥ β⁻

1 1+αψ

1+2α

1 a.e.inU; therefore, for some positive constant c⁰⁰⁰independent ofζ,u≥c⁰⁰⁰d

2 1+α

U a.e.inU. In particular, u≥c⁰⁰⁰d_U^1+2α a.e.in Aδ

8. (2.3)

From (2.1), (2.3), and (2.2), we get u ≥ cd_Ω^1+2α a.e.in Ω, withc := _min{c⁰⁰,c⁰⁰⁰}and the lemma follows.

References

[1] T. Godoy, A. Guerin, Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems, Electron. J. Qual. Theory Differ. Equ. 2017, No. 100, 1–30.

https://doi.org/10.14232/ejqtde.2017.1.100;MR3750159

[2] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second or- der, Springer-Verlag, Berlin Heidelberg New York, 2001. https://doi.org/10.1007/

978-3-642-96379-7;MR1814364