Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

Solutions positives des équations elliptiques de type divergence du second ordre superlinéaires sur des domaines coniques

Vladimir Kondratiev

, Vitali Liskevich

, Vitaly Moroz

aDepartment of Mathematics and Mechanics, Moscow State University, Moscow 119 899, Russia bSchool of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Received 3 July 2003; received in revised form 25 March 2004; accepted 7 May 2004 Available online 17 September 2004

Abstract

We study the problem of the existence and nonexistence of positive solutions to the superlinear second-order divergence type elliptic equation with measurable coefficients−∇ ·a· ∇u=u^p(∗),p >1, in an unbounded cone-like domainG⊂R^N (N3). We prove that the critical exponent p^∗(a, G)=inf{p >1: (∗)has a positive supersolution at infinity inG} for a nontrivial cone-like domain is always in(1,_N^N₋₂)and depends both on the geometry of the domainGand the coefficientsaof the equation.

2004 Elsevier SAS. All rights reserved.

Résumé

Nous étudions le problème d’existence ou non existence de solutions positives d’équations elliptiques de type divergence du second-ordre superlinéaires à coefficients mesurables −∇ ·a· ∇u=u^p (∗),p >1, sur un domaine coniqueG⊂R^N (N3). Nous prouvons que l’exposant critiquep^∗(a, G)=inf{p >1: (∗)a une supersolution positive à l’infini dansG}pour un domaine conique non–trivial est toujours dans(1,_N^N₋₂), dépend à la fois de la géometrie du domaineGet des coefficients ade l’équation.

2004 Elsevier SAS. All rights reserved.

MSC: 35J60; 35B05; 35R45

* Corresponding author.

E-mail addresses: kondrat@vnmok.math.msu.su (V. Kondratiev), v.liskevich@bristol.ac.uk (V. Liskevich), v.moroz@bristol.ac.uk (V. Moroz).

0294-1449/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2004.03.003

Keywords: Superlinear elliptic equations; Cone-like domains; Positive solutions; Nonexistence; Asymptotic behavior of solutions

1. Introduction

We study the existence and nonexistence of positive solutions and supersolutions to the superlinear second-order divergence type elliptic equation

−∇ ·a· ∇u=u^p inG. (1.1)

Here p > 1, G⊂ R^N (N 3) is an unbounded domain (i.e., connected open set) and −∇ ·a · ∇ :=

−_N

i,j=1 ∂

∂xi(a_ij(x)_∂x^∂

j)is a second order divergence type elliptic expression. We assume throughout the pa- per that the matrixa=(a_ij(x))^N_i,j=1is symmetric measurable and uniformly elliptic, i.e. there exists an ellipticity constantν=ν(a) >0 such that

ν⁻¹|ξ|² N

i,j=1

aij(x)ξiξjν|ξ|², for allξ∈R^Nand almost allx∈G. (1.2) The qualitative theory of semilinear equations of type (1.1) in unbounded domains of different geometries has been extensively studied because of various applications in mathematical physics and the rich mathematical structure. One of the features of Eq. (1.1) in unbounded domains is the nonexistence of positive solutions for certain values of the exponentp. Such nonexistence phenomena have been known at least since the celebrated paper by Gidas and Spruck [14], where it was proved that the equation

−u=u^p (1.3)

has no positive classical solutions inR^N(N3) for 1p <^N_N⁺₋²₂. Though this results is sharp (forp^N_N⁺₋²₂there are classical positive solutions), the critical exponentp^∗=^N_N⁺₋²₂ is highly unstable with respect to any changes in the statement of the problem. In particular, for anyp∈(_N^N₋₂,^N_N⁺₋²₂] one can produce a smooth potentialW (x) squeezed between two positive constants such that equation−u=W (x)u^phas a positive solution inR^N ([34], see also [11] for more delicate results). If one looks for supersolutions to (1.3) inR^N or studies (1.3) in exterior domains then the value and the properties of the critical exponent change. The following result is well-known (see, e.g. [4,6]). IfN3 and 1< p_N^N₋₂then there are no positive supersolutions to (1.3) outside a ball inR^N. The value of the critical exponentp^∗=_N^N₋₂ is sharp in the sense that (1.1) has (infinitely many) positive solutions outside a ball for anyp > p^∗. This statement has been extended in different directions by many authors (see, e.g.

[3,5,7,8,10,12,17–20,25,32,35,37,38]). In particular, in [17] it was shown that the critical exponentp^∗=_N^N₋₂ is stable with respect to the change of the Laplacian by a second-order uniformly elliptic divergence type operator with measurable coefficients, perturbed by a potential, for a sufficiently wide class of potentials (see also [18] for equations of type (1.3) in exterior domains in presence of first order terms).

In this paper we develop a new method of studying nonexistence of positive solutions to (1.1) in cone-like domains (as a model example of unbounded domains inR^Nwith nontrivial geometry). The method is based upon the maximum principle and asymptotic properties at infinity of the corresponding solutions to the homogeneous linear equation. This approach was first proposed in [17]. In the framework of our method we are able to establish the nonexistence results for (1.1) with measurable coefficients in the cone-like domains without any smoothness of the boundary in the setting of the most general definition of weak supersolutions.

We say thatuis a solution (supersolution) to Eq. (1.1) ifu∈H_loc¹ (G)and

G

∇u·a· ∇ϕ dx=()

G

u^pϕ dx for all 0ϕ∈H_c¹(G),

whereH_c¹(G)stands for the set of compactly supported elements fromH_loc¹ (G). By the weak Harnack inequality for supersolutions (see, e.g. [15, Theorem 8.18]) any nontrivial nonnegative supersolution to (1.1) is positive inG.

We say that Eq. (1.1) has a solution (supersolution) at infinity if there exists a closed ballB¯ρ centered at the origin with radiusρ >0 such that (1.1) has a solution (supersolution) inG\ ¯Bρ.

We define the critical exponent to Eq. (1.1) by p^∗=p^∗(a, G)=inf

p >1: (1.1) has a positive supersolution at infinity inG . In this paper we study the critical exponentp^∗(a, G)in a class of cone-like domains

CΩ=

(r, ω)∈R^N: ω∈Ω, r >0 ,

where(r, ω)are the polar coordinates inR^N andΩ ⊆S^N−1 is a subdomain (a connected open subset) of the unit sphereS^N−1inR^N. The following proposition collects some properties of the critical exponent and positive supersolutions to (1.1) on cone-like domains.

Proposition 1.1. LetΩ⊂Ω are subdomains ofS^N−1. Then (i) 1p^∗(a,CΩ)p^∗(a,CΩ)_N^N₋₂;

(ii) Ifp > p^∗(a,CΩ)then (1.1) has a positive supersolution at infinity inCΩ; (iii) Ifp > p^∗(a,CΩ)then (1.1) has a positive solution at infinity inCΩ.

Remark 1.2. Assertion (i) follows directly from the definition of the critical exponentp^∗(a, G)and the fact that p^∗(a,R^N)=_N^N₋₂, see [17]. Property (ii) simply means that the critical exponentp^∗(a, G)divides the semiaxes (1,+∞)into the nonexistence zone(1, p^∗)and the existence zone(p^∗,+∞). Existence (or nonexistence) of a positive solution at the critical valuep^∗itself is a separate issue. Property (iii) says that the existence of a positive supersolution at infinity implies the existence of a positive solution at infinity. More precisely, we prove that if (1.1) has a supersolutionu >0 inC_Ω^ρ then for anyr > ρit has a solutionw >0 inC_Ω^r such thatwu.

The value of the critical exponent for the equation−u=u^pinCΩ withΩ⊆S^N−1satisfying mild regularity assumptions was first established by Bandle and Levine [4] (see also [3]). They reduce the problem to an ODE by averaging overΩ. The nonexistence of positive solutions without any smoothness assumptions onΩ has been proved by Berestycki, Capuzzo-Dolcetta and Nirenberg [5] by means of a proper choice of a test function.

Letλ₁=λ₁(Ω)0 be the principal eigenvalue of the Dirichlet Laplace–Beltrami operator−_ω inΩ. Let α₋=α₋(Ω) <0 be the negative root of the equation

α(α+N−2)=λ1(Ω).

The result in [4,5] reads as follows.

Theorem 1.3. LetΩ⊆S^N−1be a domain. Thenp^∗(id,CΩ)=1−_α²₋, and (1.1) has no positive supersolutions at infinity inCΩ in the critical casep=p^∗(id,CΩ).

Applicability of both ODE and test function techniques seems to be limited to the case of radially symmetric matricesa=a(|x|), whereas the method of the present paper is suitable for studying Eq. (1.1) with a general uniformly elliptic measurable matrixa. It is extendable as far as the maximum principle is valid and appropriate asymptotic estimates are available (see the proof of Theorem 1.6 below). Advantages of this approach are its transparency and flexibility. As a first demonstration of the method we give a new proof of Theorem 1.3, which has its own virtue being considerably less technical then in [4,5]. As a consequence of Theorem 1.3 we derive the following result, which says that in contrast to the case of exterior domains the value of the critical exponent on a fixed cone-like domain essentially depends on the coefficients of the matrixaof the equation.

Theorem 1.4. LetΩ⊂S^N−1be a domain such thatλ₁(Ω) >0. Then for anyp∈(1,_N^N₋₂)there exists a uniformly elliptic matrixapsuch thatp^∗(ap,CΩ)=p.

Remark 1.5. The matrixapcan be constructed in such a way that (1.1) either has or has no positive supersolutions at infinity inCΩ in the critical casep=p^∗(ap,CΩ), see Remark 4.5 for details.

The main result of the paper asserts that Eq. (1.1) with arbitrary uniformly elliptic measurable matrixa on a

“nontrivial” cone-like domain always admits a “nontrivial” critical exponent.

Theorem 1.6. LetΩ⊆S^N−1be a domain andabe a uniformly elliptic matrix. Thenp^∗(a,CΩ) >1. If the interior ofS^N−1\Ω is nonempty thenp^∗(a,CΩ) <_N^N₋₂.

Remark 1.7. It is not difficult to see that in the caseN=2 Eq. (1.1) has no positive solutions outside a ball for anyp >1. However, whenCΩ is a “nontrivial” cone-like domain inR², that isS¹\Ω= ∅, then all the results of the paper remain true with minor modifications of some proofs.

The rest of the paper is organized as follows. In Section 2 we discuss the maximum and comparison principles in a form appropriate for our purposes and study some properties of linear equations in cone-like domains. Propo- sition 1.1 is proved in Section 3. Section 4 contains the proof of Theorems 1.3 and 1.4. The proof of Theorem 1.6 as well as some further remarks are given in Section 5.

2. Background, framework and auxiliary facts

LetG⊆R^N be a domain in R^N. Throughout the paper we assume thatN3. We writeGGif G is a bounded subdomain ofGsuch that clG⊂G. By · pwe denote the standard norm in the Lebesgue spaceL^p. Byc, c1, . . .we denote various positive constants whose exact value is irrelevant.

LetS^N−1= {x∈R^N: |x| =1}andΩ⊆S^N−1be a subdomain ofS^N−1. Here and thereafter, for 0ρ < R +∞, we denote

C^(ρ,R)_Ω :=

(r, ω)∈R^N: ω∈Ω, r∈(ρ, R)

, C_Ω^ρ :=C_Ω^(ρ,+∞). Accordingly,CΩ=C_Ω⁰ andC_SN−1=R^N\ {0}.

Maximum and comparison principles. Consider the linear equation

−∇ ·a· ∇u−V u=f inG, (2.1)

wheref ∈H_loc¹ (G)and 0V ∈L¹_loc(G)is a form-bounded potential, that is

G

V u²dx(1−)

G

∇u·a· ∇u dx for all 0u∈H_c¹(G) (2.2)

with some∈(0,1). A solution (supersolution) to (2.1) is a functionu∈H_loc¹ (G)such that

G

∇u·a· ∇ϕ dx−

G

V uϕ dx=()f, ϕ for all 0ϕ∈H_c¹(G),

where·,·denotes the duality betweenH_loc⁻¹(G)andH_c¹(G). Ifu0 is a supersolution to

−∇ ·a· ∇u−V u=0 inG, (2.3)

thenuis a supersolution to−∇ ·a·∇u=0 inG. Thereforeusatisfies on any subdomainGGthe weak Harnack inequality

inf

Gu CW

mes(G)

G

u dx,

whereCW=CW(G, G) >0. In particular, every nontrivial supersolutionu0 to (2.3) is strictly positive, that is u >0 inG.

We define the spaceD¹₀(G)as the completion ofC_c^∞(G)with respect to the normu_D¹

0(G):= ∇u2. The spaceD¹₀(G)is a Hilbert and Dirichlet space, with the dualD⁻¹(G), see, e.g. [13]. This implies, amongst other things, thatD₀¹(G)is invariant under the standard truncations, e.g.v∈D¹₀(G)implies thatv⁺=v∨0∈D¹₀(G), v⁻= −(v∧0)∈D₀¹(G). By the Sobolev inequalityD₀¹(G)⊂L^N2N−2(G). The Hardy inequality

R^N

|∇u|²dx(N−2)² 4

R^N

|u|²

|x|²dx for allu∈H_c¹(R^N), (2.4)

implies that D₀¹(G)⊂L²(G,|x|⁻²dx). Since the matrix a is uniformly elliptic and the potential V is form bounded, the quadratic form

Q(u):=

G

∇u·a· ∇u dx−

G

V u²dx

defines an equivalent norm√

Q(u)onD¹₀(G). The following lemma is a standard consequence of the Lax–Milgram Theorem.

Lemma 2.1. Letf∈D⁻¹(G). Then the problem

−∇ ·a· ∇v−V v=f, v∈D₀¹(G), has a unique solution.

The following two lemmas provide the maximum and comparison principles for Eq. (2.1), in a form suitable for our framework. We give the full proofs for completeness, though the arguments are mostly standard.

Lemma 2.2 (Weak Maximum Principle). Letv∈H_loc¹ (G)be a supersolution to Eq. (2.3) such thatv⁻∈D¹₀(G).

Thenv0 inG.

Proof. Let(ϕ_n)⊂C_c^∞(G)be a sequence such that∇(v⁻−ϕ_n)²₂→0. For everyn∈N, setv_n:=0∨ϕ_n∧v⁻. Since 0v_nv⁻∈D₀¹(G)and

G

∇(v⁻−vn)²dx=

{0ϕnv⁻}

∇(v⁻−ϕn)²dx+

{ϕn0}

|∇v⁻|²dx

G

∇(v⁻−ϕn)²dx+

{ϕn0}

|∇v⁻|²dx→0,

by the Lebesgue dominated convergence, we conclude that∇(v⁻−vn)²₂→0 (cf. [13, Lemma 2.3.4]). Taking (vn)as a sequence of test functions we obtain

0

G

∇v·a· ∇v_ndx−

G

V vv_ndx

= −

G

∇v⁻·a· ∇vndx+

G

V v⁻vndx→ −

G

∇v⁻·a· ∇v⁻dx+

G

V|v⁻|²dx0.

Thus we conclude thatv⁻=0. 2

Lemma 2.3 (Weak Comparison Principle). Let 0u∈H_loc¹ (G),v∈D₀¹(G)and

−∇ ·a· ∇(u−v)−V (u−v)0 inG.

ThenuvinG.

Remark 2.4. Note that the assertion of Lemma 2.3 follows from Lemma 2.2 if one assumes in addition that u∈H¹(G).

Proof. Let(Gn)_n∈N be an exhaustion of G, i.e. an increasing sequence of bounded smooth domains such that GnGn+1Gand

n∈NGn=G. Letv∈D¹₀(G). Letf ∈D⁻¹(G)be defined by duality as f:= −∇ ·a· ∇v−V v.

Letvn∈D₀¹(Gn)be the unique weak solution to the linear problem

−∇ ·a· ∇vn−V vn=f, vn∈D¹₀(Gn).

Then

−∇ ·a· ∇(u−vn)−V (u−vn)0 inGn, with

u−vn∈H¹(Gn), 0(u−vn)⁻v_n⁺∈D₀¹(Gn).

Therefore(u−v_n)⁻∈D₀¹(G_n). By Lemma 2.2 we conclude that(u−v_n)⁻=0, that isv_nu. Letv¯_n∈D₀¹(G)be defined asv¯_n=v_nonG_n,v¯_n=0 onG\G_n. To complete the proof of the lemma it suffices to show thatv¯_n→v inD₀¹(G). Indeed,

Q(v¯_n)=

G

∇ ¯v_n·a· ∇ ¯v_n−

G

V| ¯v_n|²= f, v_ncfD⁻¹(G)¯v_nD¹

0(G),

where·,·stands for the duality betweenD¹₀(G)andD⁻¹(G). Hence the sequence(v¯n)is bounded inD¹₀(G).

Thus we can extract a subsequence, which we still denote by(v¯_n), that converges weakly tov_∗∈D¹₀(G). Now let ϕ∈H_c¹(G). Then for alln∈Nlarge enough, we have that Supp(ϕ)⊂G_nand

G

∇ ¯v_n·a· ∇ϕ−

G

Vv¯_nϕ=

G_n

∇v_n·a· ∇ϕ−

G_n

V v_nϕ= f, ϕ. By the weak continuity we conclude that

G

∇v_∗·a· ∇ϕ−

G

V v_∗ϕ= f, ϕ. Thereforev_∗∈D₀¹(G)satisfies

−∇ ·a· ∇v−V v=f, v∈D₀¹(CΩ).

Hencev_∗=v. Furthermore,

Q(v¯n−v)= f,v¯n −2f, v + f, v.

Sincef,v¯_n → f, vit follows thatv¯_n→vinD¹₀(G). 2

Minimal positive solution in cone like domains. LetΩ⊆S^N−1be a domain. Consider the equation

−∇ ·a· ∇u−V u=0 inCΩ, (2.5)

where 0V ∈L¹_loc(C_Ω^ρ)is a form-bounded potential. We say thatv >0 is a minimal positive solution to (2.5) inC_Ω^ρ ifvis a solution to (2.5) inC_Ω^ρ and for any positive supersolutionu >0 to (2.5) inC_Ω^r withr∈(0, ρ)there existsc >0 such that

ucv_ψ inC_Ω^ρ.

Note, that such definition of a minimal positive solution, adopted for the framework of cone-like domains, is slightly different from the notion of the minimal positive solution at infinity introduced by Agmon [1] (see also [22–24,28,29]).

Below we construct a minimal positive solution to (2.5) inC_Ω^ρ. Let 0ψ∈C_c^∞(Ω)andθ_ρ∈C^∞[ρ,+∞)be such thatθ_ρ(ρ)=1, 0θ_ρ1 andθ_ρ=0 forrρ+with some >0. Thusf_ψ:= ∇ ·a· ∇(ψθ_ρ)∈D⁻¹(C_Ω^ρ).

Letw_ψbe the unique solution to the problem

−∇ ·a· ∇w−V w=f_ψ, w∈D₀¹(C_Ω^ρ), (2.6)

which is given by Lemma 2.1. Setv_ψ:=w_ψ+ψθ_ρ. Thenv_ψ is the solution to the problem

−∇ ·a· ∇v−V v=0, v−ψθ_ρ∈D¹₀(C_Ω^ρ). (2.7)

By the weak Harnack inequalityvψ>0 inC_Ω^ρ. Notice thatvψ actually does not depend on the particular choice of the functionθρ(this easily follows, e.g., from Lemma 2.2).

Lemma 2.5.v_ψ is a minimal positive solution to Eq. (2.5) inC_Ω^ρ.

Proof. ChooseΩΩ such that Supp(ψ)Ω. Let >0 be such thatθρ=0 for allrρ+. Letu >0 be a positive supersolution to (2.5) inC_Ω^r withr∈(0, ρ). By the weak Harnack inequality there existsm=m(Ω, ) >0 such that

u > m inC_Ω^{(ρ,ρ +)}.

Choosec >0 such thatcψ < m. Thenu−cψθρ0 inC_Ω^ρ,cwψ∈D₀¹(C^ρ_Ω)and (−∇ ·a· ∇ −V )

(u−cψθ_ρ)−cw_ψ =(−∇ ·a· ∇ −V )u0 inC_Ω^ρ. By Lemma 2.3 we conclude thatu−cψθρcwψ, that isucvψ inC_Ω^ρ. 2

Remark 2.6. LetΓ_a(x, y)be the positive minimal Green function to the equation−∇ ·a· ∇u=0 inR^N. Then for any domainΩ⊆S^N−1the functionΓ_a(x,0)is a positive solution to

−∇ ·a· ∇u=0 inCΩ. (2.8)

By Lemma 2.3 and the classical estimate [21] we conclude that any minimal positive solutionvψ to (2.8) inC_Ω^ρ obeys the upper bound

v_ψc₁Γ_a(x,0)c₂|x|^2−N inC_Ω^ρ. (2.9)

Nonexistence Lemma. The next lemma (compare [17], [30, p. 156]) is the key tool in our proofs of nonexistence of positive solutions to nonlinear equation (1.1).

Lemma 2.7 (Nonexistence Lemma). Let 0V ∈L¹_loc(C_Ω^ρ)satisfy

|x|²V (x)→ ∞ asx∈C_Ω^ρ and|x| → ∞ (2.10)

for a subdomainΩ⊆Ω. Then the equation

−∇ ·a· ∇u−V u=0 inC_Ω^ρ (2.11)

has no nontrivial nonnegative supersolutions.

The proof is based upon the following well-known result (see, e.g., [1, Theorem 3.3]).

Lemma 2.8. Let G⊂R^N be a bounded domain and λ₁=λ₁(G) >0 be the principal Dirichlet eigenvalue of

−∇ ·a· ∇inG. Ifµ > λ₁then the equation

−∇ ·a· ∇u=µu inG (2.12)

has no positive supersolutions.

Proof of Lemma 2.7. Letλ₁(C_Ω^(ρ,2ρ)) >0 be the principal Dirichlet eigenvalue of−∇ ·a· ∇onC_Ω^(ρ,2ρ). Rescaling the equation−∇ ·a· ∇v=λvfromC_Ω^(ρ,2ρ)toC_Ω^(1,2)one sees that

c⁻¹

ρ²λ1(C_Ω^(1,2))λ1(C_Ω^(ρ,2ρ)) c

ρ²λ1(C_Ω^(1,2)),

wherec=c(a) >0 depends on the ellipticity constant of the matrixaand does not depend onρ >0.

Letu0 be a supersolution to (2.11). Then (2.10) implies that for someR1 one can findµ >0 such that V (x)µcλ₁(C_Ω^(1,2) )R⁻²inC_Ω^(R,2R) . Henceuis a supersolution to

−∇ ·a· ∇u=µu inC_Ω^(R,2R)

withµ > λ1(C_Ω^(R,2R) ). By Lemma 2.8 we conclude thatu=0 inC_Ω^(R,2R) . Therefore by the weak Harnack inequality u=0 inC_Ω^ρ. 2

3. Proof of Proposition 1.1

Property (i) is obvious. We need to prove (ii) and (iii).

(ii) Letp0p^∗(a,C_Ω^ρ)be such that Eq. (1.1) with exponentp0has a positive supersolutionu >0 inC_Ω^ρ. Let p > p0andα=_p^p₀⁻₋¹₁>1. Setv:=u^1/α. By the weak Harnack inequalityu >0 inC_Ω^ρ. Henceu^−s∈L^∞_loc(C_Ω^ρ)for anys >0. Therefore∇v=α⁻¹u^1/α−1∇u∈L²_loc(C^ρ_Ω), that isv∈H_loc¹ (C_Ω^ρ).

Let 0ϕ∈C_c^∞(C^ρ_Ω). Then

CΩ^ρ

∇v^α·a· ∇ϕ dx=α

CΩ^ρ

v^α−1∇v·a· ∇ϕ dx

=α

CΩ^ρ

∇v·a· ∇(v^α−1ϕ) dx−α(α−1)

C^ρΩ

∇v·a· ∇v(v^α−2ϕ) dx

α

CΩ^ρ

∇v·a· ∇(v^α−1ϕ) dx.

Notice, thatv^α−1=u^1−1/α∈H_loc¹ (C_Ω^ρ)by the same argument as above. Thereforev^α−1ϕ∈H_c¹(C_Ω^ρ). We shall prove that the set

Kv=

v^α−1ϕ, 0ϕ∈H_c¹(C_Ω^ρ)

is dense in the cone of nonnegative functions inH_c¹(C_Ω^ρ). Indeed, let 0ψ∈H_c¹(C_Ω^ρ). Letψ_n∈C₀^∞(C^ρ_Ω)be an approximating sequence such that∇(ψ_n−ψ)2→0. Setϕ_n=v^1−αψ_n⁺. It is clear that 0ϕ_n∈Kv⊂H_c¹(C^ρ_Ω) and∇(v^α−1ϕn−ψ)2→0.

Sincev^α=uandu >0 is a supersolution of (1.1), we obtain that α

CΩ^ρ

∇v·a· ∇(v^α−1ϕ) dx

CΩ^ρ

v^αp0ϕ dx=

C^ρΩ

v^p(v^α−1ϕ) dx

for any 0ϕ∈H_c¹(C_Ω^ρ). Thusα^1/(1−p)vis a supersolution to Eq. (1.1) inC_Ω^ρ with exponentp > p0.

(iii) The existence of a (bounded) positive solution to Eq. (1.1) withp > _N^N₋₂ inB_r^c for anyr >0 has been proved in [17]. We shall consider the casep_N^N₋₂.

Letu >0 be a supersolution to (1.1) with exponentpN/(N −2)inC_Ω^ρ. Fixψ∈C_c^∞(Ω)andr > ρ. Let v_ψ>0 be a minimal positive solution in to−∇ ·a· ∇v=0 inC_Ω^r , as constructed in (2.6), (2.7). Thenucv_ψ in C^r_Ω by Lemma 2.3. Without loss of generality we assume thatc=1. Thusvψ>0 is a subsolution to (1.1) inC_Ω^r andvψuinC_Ω^r . We are going to show that (1.1) has a positive solutionwinC_Ω^r such thatvψwuinC_Ω^r .

Let(Gn)n∈Nbe an exhaustion ofC_Ω^r . Consider the boundary value problem −∇ ·a· ∇w=w^p inG_n,

w=v_ψ on∂G_n. (3.1)

SinceG_nC_Ω^r is a smooth bounded domain andv_ψ∈C_loc^0,γ(C_Ω^r ), the problem (3.1) is well-posed. Clearly,v_ψu is still a pair of sub and supersolutions for (3.1). Notice that we do not assume thatu∈H¹(Gn)is bounded. How- ever, sincep_N^N₋₂<^N_N⁺₋²₂, one can use anH¹-version of sub and supersolution method, see e.g. [9, Theorem 2.2].

Thus there exists a weak solutionwn∈H¹(Gn)of (3.1) such thatvψwnuinGn.

Consider a sequence(w_n)_n>1inG₁. Choose a functionθ∈C^∞_c (G₂)such that 0θ1 andθ=1 on G₁. Usingθ²w_n∈H_c¹(G₂)as a test function we obtain

G₂

w_n^p+1θ²=

G₂

θ²∇wn·a· ∇wndy+2

G₂

θ wn∇wn·a· ∇θ dy.

Thus, by standard computations 1

2

G2

θ²∇w_n·a· ∇w_ndx2

G2

w_n²∇θ·a· ∇θ dx+

G2

θ²w_n^p+1dx

2c₁(∇θ )²_∞

G₂

u²dx+

G₂

u^p+1dx.

We conclude that(w_n)is bounded inH¹(G₁). By the constructionv_ψw_nu∈H¹(G₁)for alln∈N. Therefore (w_n)has a subsequence, denoted by(w_n1(k))_k∈N, which converges to a functionw⁽¹⁾∈H¹(G₁)weakly inH¹(G₁), strongly in L²(G₁)and almost everywhere in G₁. Hence it is clear that w⁽¹⁾ is a solution to (1.1) inG₁ and vψw⁽¹⁾u.

Now we proceed by the standard diagonal argument (see, e.g., [26, Theorem 2.10]). At the second step, consider a sequence(w_n1(k))_k∈N inG₂ (assuming that n₁(1) >2). In the same way as above we obtain a subsequence (w_n2(k))_k∈N that converges to a function w⁽²⁾ ∈H¹(G₂), which is a solution to (1.1) in G₂. Moreover,v_ψ w⁽²⁾uinG₂andw⁽²⁾=w⁽¹⁾ inG₁. Continuing this process, for each fixedm >2 we construct a subsequence (wn_m(k))k∈N(withnm(1) > m) that converges weakly tow^(m)∈H¹(Gm)which is a solution to (1.1) inGm and such thatvψw^(m)uinGm,w^(m)=w^(m−1)inGm−1.

By the diagonal process(w_nm(m))_m∈N is a subsequence of(w_nm(k))_k∈N for everym∈N. Thus for each fixed k∈N the sequence(w_nm(m))converges weakly to w^(k) in H¹(G_k). Let w_∗ be the weak limit of (w_nm(m)) in H_loc¹ (C_Ω^r ). Thenw_∗is a solution of (1.1) inC_Ω^r such thatv_ψw_∗uinC_Ω^r . 2

Remark 3.1. The constructed solutionw_∗is actually locally Hölder continuous. Indeed, sincep_N^N₋₂<^N_N⁺₋²₂ we conclude by the Brezis–Kato estimate (see, e.g. [33, Lemma B.3]) thatw_∗∈L^s_loc(C_Ω^r )for anys <∞. Then

−∇ ·a· ∇w_∗=w^p_∗ ∈L^s_loc(C^r_Ω)for anys <∞. Hence the standard elliptic estimates imply thatw_∗∈C_loc^0,γ(C_Ω^r ).

4. Proof of Theorems 1.3 and 1.4

In this section we study positive supersolutions at infinity to the model equation

−u=u^p inCΩ, (4.1)

wherep >1 andΩ is a subdomain ofS^N−1. Recall, that λ₁ denotes the principal eigenvalue of the Dirichlet Laplace–Beltrami operator−_ωinΩandα₋stands for the negative root of the equationα(α+N−2)=λ₁.

Existence of positive supersolutions to (4.1) withp > p^∗(id,CΩ)=1−2/α₋can be easily verified. Namely, by direct computations one can find supersolutions of the formu=cr^2/(1−p)φ, where φ >0 is the principal eigenfunction of−ω onΩ(see also [4,3] for a direct proof of the existence of positive solutions). We are going to prove absence of positive supersolutions to (4.1) inC_Ω^ρ forp∈(1,1−2/α₋). Notice that ifu >0 is a solution to (4.1) inC_Ω^ρ then, by the scaling properties of the Laplacian,ρ^p−21u(ρx)is a solution to (4.1) inC_Ω¹. So in what follows we fixρ=1.

Minimal solution estimate. Here we derive the sharp asymptotic at infinity of the minimal solutions to the equa- tion

−u−V (ω)

|x|² u=0 inCΩ (4.2)

withV ∈L^∞(Ω). Let −ω be the Dirichlet Laplace–Beltrami operator in L²(Ω)and 0V ∈L^∞(Ω). Let (˜λ_k)_k∈N be the sequence of Dirichlet eigenvalues of−_ω−V, such thatλ˜₁<λ˜₂λ˜₃· · ·. By (φ˜_k)_k∈N we denote the corresponding orthonormal basis of eigenfunctions inL²(Ω), withφ˜₁>0.

From now on we assume thatλ˜₁>−(N−2)²/4. Then the roots of the quadratic equationα(α+N−2)= ˜λ_k are real for eachk∈N. Byα˜_kwe denote the smallest root of the equation, i.e.,

˜

α_k:= −N−2

2 −

(N−2)² 4 + ˜λ_k.

Notice that sinceλ˜1>−(N−2)²/4 it follows from the Hardy inequality (2.4) that the potentialV (ω)|x|⁻²is form bounded.

Lemma 4.1. Letψ∈C_c^∞(Ω). Then

v_ψ(x)=^∞

k=1

ψ_kr^α˜kφ˜_k(ω), whereψ_k=

Ω

ψ(ω)φ˜_k(ω) dω, (4.3)

is a minimal positive solution to Eq. (4.2) inC_Ω¹.

Proof. Setvk(x):=r^α˜kφ˜k(ω). Then a direct computation gives that

−vk−V (ω)

|x|² vk=0 inC¹Ω.

Recall that∇ =ν_∂r^∂ +¹_r∇ω, whereν=_|^x_x|∈R^N. Since

Ω

|∇ωφ˜k|²dω−

Ω

V (ω)| ˜φk|²dω= ˜λk,

we obtain

∇v_k²_L2

CΩ^ρ

|∇v_k|²−V (ω)

|x|² |v_k|²

dx

= ∞

1

Ω

∂

∂rr^α˜kφ˜k(ω)

²+|r^α˜k∇ωφ˜_k(ω)|²

r² −V (ω)|r^α˜kφ˜_k|² r²

r^N−1dω dr

= ∞

1

r^2α˜k+N−3(α˜_k²+ ˜λ_k) dr= α˜_k²+ ˜λ_k

2−N−2α˜_k = − ˜α_k,

where >0 is the constant in (2.2). Now it is straightforward thatvk− ˜φkθ1∈D₀¹(C_Ω¹), sovksolves the problem

−v−V (ω)

|x|² v=0, v− ˜φkθ1∈D₀¹(CΩ¹).

Hence we have ∇v_ψ²₂

CΩ¹

|∇v_ψ|²−V (ω)

|x|² |v_ψ|²

dx=^∞

k=1

ψ_k²(− ˜α_k)

N−2

2 ψ²₂+ ψ2 Ω

|∇ωψ|²−V (ω)ψ²+

N−2 2

2

ψ²

dω ¹

2

.

Hencev_ψ−ψθ₁∈D¹₀(C_Ω¹), sov_ψ solves the problem

−v−V (ω)

|x|² v=0, v−ψθ₁∈D₀¹(C_Ω¹).

By the uniqueness we conclude thatv_ψ defined by (4.3) coincides with the minimal solutionv_ψ as constructed in (2.6), (2.7). 2