existence of solutions to a singular elliptic equation

Existence of Solutions to a Singular EllipticEquation

Marcelo Montenegro

Abstract. We study the equation −Δu = (− 1uβ +λup)χ{u>0} in Ω with Dirichlet

boundary condition, where 0 < p < 1 and 0 < β < 1. We regularize the term1/uβ near u ∼ 0 by using a function gε(u) which pointwisely tends to 1/uβ asε → 0. When the parameter λ > 0 is large enough, the corresponding energyfunctional has critical points uε. Letting ε → 0, then uε converges to a solutionof the original problem, which is nontrivial, nonnegative and vanishes at someportion of Ω. There are two nontrivial solutions.

Mathematics Subject Classification (2010). Primary 34B16; Secondary 35J20,35B65.

Keywords. Singular problems, multiple solutions, variational methods, estimates.

1. Introduction

The purpose of this note is to find solutions to the singular problem (1). We describehow to approximate the singular equation by nonsingular equations. We obtain asolution to each nonsingular problem and estimates guaranteeing that the limitingfunction is a solution of the original problem.

The following problem was studied in [3]{−Δu = χ{u>0}

(− u−β + λup)

in Ω

u = 0 on ∂Ω.(1)

The expression χ{u>0} denotes the characteristic function corresponding to the set{u > 0}. Hereafter, Ω ⊂ IRN , N ≥ 1, is a bounded smooth domain, 0 < β < 1 and0 < p < 1.

Theorem 1.1. There exists a maximal solution for every λ > 0. There is constantλ∗ > 0 such that for λ > λ∗ the maximal solution is positive. And for λ < λ∗, themaximal solution vanishes on a set of positive measure.

The author was supported by CNPq and FAPESP..

Milan J. Math. Vol. 79 (2011) 293–301DOI 10.1007/s00032-011-0152-9Published online June 29, 2011© 2011 Springer Basel AG Milan Journal of Mathematics

294 M. Montenegro Vol. 79 (2011)

It is possible to solve problem (1) by perturbing the equation as

−Δu +u

(u + ε)1+β= up. (2)

The solutions uε ↘ u pointwisely and∫Ωu(−Δϕ) +

∫{u>0}

1uβ

ϕ ≤ λ

∫Ωupϕ, (3)

∀ϕ ∈ C2(Ω), ϕ ≥ 0, ϕ = 0 on ∂Ω.There are two approaches to show that u is indeed a solution of (1), thus

proving Theorem 1.1. Relation (3) tells us that u is a maximal subsolution. We then

regularize it and show that u ∈ C1, 1−β

1+β and indeed solve the problem (1). In doingthis, we need to obtain the local estimate

|∇u| ≤ Cu1−β2 in Ω′ ⊂⊂ Ω.

One of the main ingredients to prove it is the Harnack type lemma below.

Lemma 1.2. For every ball Br(p) ⊂ Ω there are constants c0, τ > 0 depending onlyon n and β such that

if −∫∂Br(p)

u ≥ c0r2

1+β , then u(x) ≥ τ−∫∂Br(p)

u a.e. in Br/2(p)

The second approach relies on an estimate for uε obtained by the maximumprinciple, namely

|∇uε| ≤ Cu1−β2

ε in Ω′ ⊂⊂ Ω. (4)

The idea to obtain such estimate is to define v = |∇uε|2u1−βε

ϕ21, where ϕ1 is the first

eigenfunction of the Laplacian with zero boundary condition. The function v hasa maximum at x0 ∈ Ω, and then Δv(x0) ≤ 0. If the estimate is not true, it ispossible to reach a constant C > 0 independently of ε such that supΩ v > C andby computation Δv(x0) > 0, obtaining a contradiction. Using the estimate andmultiplying the equation by an adequate test function, we let ε→ 0 in the equationto get a weak solution.

Similar methods can be used when dealing with other equations. The followingproblem was studied in [5]{

−Δu = χ{u>0}(log u + λup

)in Ω

u = 0 on ∂Ω

Both approaches described above work in this case and a result analogous to theorem(1.1) holds true. The estimate obtained for the maximal subsolution (which is asolution after regularizing it) is |∇u| ≤ Cu in Ω′ ⊂⊂ Ω and u ∈ C1,1, a betterregularity than the one for (1). This is roughly explained since logu is less singularthan −1/uβ . The estimate by maximum principle is |∇uε| ≤ Cuε in Ω′ ⊂⊂ Ω.

Vol. 79 (2011) Existence of Solutions to a Singular Elliptic Equation 295

2. The variational approach

We use variational methods to show that (1) has two weak nonnegative solutionswhen the parameter λ > 0 is large. This result complements Theorem 1.1, see [6]where complete results and proofs are presented.

We define the perturbation

gε(u) =

⎧⎨⎩uq

(u + ε)q+βfor u ≥ 0

0 for u < 0,(5)

where 0 < q < p < 1 and the corresponding perturbed problem{−Δu + gε(u) = λup in Ω

u = 0 on ∂Ω.(6)

The above perturbation (5)–(6) is different from (2), but it is adequate in the vari-ational characterization described below.

Since gε ≥ 0 and is continuous, then Gε(u) =∫ u0 gε(s)ds ≥ 0. We define the C1

functional Iε : H10 (Ω) → IR corresponding to (6) by

Iε(u) =12

∫Ω|∇u|2 +

∫ΩGε(u)− λ

p + 1

∫Ω(u+)p+1.

Our aim is to show that Iε satisfies the assumptions of the Mountain Pass Theorem.This allows us to find two distinct nontrivial solutions of problem (6). Letting ε→ 0these two solutions do not tend to zero neither collapse at the same limit, they tendto two distinct nontrivial solutions of (1). For that matter, the main ingredient is agradient estimate for solutions uε of (6) that allows us to conclude that uε tend to asolution u of (1) as ε→ 0, according to Sections 3, 4 and 5. When taking the limitwe need to be careful since the gradient estimate provided by Lemma 4.1 is local.We state our existence result.

Theorem 2.1. There is a λ0 > 0 such that problem (1) has two distinct nontrivialnonnegative solutions for λ > λ0.

We are unable to prove that the solutions of Theorem 2.1 are positive. Weconjecture that one of them is positive and the other one vanishes somewhere in Ω.

This free boundary solution could be used to produce infinitely many solutionson a domain Ω with finitely many separated bumps supported on balls in the interiorof Ω. One of the achievements of our results is the variational characterization ofthe solutions. Free boundary solutions appear in [1]. See also [2, 4] for other typesof singular equations.

3. Two solutions of the perturbed problem

We proceed to show that the perturbed functional Iε has two nontrivial criticalpoints, a global minimum and a mountain pass, whenever λ > 0 is large and ε > 0 issufficiently small. We need to prove estimates for the associated critical levels which


are independent of the value of the parameter ε. Allowing us to show that weaklimits of the critical points of the perturbed functional, obtained by making ε→ 0,converge to nontrivial and distinct functions in H1

0 (Ω).Denoting by ϕ1 > 0 the first normalized eigenfunction of the operator −Δ in

H10 (Ω), we may state our first preliminary result.

Lemma 3.1. There exist λ0 > 0 and a1, b1 > 0 such that, for every λ ≥ λ0 and everyε > 0, we have

max0≤s≤1

Iε(sϕ1) ≤ a1 <∞ (7)

and

Iε(ϕ1) ≤ −b1 < 0. (8)

Proof. From (5), we obtain

gε(t) ≤ |t|−β for every t �= 0. (9)

Therefore, since 0 < β < 1, we obtain |Gε(t)| ≤ |t|1−β/(1 − β), for every t ∈ IR.Consequently, for 0 ≤ s ≤ 1,

Iε(sϕ1) ≤ s2

2+

s1−β

(1− β)

∫Ωϕ1−β1 − λsp+1

p + 1

∫Ωϕp+11 ,

since ‖ϕ1‖H10

= 1. The estimates (7) and (8) follow immediately from the aboveinequality. �

Next lemma implies, in particular, that the functional Iε is coercive and boundedfrom below. Combined with Lemma 3.1, it will be used to show that this functionalhas two nontrivial critical points.

Lemma 3.2. Given λ > 0, there exist a2, b2 > 0 and 0 < ρ < 1 such that, for every0 < ε < 1,

Iε(u) ≥ a2 > 0 for every such that ‖u‖H10

= ρ, (10)

Iε(u) →∞ as ‖u‖H10→∞ (11)

and

Iε(u) ≥ −b2 > −∞ for every u ∈ H10 (Ω). (12)

Proof. Given 0 < ε < 1, from (5), we have that gε(t) ≥ tq/(t+1)q+β for every t ≥ 0.Since 0 < q < p, we may find δ = δ(λ) > 0 such that

gε(t) ≥ λtp for every 0 ≤ t ≤ δ. (13)

Taking 0 < ρ < 1 sufficiently small, we obtain Iε(u) ≥ a2 := ρ2/4 for every u suchthat ‖u‖H1

0= ρ.

We also obtain that for every ε > 0,

Iε(u) ≥ 12‖u‖2H1

0 (Ω) − C3‖u‖p+1H1

0 (Ω) for every u ∈ H10 (Ω).

The above estimate and 0 < p < 1 imply that (11) and (12) are true. �


Given a Banach space E, we recall that a functional Φ ∈ C1(E, IR) satisfies thePalais-Smale (PS) condition if every sequence (un) ⊂ E, satisfying Φ(un) → c and‖Φ′(un)‖ → 0 as n→∞, has a convergent subsequence.

From now on in this Section we fix λ, where λ ≥ λ0 > 0 and λ0 is given byLemma 3.1. Next proposition provides the existence of two critical points for thefunctional Iε.

Proposition 3.3. Suppose 0 < ε < 1. Then the functional Iε possesses a globalminimum u1ε and a mountain pass critical point u2

ε satisfying

−∞ < −b2 ≤ c1ε := Iε(u1ε) ≤ −b1 < 0 (14)

and

0 < a2 ≤ c2ε := Iε(u2ε) ≤ a1 <∞; (15)

where a1, b1 and a2, b2 are given by Lemmas 3.1 and 3.2, respectively, and do notdepend on 0 < ε < 1.

Proof. First we claim that the functional Iε satisfies the (PS) condition. Indeed,given a sequence un in H1

0 (Ω) satisfying Iε(un) → c and ‖I ′ε(un)‖ → 0, as n →∞, by Lemma 3.2-(11), we assert that un is a bounded sequence. Observing thatthe nonlinear term f(t) = λ(t+)p − gε(t) is continuous and has subcritical growthat infinity, we use the Sobolev Imbedding Theorem to derive that un possesses aconvergent subsequence. The claim is proved.

By the above claim, Lemma 3.1-(8) and Lemma 3.2-(12), we conclude that thefunctional Iε has a global minimum u1ε satisfying (14). We define

c2ε := infγ∈Γ

max0≤t≤1

Iε(γ(t)), (16)

whereΓ := {γ ∈ C([0, 1], H1

0 (Ω)); γ(0) = 0, γ(1) = ϕ1}, (17)

By Lemma 3.1, Lemma 3.2-(10) and the fact that Iε satisfies the (PS) condition, wemay invoke the Mountain Pass Theorem to conclude that c2ε is a critical level of thefunctional Iε and that the associated critical point u2

ε satisfies (15). �

Let u be a critical point of the functional Iε. Then, setting u− := u+ − u, wehave

0 = I ′ε(u)u− = −∫

Ω|∇u−|2 +

∫Ωgε(u)u− − λ

∫Ω(u+)pu− = −‖u−‖2H1

0 (Ω).

Thus u− ≡ 0, and one concludes that u ≥ 0. Consequently, u is a nonnegative weaksolution of the perturbed problem (6). We also note that by standard regularityargument, the weak solutions of (6) are classical solutions. Next lemma provides ana priori bound in H1

0 (Ω) and in L∞(Ω) for the solutions u of (6)

Lemma 3.4. There exists S > 0, independent of 0 < ε < 1, such that, for everysolution u of (6),

‖u‖H10 (Ω) ≤ S (18)


and

‖u‖L∞(Ω) ≤ S. (19)

Given a sequence εn in the interval (0, 1), we denote by u1n and u2n, respectively,

the two solutions u1εn and u2εn of (6) provided by Proposition 3.3.

Proposition 3.5. Suppose εn ⊂ (0, 1) is a sequence such that εn → 0 as n → ∞.Then u1n and u2

n have subsequences which converge weakly in H10 (Ω) to u1 and u2,

respectively. Moreover, u1 and u2 are nontrivial and distinct.

Our aim now is to get estimates for solutions of (6) and prove that in the limitas ε → 0 the functions u1 and u2, given by Proposition 3.5, are indeed solutions of(1).

4. Gradient estimates

In this section we shall obtain a local gradient estimate for solutions uε to theperturbed equation (6), see [6].

Let ψ be such that

ψ ∈ C2(Ω), ψ > 0 in Ω, ψ = 0 on ∂Ω and|∇ψ|2ψ

is bounded in Ω. (20)

Observe that an example is ψ = ϕ21.

Lemma 4.1. If uε is a solution of (6), then there is a constant M > 0 independentof ε ∈ (0, 1) such that

ψ(x)|∇uε(x)|2 ≤M(uε(x)1−β + uε(x)) for every x ∈ Ω, (21)

where M depends only on Ω, N , β, ψ and S. Notice that from (19) we have‖uε‖L∞(Ω) ≤ S.

5. Taking the limit

In this section we prove Theorem 2.1 by letting ε → 0. We use the above estimate(21) to prove that an arbitrary solution uε of (6) converges to a solution of (1). Withthis, we obtain that u1 and u2 are distinct solutions of (1).We proof now Theorem 2.1.

Proof. Let uε be a solution of problem (6) and ϕ ∈ C1c (Ω), hence∫

Ω∇uε∇ϕ =

∫Ω(−gε(uε) + λupε)ϕ.

Let η ∈ C∞(IR), 0 ≤ η ≤ 1, η(s) = 0 for s ≤ 1/2, η(s) = 1 for s ≥ 1. For m > 0 thefunction �(.) := ϕ(.)η(uε(.)/m) belongs to C1

c (Ω).Since uε is a critical point of Iε, we obtain∫

Ω|∇uε|2 +

∫Ωgε(uε)uε =

∫Ωλup+1

ε .


Thus ‖uε‖H10

is bounded independently of ε, see the proof of Lemma 3.4. For asequence εn → 0 which for the sake of notation we will continue to denote by ε, wehave ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

uε ⇀ u weakly in H10 (Ω);

uε → u strongly in Lσ(Ω);

uε → u a.e in Ω;

|uε| ≤ h a.e in Ω for someh ∈ Lσ(Ω),

(22)

where 1 ≤ σ < 2N/(N − 2) if N ≥ 3 (1 ≤ σ <∞ if N = 2).By Lemma 4.1, |∇uε| is locally bounded independently of ε. Thus for a sequence

εn which we keep denoting by ε, we have uε → u in C0loc(Ω), and the set Ω+ = {x ∈

Ω : u(x) > 0} is open. Let Ω be an open set such that support(ϕ) ⊂ Ω and Ω ⊂ Ω.Let Ω0 = Ω+ ∩ Ω. For every m > 0 there is an ε0 > 0 such that

uε(x) ≤ m/2 for every x ∈ Ω \ Ω0 and 0 < ε ≤ ε0. (23)

Replacing ϕ by � we obtain∫Ω∇uε∇(ϕη(uε/m)) =

∫˜Ω(−gε(uε) + λupε)ϕη(uε/m). (24)

We split the previous integral as

Pε :=∫

Ω0

(−gε(uε) + λupε)ϕη(uε/m)

and

Qε :=∫˜Ω\Ω0

(−gε(uε) + λupε)ϕη(uε/m).

Clearly, Qε = 0, whenever 0 < ε ≤ ε0 by (23) and the definition of η. Notice that

Pε →∫

Ω0

(−u−β + λup)ϕη(u/m) as ε→ 0. (25)

Indeed, uε → u uniformly in Ω0. If u ≤ m/4, for ε > 0 sufficiently small, we haveuε ≤ m/2. So the integral Pε restricted to this set is zero. For u > m/4, thenuε ≥ m/8 for ε > 0 small enough. We then apply the Dominated ConvergenceTheorem as ε→ 0 to get (25).

We now take the limit in m to conclude that∫Ω0

(−u−β + λup)ϕη(u/m) →∫

Ω0

(−u−β + λup)ϕ as m→ 0, (26)

since η(u/m) ≤ 1 and −u−β + λup ∈ L1(Ω), according to Lemma 5.1 below.Observing the integral on the left side of (24), we set∫

Ω∇uε∇(ϕη(uε/m)) :=

∫˜Ω(∇uε∇ϕ)η(uε/m) + Jε. (27)

Clearly, ∫˜Ω(∇uε∇ϕ)η(uε/m) →

∫˜Ω(∇u∇ϕ)η(u/m) as ε→ 0,


since uε ⇀ u in H10 (Ω) and uε → u uniformly in Ω. Consequently, by the Dominated

Convergence Theorem,∫˜Ω(∇u∇ϕ)η(u/m) →

∫˜Ω∇u∇ϕ as m→ 0. (28)

We claim that

Jε :=∫˜Ω

|∇uε|2m

η′(uε/m)ϕ→ 0 as ε→ 0 (and then as m→ 0). (29)

By the estimate |∇uε|2 ≤M(u1−βε + uε) in Ω provided by Lemma 4.1, the fact thatη(u/m) ≤ 1, Lemma 3.4 and the Dominated Convergence Theorem, we obtain

lim supε→0

|Jε| ≤M limε→0

∫˜Ω∩{m

2≤uε≤m}

(u1−βε + uε)m

|η′(uε/m)ϕ| =

= M

∫˜Ω∩{m

2≤u≤m}

(u1−β + u)m

|η′(u/m)ϕ|.

Letting m → 0 in the above estimate, we may invoke Lemma 5.1, the fact thatη′(u/m) is uniformly bounded and the Dominated Convergence Theorem to concludethat (29) must hold. The claim is proved.

As a direct consequence of (24), (26),(27), (28) and (29), we have∫Ω∇u∇ϕ =

∫Ω∩{u>0}

(− 1uβ

+ λup)ϕ

for every ϕ ∈ C1c (Ω). This concludes the proof of Theorem 2.1. �

We need the following lemma to justify a calculation in the proof of Theorem2.1.

Lemma 5.1. Let Ω+ = {x ∈ Ω : u(x) > 0}. The function1uβ

χΩ+ belongs to L1loc(Ω).

References

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nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222.

[3] J. Davila and M. Montenegro, Positive versus free boundary solutions to a singular

elliptic equation, J. Anal. Math. 90 (2003), 303–335.

[4] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem,Proc. Amer. Math. Soc. 111 (1991), 721–730.

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[6] M. Montenegro and E. A. B. Silva, Two solutions for a singular elliptic equation by

variational methods, to appear in Annali della Scuola Normale Superiore di Pisa.

Marcelo MontenegroUniversidade Estadual de Campinas, IMECCDepartamento de MatematicaRua Sergio Buarque de Holanda, 651Campinas, SP, CEP 13083-970Brazile-mail: [email protected]

Received: March 31, 2011.

existence of solutions to a singular elliptic equation

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