(1)SINGULAR NONLINEAR ELLIPTIC EQUATIONS IN R C

SINGULAR NONLINEAR ELLIPTIC EQUATIONS IN R

C. O. ALVES, J. V. GONCALVES^∗ AND L. A. MAIA

Abstract. This paper deals with existence, uniqueness and regularity of positive generalized solutions of singular nonlinear equations of the form

−∆u+a(x)u=h(x)u^−γ inR^N where a, hare given, not necessarily con- tinuous functions, andγ is a positive number. We explore both situations wherea, hare radial functions, withabeing eventually identically zero, and cases where no symmetry is required from eithera or h. Schauder’s fixed point theorem, combined with penalty arguments, is exploited.

1. Introduction

This paper addresses existence, uniqueness and regularity questions on gen- eralized solutions of the singular nonlinear elliptic problem

(∗)

−∆u+a(x)u=h(x)u^−γ in R^N u >0 in R^N

where a, h are nonnegative L^∞_loc functions, h ≡ 0, (eventually we consider a≡0),γ >0 andN ≥3. We point out that the search of positive solutions of the Dirichlet problem for the equation

−∆u+a(x)u=h(x)u^−γ in Ω

where Ω is a smooth bounded domain has deserved the attention of many authors. Nowosad [1] studied a related Hammerstein equation, namely

u(x) = ₁

0 K(x, y)(u(x))^−γdy,

1991Mathematics Subject Classification. 35J60.

Key words and phrases. singular nonlinear elliptic equations, Schauder’s fixed point theorem, existence, uniqueness, regularity, positive solutions.

∗ Partially supported by CNPq/Brasil.

Received: April 8, 1998.

c

1996 Mancorp Publishing, Inc.

411

where γ =1, ₀¹K(x, y)dy ≥ δ > 0 and K(x, y) is positive semidefinite.

Nowosad’s work was extended by Karlin and Nirenberg [2] where more gen- eral Hammerstein equations were considered including the caseγ >0 in the equation above. Crandall-Rabinowitz and Tartar [3] studied the Dirichlet problem

Lu=f(x, u) in Ω, u= 0 on ∂Ω

whereL is a linear second order elliptic operator andf : Ω×(0,+∞)→R is singular in the sense that f(x, r) → ∞ as r → 0⁺. Examples such as f(x, r) =r^−γ with γ >1,γ <1 or γ =1 were covered.

There is by now an extensive literature on singular elliptic problems. With respect to the case of bounded domains Ω ⊂ R^N we would like to further mention Gomes [4], Lazer and McKenna [5], Cac and Hernandez [8], Chen [9], Lair and Shaker [10], Shangbin [13] while for the case Ω =R^N we recall Kuzano and and Swanson [11], Lair and Shaker [12,14]. This reference list is far from complete. In the earlier papers concerning Ω = R^N, h(x) is assumed at least continuous and several techniques are developed such as the method of lower and upper solutions. In this paper we assumeh(x) only integrable and use the Schauder fixed point theorem and elliptic estimates.

Singular equations appear in the theory of heat conduction in electrically conducting materials, (Fulks and Maybee [6]), in binary communications by signals (Nowosad [1]) and in the theory of pseudoplastic fluids (Nachman and Callegari [7]).

The following condition on a will be required in the first one of our main results stated below:

(a)_R a(x)≥a₀ for|x| ≥R for somea₀, R >0.

In what follows we take γ, α∈(0,1) and h∈L^θ∩L² where θ≡ _2−(1−γ)² . Theorem 1. Assume (a)R. Then (∗) has a unique solutionu∈ D^1,2∩W_loc^2,p where1< p <∞with a(x)u²<∞. Ifa, hare radial functions the solution is radial, as well, and in fact, u(x)→0 as|x| → ∞. Moreover if a, h∈C_loc^α thenu∈C_loc^2,α.

In our second result we take a ≡ 0 and h radially symmetric that is, we study the problem

(∗)_o

−∆u = h(|x|)u^−γ in R^N u >0 in R^N.

This problem shall be treated by first perturbing the equation by a radially symmetric term, then using the earlier result in the case a, h are radial functions and finally taking limits.

Theorem 2. Let a ≡ 0 and let h be radially symmetric. Then (∗)_o has a unique radially symmetric solution u ∈ D^1,2 ∩W_loc^2,p, 1 < p < ∞ and u(x)→0 as |x| → ∞. Moreover, ifh∈C_loc^α then u∈C_loc^2,α.

2. Preliminaries

The main goal in this section is to prove theorem 1. For that purpose let >0 and consider the problem

(2.1)

−∆u+a(x)u= _(u+)^h(x)γ in R^N u >0 in R^N.

We are going to show by applying the Schauder fixed point theorem that (2.1) has a solutionu∈W_loc^2,p, 1< p <∞, and then by passing to the limit as →0 we arrive at a solution of (∗).

In order to deal with a first step namely, existence of a solution of (2.1), let f ∈L² and consider the linear equation

(2.2) −∆u+a(x)u=f(x) in R^N.

Recalling that the Hilbert space D^1,2 is defined as the closure of C₀^∞ with respect to the gradient norm ϕ²₁=|∇ϕ|² we introduce the space

E≡

u∈ D^1,2 | au² <∞

which endowed with the inner product and norm given respectively by u, v= (∇u.∇v+auv) andu²=u, u

is itself a Hilbert space. Under condition (a)_R it follows that u ∈ E iff u∈W^1,2(R^N).

Yet iff ∈L²it follows by minimizing overEthe energy functional associated with (2.2),

I(u) = 1

2u²− fu that (2.2) has a weak solution u∈E, that is,

(∇u∇ϕ+auϕ) = f(x)ϕ, ϕ∈E.

The solution u is, in fact, unique. Letting S : L² → E be the solution operator associated to (2.2) that is Sf = u for f ∈ L² it follows that S is linear and moreover

Sf ≤C|f|₂, f ∈L²

for some C > 0. In addition, splitting u into u⁺−u⁻ where u^± are re- spectively the positive and negative parts of u, taking ϕ=−u⁻ above and noticing that u⁻∈E we infer that

Sf ≥0 whenever f ≥0.

Now letu∈L² with u≥0. Since

(2.3) 0≤ h(x)

(u+)^γ ≤ h(x) ^γ

and ^h(x)γ ∈L² the operator

T u≡S

h(x) (u+)^γ

is continuous inL², and as a matter of fact, letting w≡T(0) we have w=S

h(x) ^γ

. Considering

K≡v∈L² |0≤v≤w a.e. in R^N we shall prove that the following result holds true.

Lemma 3. The set K ⊂ L² is closed, convex and bounded and moreover T(K)⊂K and T(K) is a compact subset of L².

Using lemma 3 and the Schauder fixed point theorem there is someu ∈K satisfying

u =S

h(x) (u+)^γ

that is

(∇u∇ϕ+auϕ) = _(u^h(x)ϕ₊₎γ, ϕ∈E u≥0 a.e. inR^N, u ∈E.

Now since by (2.3)

h(x)

(u+)^γ ∈L^∞_loc

it follows by the elliptic regularity theory thatu ∈W_loc^2,p, 1 < p <∞, and further ifB⊂R^N is a ball, then

−∆u+a(x)u= h(x)

(u+)^γ a.e. in B.

In fact, it follows by the maximum principle that u >0 in B and so −∆u+a(x)u = _(u^h(x)₊₎γ a.e. in R^N

u >0 in R^N.

On the other hand, if f ∈L²_rad we get by minimizing the functionalI above over the space

E_rad ≡

u∈W_rad^1,2 |

a(r)u²<∞

which endowed with the inner product and norm given above is also a Hilbert space, a weak solutionu∈E_rad of (2.2) that is

(∇u∇ϕ+auϕ) =

f(x)ϕ, ϕ∈E_rad.

The solution is also unique and as before the solution operator associated to (2.2), namelyS:L²_rad→E_rad satisfies

Sf ≤C|f|₂

forf ∈L²_rad and further

Sf ≥0 wheneverf ≥0.

Letting

K ≡v∈L²_rad|0≤v ≤w a.e. in R^N

we have a corresponding symmetric variant of lemma 3 and so there is some u∈E_rad with

(∇u∇ϕ+a(r)uϕ) =

h(r)

(u+)^γϕ, ϕ∈E_rad. Proof of Lemma 3.

It is easy to show that K is convex, closed and bounded. So we will only show that T(K)⊂K and T(K) is compact inL². Ifv∈K then

T(0)−T(v) =S

h 1

^γ − 1 (v+)^γ

≥0 that is T(v)≤w and henceT(K)⊂K.

In order to show thatT(K)⊂L² is compact let vn be a sequence in T(K) sayv_n=T(u_n) for some u_n∈K. By (2.3)

h(x)

(u_n+)^γ is bounded in L² so that

T(u_n) =S

h(x) (u_n+)^γ

is bounded in E.

Thus, passing to subsequences,

T(un) $ v for some v∈E and

T(un)→ v a.e. in R^N.

On the other hand, since 0 ≤T(u_n) ≤ w it follows by Lebesgue’s theorem that

T(u_n)→v in L².

showing that T(K) is compact in L², ending the proof of lemma 3. The radial case is handled similarly.

The next result states that the familyu increases when decreases.

Lemma 4. If 0< < then u ≤u in R^N. Proof of Lemma 4.

Lettingω≡u −u we get

−∆ω+a(x)ω=h(x)

1

(u+)^γ − 1 (u+)^γ

a.e. in R^N

which gives

|∇ω⁺|²+a(x)ω⁺²= h(x)

1

(u +)^γ − 1 (u+)^γ

ω⁺≤0 showing that ω⁺ =0 and thus u ≤u a.e. in R^N, finishing the proof of lemma 4.

3. Proofs of Main Results Proof of Theorem 1.

Step 1(the non-symmetric case).

Let_n>0 be a decreasing sequence converging to 0 and set u_n =u_n. We claim that

un is bounded.

Indeed,

(3.1) |∇u_n|²+a|u_n|²= h(x)un

(u_n+_n)^γ ≤ h(x)u^1−γ_n ≤C|h|_θu_n^1−γ for some C > 0, showing that un is bounded in E. Hence, passing to subsequences, we have

u_n$ u in E, and u_n→u a.e. in R^N.

Moreover since by lemma 4 0< u₁ ≤u_n in R^N we infer that ifϕ∈E has compact support thensupp(ϕ)⊂B for some ball B ⊂R^N and

|h(x)ϕ|

(u_n+_n)^γ ≤H(x) for some H ∈L¹ which gives, by applying Lebesgue’s theorem to

(∇u_n∇ϕ+au_nϕ) = h(x)ϕ (u_n+_n)^γ that

(∇u∇ϕ+auϕ) = ^h(x)ϕ_uγ

u≥u₁ >0 in R^N. Using the regularity theory again we arrive at







−∆u+a(x)u=h(x)u^−γ a.e. in R^N u∈W_loc^2,p, 1< p <∞

u >0 in R^N.

In order to prove uniqueness let M ∈C₀^∞such that

M(x) = 1 if |x| ≤1, M(x) = 0 if |x| ≥2 and 0≤M ≤1.

Given ϕ∈E, an integerj ≥1 and letting ϕ^j(x)≡M(x

j)ϕ(x), x∈R^N

it follows thatϕ^j ∈E and supp(ϕ^j) is compact. Moreover as we will show in the Appendix

(3.2) ϕ^j →ϕ in E.

Now assume u, v are two solutions of (∗) and letwj ≡u^j −v^j. Then u−v, u^j−v^j = (∇(u−v)∇w_j+a(x)(u−v)w_j)

= h(x)_u¹γ −_v¹γ

wj. Assuming, by contradiction, thatu=v and once

u−v, u^j−v^j→ u−v²

we have

h(x) 1

u^γ − 1 v^γ

w_j >0 for large values of j. On the other hand,

h(x)

1 u^γ − 1

v^γ

w_j ≤

Ωj

h(x)u^1−γ+

Ωj

h(x)v^1−γ

where Ωj ≡B2j\Bj. Therefore, passing to the limit asj → ∞and noticing that the two integrals in the right hand side tend to zero we get a contra- diction, that isu=v.

Assume now,h∈C_loc^α . Then by the elliptic regularity theory more precisely, interior elliptic estimates, we get u ∈ C_loc^2,α. This proves theorem 1 (in the case of Step 1).

Step 2(the symmetric case: a, h are radial).

From section 2 we have found by Schauder’s Theorem some radial function u∈K,u =0 satisfyingu=T u, which means

(3.3) (∇u∇v+a(r)uv) = h(r)

(u+)^γv, v∈E_rad. We will show next that u∈W_loc^2,p(R^N\{0}) for 1< p <∞,and

−∆u+a(r)u = h(r)

(u+)^γ a.e. in R^N\{0}.

Indeed, changing variables we get from (3.3)

S

_∞

0

uv+a(r)uvr^N−1drdS=

S

_∞

0

h(r)

(u+)^γvr^N−1drdS where S⊂R^N is the unit sphere. Making

v≡r^−(N−1)ψ, r >0, ψ ∈C₀^∞(0,∞)

we have _∞

0

r^(N−1)u r^−(N−1)ψ+auψ

dr= _∞

0

h(r)

(u+)^γψ(r)dr, for ψ∈C₀^∞(0,∞),and labelling

h(r)

(u+)^γ −a(r)u≡H(r), r > 0 we get

− 1

r^N−1(r^(N−1)u) =H(r) in (0, ∞)

in the distribution sense. But since a, h, u ∈ L^p_loc(0,∞), 1 < p < ∞ it follows that H ∈ L^p_loc(0,∞) and using the regularity theory we infer that u∈W_loc^2,p(0,∞) and

− 1

r^N−1(r^(N−1)u)=H(r) a.e. in (0, ∞). By the maximum principle,

u>0 in (0,∞). Sinceu∈W_loc^2,p(R^N\{0}) and

−∆u =− 1

r^N−1(r^(N−1)u) we also have

−∆u+a(r)u = h(r)

(u+)^γ a.e. in R^N\{0}.

Now, let n >0 such that n → 0 and label un ≡un. Following the proof of lemma 4 we haveu_n≥u₁ >0.On the other hand we claim that

unis bounded.

Indeed, as in (3.1) we have

|∇un|²+a|un|²≤C |h|_θun^1−γ

so thatu_n is bounded in E_rad. Passing to subsequences we have u_n$ u inE_rad, and u_n→u a.e. in R^N.

On the other hand, if v ∈ Erad has compact support then, as in section 1, applying Lebesgue’s Theorem to

(∇u_n∇v+a(r)u_nv) = h(r) (u_n+_n)^γv,

gives

(∇u∇v+a(r)uv) = h(r) u^γ v.

Now changing variables, making again v ≡ r^−(N−1)ψ where r > 0 and ψ∈C₀^∞(0,∞) and arguing as above we obtainu∈W_loc^2,p(R^N\ {0}) and

− 1

r^N−1(r^(N−1)u)+a(r)u= h(r)

u^γ a.e. in (0,∞) and in addition,

−∆u+a(r)u= h(r)

u^γ a.e. in R^N\{0}.

So, if ϕ∈C₀^∞(R^N\ {0}) then

(∇u∇ϕ+a(r)uϕ) =

h(r) u^γ ϕ that is

−∆u+a(r)u= h(r)

u^γ in R^N\{0}

in the distribution sense. Next we show thatu∈W_loc^2,p(R^N) and

(∇u∇ϕ+a(r)uϕ) = h(r)

u^γ ϕ, ϕ∈C₀^∞(R^N).

Indeed, letη∈C^∞(R^N) such that

η(x) =0 for |x| ≤1, and η(x) =1 for |x| ≥2 and let

ψ(x)≡η(x

), >0.

If ϕ∈C₀^∞(R^N) then ψϕ∈C₀^∞(R^N\ {0}) and from above

(∇u∇(ψϕ) +a(r)u(ψϕ)) =

h(r) u^γ (ψϕ) so that

(ψ∇u∇ϕ+ϕ∇u∇ψ+a(r)uψϕ) = h(r) u^γ ψϕ.

Making→0 and using Lebesgues’s Theorem we infer that

ψ∇u∇ϕ→ ∇u∇ϕ,

a(r)uψϕ→

a(r)uϕ

and

h(r)

u^γ ψϕ→ h(r) u^γ ϕ.

Claim. ϕ∇u∇ψ →0.

Assuming the Claim has been proved we have

(∇u∇ϕ+a(r)uϕ) = h(r) u^γ ϕ

and sincea, h∈L^∞_locwe get by the regularity theory that u∈W_loc^2,p(R^N) for 1< p <∞and

−∆u+a(r)u= h(r)

u^γ a.e. in R^N

and if in addition a, h ∈ C_loc^α then u ∈ C_loc^2,α by the interior Schauder esti- mates.

Verification of the Claim.

Using Schwarz inequality we have

| ϕ∇u∇ψ| ≤ |ϕ|∞

|x|≤2|∇u|²¹² _|x|≤2|∇ψ|²¹²

≤ |ϕ|_∞|∇η|_2|x|≤2|∇u|^212N−22 whereN ≥3. Letting→0 shows the Claim.

As for the uniqueness the argument in the proof of theorem 1 (Step 1) applies ending the proof of theorem 1 (in case of Step 2). The proof of theorem 1 is finished.

Proof of Theorem 2.

In order to solve (∗)0 we consider the family of problems (3.4)

−∆u+ ¹_ku=h(|x|)u^−γ in R^N u >0 in R^N.

where k≥ 1 is an integer. Making a(x) ≡ ¹_k in theorem 1 (radial case), it follows that (3.4) has a solution u_k∈H_rad¹ ∩W_loc^2,p, 1< p <∞ satisfying

|∇u_k|²+1 ku²_k=

h(r)u^1−γ_k .

Using both H¨older’s inequality and the continuous embedding D^1,2 → L^2∗ in the equality above we infer that

(3.5)

|∇uk|² ≤C1 for someC1>0.

By a well known property of radial functionsu∈D^1,2, namely

|u(x)| ≤ C2

|x|^N−22 u_D^1,2, x=0 for some C2 >0 we get

(3.6) 0≤u_k(x)≤ C

|x|^N−22 , x=0 for some C >0.

We shall need the following result which says that the sequence u_k increases with k.

Lemma 5. If k < k then u_k≤u_k, a.e. inR^N.

By the boundedness ofu_kinD^1,2 and lemma 5 there is some radial function u∈D^1,2 such that

u_k$ u inD^1,2, u_k→u a.e. in R^N and

u₁ ≤u₂ ≤, ...,≤u_k ≤, ...,≤u a.e. in R^N. Now if ϕ∈C₀^∞(R^N) then

(3.7) ∇u_k∇ϕ+1

ku_kϕ

=

hu^−γ_k ϕ.

Let Ω⊂R^N be a bounded domain such thatsupp(ϕ)⊂Ω. Then

|hu^−γ_k ϕ| ≤hu^−γ₁ |ϕ| ∈L^p(Ω), 1≤p <∞

and

hu^−γ_k ϕ→ hu^−γϕ.

On the other hand, using (3.6) we get 1 k

u_kϕ→0.

Passing to the limit in (3.7) gives

∇u∇ϕ= hu^−γϕ.

Since 0 < u1 ≤u and u1 ∈W_loc^2,p(R^N) it follows thathu^−γ ∈L^p_loc(R^N) and by the regularity theoryu∈W_loc^2,p(R^N). In additionu∈C_loc^2,αwhenh∈C_loc^α . This proves Theorem 2.

Proof of Lemma 5.

Lettingω=uk−u_k we have

|∇ω⁺|²+ _k¹(ω⁺)² ≤ ∇ω∇ω⁺+_k¹ωω⁺

≤ h 1

u^γ_k −_u¹γ k

ω⁺ showing thatω⁺=0 and so ω≤0, ending the proof of lemma 5.

4. Appendix Verification of (3.2).

Indeed,

a|ϕ^j −ϕ|² ≤4aϕ²∈L¹ anda|ϕ^j−ϕ|² →0 a.e.in R^N so that by Lebesgue’s theorem

a|ϕ^j−ϕ|²→0.

Now ∂ϕ^j

∂xi = 1 j

∂

∂xiM x

j

ϕ+M x

j ∂ϕ

∂xi. Hence

|^∂ϕ_∂x^j_i −_∂x^∂ϕ_i|² = |¹_j∂x^∂_iM^x_jϕ+M^x_j∂x^∂ϕ_i −_∂x^∂ϕ_i|²

≤ C|_j¹2 ∂

∂xiM^x_jϕ|²+|M^x_j∂x^∂ϕ_i −_∂x^∂ϕ_i|². Arguing as above we infer that

M x

j ∂ϕ

∂x_i → ∂ϕ

∂x_i in L². It remains to show that

|1 j²

∂

∂x_iM x

j

ϕ|²→0.

At first we remark that |_j¹2 ∂

∂xiM^x_jϕ|² = _B2j\Bj|_j¹2 ∂

∂xiM^x_jϕ|²

≤ _j^C2

B2j\Bjϕ².

Now using H¨older inequality with exponents _N^N₋₂ and ^N₂ in the last integral we obtain

|_j¹2 ∂

∂xiM^x_jϕ|² ≤ _j^C2

B2j\Bj1dx^N2 _B2j\Bj|ϕ|^2∗21∗

≤ _j^C2

B2j1dx^N2 _Bc 2j|ϕ|^2∗

¹

2∗

≤ ^CωN_j^N22^(2j)2

B^c_2j|ϕ|^{2∗ 1}

2∗

whereω_N denotes the volume of the unit sphere ofR^N. Next passing to the limit we get

|1 j²

∂

∂xiM x

j

ϕ|²→0.

This shows that ϕ^j →ϕinE proving (3.2).

References

[1] P. Nowosad,On the integral equationKf =_f¹ arising in a problem in communications, J. Math Anal Appl.14(1966), 484–492.

[2] S. Karlin and L. Nirenberg, On a theorem of P. Nowosad, J. Math. Anal. App. 17 (1967), 61–67.

[3] M. Crandall, P. Rabinowitz and L. Tartar, On a Dirichlet problem with singular nonlinearity, Comm. Partial Differential Equations,2(1977), 193–222.

[4] S. M. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal. 17 (1986), 1359–1369.

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

[6] W. Fulks and J. S. Maybee,A singular nonlinear equation, Osaka Math J.12(1960), 1–19.

[7] A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math.28(1986), 271–281.

[8] N. P. Cac and G. Hernandez,On a singular elliptic boundary value problem,preprint.

[9] H. Chen,On a singular nonlinear elliptic equation, Nonlinear Anal.29(1997), 337–

345.

[10] A. Lair and A. Shaker,Uniqueness of solution to a singular quasilinear elliptic prob- lem, Nonlinear Anal.28(1997), 489–493.

[11] T. Kusano and C. Swanson, Entire positive solutions of singular semilinear elliptic equations, Japan J. Math.11(1985), 145–156.

[12] A. Lair and A. Shaker,Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. App.211(1997), 371–385.

[13] C. Shangbin,Positive solutions for Dirichlet problems associated to semilinear elliptic equations with singular nonlinearity, Nonlinear Anal.21(1993), 181–190.

[14] A. Shaker,On singular semilinear elliptic equations, J. Math. Anal. App.173(1993), 222–228.

C. O. Alves

Departamento de Matem´atica e Estat´ıstica Universidade Federal da Paraiba

58109-970 Campina Grande, PB BRAZIL E-mail address: [email protected]

J. V. Goncalves and L. A. Maia Departamento de Matem´atica Universidade de Bras´ılia 70910-900 Bras´ılia DF, BRAZIL

E-mail addresses: [email protected], [email protected]