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BY 4.0 license Open Access Published by De Gruyter March 18, 2021

Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

  • Marie-Françoise Bidaut-Véron , Marta Garcia-Huidobro and Laurent Véron EMAIL logo

Abstract

In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem p,qMu:=-Δu+up-M|u|q=μ in a domain ΩN where μ is a nonnegative Radon measure, when p>1, q>1 and M0. We also give conditions under which nonnegative solutions of p,qMu=0 in ΩK, where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

1 Introduction

Let Ω be a bounded domain of N, N2, and let p,qM be the operator

(1.1) u p , q M u := - Δ u + | u | p - 1 u - M | u | q for all  u C 2 ( Ω ) ,

where M0 and p,q>1. We first provide an a priori estimate for a positive solution of (1.1) and its gradient in the range 1<q<p. Then we study under what conditions on the parameters any solution of

p , q M u = 0 in  Ω K ,

where K is a compact subset of Ω, can be extended as a solution of the same equation in whole Ω, and if it is the case, whether the solution is bounded or not in Ω. We also consider the Dirichlet problem with measure data

(1.2)

p , q M u = μ in  Ω ,
u = 0 in  Ω ,

where μ is a nonnegative bounded Radon measure in Ω and exhibit conditions which guarantee the existence of nonnegative solutions to this problem.

If M=0, then p,qM reduces to the Emden–Fowler operator

u p u := - Δ u + | u | p - 1 u .

Singularity problems for solutions of pu=0 have been investigated since forty years, starting with the work of Brezis and Véron [13] who gave conditions for the removability of an isolated singularity. Later on Baras and Pierre [4] extended the result in [13] to more general removable sets, introducing the good framework. They obtained a necessary and sufficient condition expressed in terms of the Bessel capacities cap2,pN (p=pp-1) both for the removability of compact subsets of Ω and the solvability of the associated Dirichlet problem with measure data. Another class of operator strongly related to p,qM is the Riccati operator

u q M u := - Δ u - M | u | q .

The Dirichlet problem with measure data

- Δ u - M | u | q = μ in  Ω ,
u = 0 on  Ω

has been studied by Maz’ya and Verbitsky [19] and Hansson, Maz’ya and Verbitsky [16] when q>2 (and also in N when q>1) and Phuc [23]. Their results necessitate an extensive use of Riesz potentials.

When M>0, there is a balance between the absorption term |u|p-1u and the source term M|u|q, and this interaction is the origin of many unexpected new effects. In the study of singularity problems the effect of the diffusion can be neglectable compared to the two nonlinear terms. The scale of the two opposed reaction terms depends upon the position of q with respect to 2pp+1. This is due to the fact that if q=2pp+1, then (1.1) is equivariant with respect to the scaling transformation T defined for >0 by

T [ u ] ( x ) = 2 p - 1 u ( x ) = 2 - q q - 1 u ( x ) .

If q<2pp+1, the absorption term is dominant and the behavior of the singular solutions is modelled by the equation pu=0 studied in [27]. If q>2pp+1, the diffusion is negligible and the behavior of the singular solutions is modelled by positive separable solutions of p,qMu=0, where p,qM is an eikonal-type operator defined by

(1.3) p , q M u = u p - M | u | q .

This problem is studied in the forthcoming article [8]. If q=2pp+1, the coefficient M>0 plays a fundamental role in the properties of the set of solutions, in particular for the existence of singular solutions and removable singularities; this is not the case when q2pp+1 since by an homothety M can be assumed to be equal to 1.

Brezis and Véron proved in [13] that isolated singularities of solutions of pu=0 are removable when pNN-2. The removability property has been extended to more general sets using a capacity framework in [4]. Using a change of variable inspired by [7], where boundary singularities of solutions of (1.3) are considered we prove a series of removability results for solutions of

(1.4) p , q M u = 0 .

Theorem 1.1.

Assume that 0ΩRN, N3, M>0, pNN-2, 1<q2pp+1 and (p,q)(NN-2,NN-1). Then any nonnegative solution uC2(Ω{0}) of (1.4) in Ω{0} belongs to Wloc1,q(Ω)Llocp(Ω), and it can be extended as a weak solution of (1.4) in Ω. Furthermore, if we assume

  1. either p N N - 2 and 1 < q < 2 p p + 1 ,

  2. or p > N N - 2 , q=2pp+1 and

    (1.5) M < m * := ( p + 1 ) ( ( N - 2 ) p - N 2 p )

then uC2(Ω).

The existence of radial singular solutions when (p,q)=(NN-2,NN-1) and M>0, or when p>NN-2, q=2pp+1 and Mm* shows the optimality of the statements (see [8]). A series of pointwise a priori estimates concerning u and u are presented in the first section. They are obtained by a combination of Keller–Osserman-type estimates, rescaling techniques and Bernstein method. They play a key role for analyzing the case p=NN-2 in the previous theorem, and will be of fundamental importance in the forthcoming paper [8].

The method introduced in the proof of Theorem 1.1 combined with the result of [4] yields a more general removability result. For such a task we denote by capk,bN the Bessel capacity relative to N with order k>0 and power b(1,). If k* it coincides with the Sobolev capacity associated to the space Wk,b(N) by Calderon’s theorem (see, e.g., [1] for a detailed presentation).

Theorem 1.2.

Let p>NN-2 and NN-2<r<p. Suppose that one of the following conditions is verified:

  1. either q = 2 p p + 1 and

    0 < M < m * ( r ) := ( p + 1 ) ( p - r p ( r - 1 ) ) p p + 1 ,

  2. or 1 < q < 2 p p + 1 and M > 0 .

Then, if K is a compact subset of Ω such that cap2,rRN(K)=0, any nonnegative solution uC2(ΩK) of (1.4) in ΩK can be extended to Ω as a solution still denoted by u in the sense of distributions in Ω. Furthermore, if r2NN-2, then uC2(Ω).

Next we obtain sufficient conditions on a positive measure in Ω in order (1.2) be solvable. In the sequel we assume that ΩN, N2, is a bounded smooth domain. We denote by 𝔐(Ω) (respectively 𝔐b(Ω)) the set of Radon measures (respectively bounded Radon measures) in Ω and by 𝔐+(Ω) (respectively 𝔐+b(Ω)) its positive cone. The total variation norm of a bounded measure μ is μ𝔐.

Since for any μ𝔐+b(Ω) the nonnegative solutions of pv=μ and qMw=μ are respectively a subsolution and a supersolution of equation (1.2) and they satisfy 0vw, the construction of v and w is the key-stone for solving (1.2). It is known that these two problems can be solved when the measure μ satisfies some continuity properties with respect to some specific capacities.

Theorem 1.3.

Assume that p>1, 1<q<2. Let μM+b(Ω). If μ satisfies

(1.6) μ ( E ) C min { cap 2 , p N ( E ) , cap 1 , q N ( E ) } for all Borel sets  E Ω ,

there is a constant c0>0 such that for any 0cc0 there exists a function uW01,q(Ω)Lp(Ω), u0, satisfying

(1.7) - Ω u Δ ζ 𝑑 x + Ω ( u p - M | u | q ) ζ 𝑑 x = c Ω ζ 𝑑 μ for all  ζ C c 2 ( Ω ¯ ) .

The condition on the measure is satisfied if W-1,q(Ω)W-2,p(Ω), and we prove the following:

Corollary 1.4.

Let NpN+pq<2 and μM+b(Ω) be such that

(1.8) μ ( E ) C cap 1 , q N ( E ) for all Borel sets  E Ω ,

for some C>0. Then there exists a constant c1>0 such that for any 0cc1 there exists a nonnegative function uW01,q(Ω)Lp(Ω) satisfying (1.7).

By comparison results between capacities we have another type of result:

Corollary 1.5.

Let NN-1q2pp+1. If μM+b(Ω) satisfies

(1.9) μ ( E ) C cap 2 , p N ( E ) for all Borel sets  E Ω ,

for some C>0, then there exists a constant c2>0 such that for any 0cc2 there exists a nonnegative function uW01,q(Ω)Lp(Ω) satisfying (1.7).

As an application of the previous results, we prove the following:

Corollary 1.6.

Let p>1, 1<q<2 and μM+b(Ω). There exists a function uW01,q(Ω)Lp(Ω) solution of (1.7) if one of the following conditions is satisfied:

  1. When p < N N - 2 and q < N N - 1 , if μ 𝔐 c 3 for some c 3 > 0 .

  2. When p < N N - 2 and N N - 1 q < 2 , if μ satisfies (1.8) for some C>0. In that case there exists c4>0 such that there must hold 0<c<c4 in problem (1.6).

  3. When p N N - 2 and q < N N - 1 , if μ 𝔐 c 4 * M - 1 q - 1 for some c 4 * = c 4 * ( N , q , Ω ) > 0 which can be estimated, and if

    μ ( E ) = 0 for all Borel sets  E Ω such that  cap 2 , p N ( E ) = 0 .

In case (i) we show in a forthcoming article [8] and by a completely different method that there is no restriction on c if μ=cδa for some aΩ. In the above mentioned article we construct many types of local or global singular solutions using methods inherited from dynamical systems.

2 Removable Singularities

Throughout this article we denote by c and C generic constants the value of which may vary from one occurrence to another even within a single string of estimates, and by cj (j=1,2,) some constants which have a more important significance and a more precise dependence with respect to the parameters.

2.1 A Priori Estimates

We give two estimates for positive solutions of (1.1) which differ according to the sign of M. If G is an open subset of N, we set dG(x)=dist(x,G)

Proposition 2.1.

Let GRN be an open subset, M>0 and 1<q<p. If uC1(G) is a nonnegative solution of (1.1), there holds

(2.1) u ( x ) c 5 max { M 1 p - q ( d G ( x ) ) - q p - q , ( d G ( x ) ) - 2 p - 1 } for all  x G ,

for some c5=c5(N,p,q)>0.

Proof.

Following the method of Keller [17] and Osserman [22], we fix xG and 0<a<dG(x), and introduce U(z)=λ(a2-|z-x|2)-b for some b>0. Then putting r=|x-z| and U~(r)=U(z), we have in Ba(x)

L U ~ = - U ~ ′′ - N - 1 r U ~ - M | U ~ | q + U ~ p
= λ ( a 2 - r 2 ) - 2 - b [ λ p - 1 ( a 2 - r 2 ) 2 - b ( p - 1 ) + 2 b ( N - 2 ( b + 1 ) ) r 2
- 2 N b a 2 - M 2 q b q λ q - 1 r q ( a 2 - r 2 ) 2 + b - q ( b + 1 ) ] .

If M>0, the two necessary conditions on b>0 to be fulfilled is order that U~ is a supersolution in B|a|(a) are

  1. 2 - b ( p - 1 ) 0 if and only if b(p-1)2,

  2. 2 + b - q ( b + 1 ) 2 - b ( p - 1 ) if and only if b(p-q)q.

The above inequalities are satisfied if

b = max { 2 p - 1 , q p - q } .

If q>2pp+1, then b=qp-q and

L U ~ λ ( a 2 - r 2 ) - 2 p - q p - q [ λ q - 1 ( λ p - q - M 2 q b q ρ q ) ( a 2 - r 2 ) 2 p - q ( p + 1 ) p - q - ( 3 b + 1 ) N a 2 ] .

There exists c51>0 depending on N, p and q such that if we choose

λ = c 5 1 max { M 1 p - q a q p - q , a 2 p ( q - 1 ) ( p - 1 ) ( p - q ) } ,

there holds

(2.2) L U ~ 0 in  B a ( x ) .

Since U~(z) when ra, we derive by the maximum principle that uU~ in Ba(x). In particular,

u ( x ) U ~ ( x ) = λ a - 2 q p - q = c 5 1 max { M 1 p - q a - q p - q , a - 2 p - 1 } .

If q2pp+1, then b=2p-1 and

L U ~ λ ( | a | 2 - ρ 2 ) - 2 p p - 1 [ λ p - 1 + 2 p - 1 ( N - 2 ( p + 1 ) p - 1 ) ρ 2 - 2 N p - 1 | a | 2
- M 2 q ( 2 p - 1 ) q λ q - 1 ρ q ( | a | 2 - ρ 2 ) 2 p - q ( p + 1 ) p - 1 ]
λ ( | a | 2 - ρ 2 ) - 2 p p - 1 [ λ p - 1 - c 2 | a | 2 - c 3 λ q - 1 M | a | 4 p - q ( p + 3 ) p - 1 ] .

Hence, if q=2pp+1, then (2.2) holds if for some c52>0 depending on N,p,q,

λ = c 5 2 max { M p + 1 p ( p - 1 ) , 1 } | a | 2 p - 1 ,

which yields

u ( x ) U ~ ( x ) = λ a - 4 p - 1 = c 5 2 max { M p + 1 p ( p - 1 ) , 1 } a - 2 p - 1 .

While if q<2pp+1, we choose

λ = c 5 3 max { M 1 p - q a 4 p - q ( p + 3 ) ( p - 1 ) ( p - q ) , a 2 p - 1 } ,

where c53>0=c53(N,p,q), which implies

u ( x ) U ~ ( x ) = λ a - 4 p - 1 = c 5 3 max { M 1 p - q a - q p - q , a - 2 p - 1 } .

By letting adG(x), we derive (2.1) with a constant c5=c53, depending on N,p,q. ∎

Corollary 2.2.

Under the assumptions of Proposition 2.1 with G=B2R{0}, there holds for xBR{0},

u ( x ) c 5 max { M 1 p - q | x | - q p - q , | x | - 2 p - 1 } .

We infer from Proposition 2.1 an estimate of the gradient of a positive solution when M>0. If we set σ=2p-q(p+1), then σ>0 (respectively σ<0) according to 2p>q(p+1) (respectively 2p<q(p+1)).

Proposition 2.3.

Let p>q>1. For any M0>0 and R>0 there exists a constant c8=c8(N,p,q,M0Rσp-1) such that for 0<MM0 there holds:

  1. If q 2 p p + 1 (then σ 0 ), any positive solution u of ( 1.1 ) in B 2 R { 0 } satisfies

    (2.3) | u ( x ) | c 8 max { M 1 p - q | x | - p p - q , | x | - p + 1 p - 1 }

    for all x B R { 0 } .

  2. If 2 p p + 1 q 2 (then σ 0 ), any positive solution u of ( 1.1 ) in N B ¯ R 2 satisfies ( 2.1 ) for all x N B R .

Proof.

(i) For 0<r<2R we set

u ( x ) = r - 2 p - 1 u r ( x r b i g g r ) = r - 2 p - 1 u r ( y ) with  y = x r .

If r2<|x|<2r, then 12<|y|<2 and ur>0 satisfies

(2.4) - Δ u r + u r p - M r 2 p - q ( p + 1 ) p - 1 | u r | q = 0 in  B 2 B 1 2 .

Since 0<Mrσp-1M(2R)σp-1M0(2R)σp-1 as σ0, it follows that

(2.5) max { | u r ( z ) | : 2 3 < | z | < 3 2 } c max { | u r ( z ) | : 1 2 < | z | < 2 } ,

where c depends on N,p,q and Rσp-1M0 (see, e.g., [15, Chapter 13]). From Proposition 2.1 there holds

max { | u r ( z ) | : 1 2 < | z | 2 } 2 2 p - 1 c 5 max { M 1 p - q r 2 p - q ( p + 1 ) ( p - 1 ) ( p - q ) , 1 }

by (2.1). Therefore

max { | u ( y ) | : r 2 < | z | < 2 r } 2 2 p - 1 c c 5 r - p + 1 p - 1 max { M 1 p - q r 2 p - q ( p + 1 ) ( p - 1 ) ( p - q ) , 1 } c 8 max { M 1 p - q | x | - p p - q , | x | - p + 1 p - 1 } ,

which implies (2.3).

(ii) For r>R we define ur as in (i). It satisfies (2.4) and since σ0, we have again

0 < M r σ p - 1 M R σ p - 1 M 0 R σ p - 1

if rR. Since 1<q<2, it follows that (2.5) holds and we derive (2.3).∎

Remark.

If q=2pp+1, the constant c8 depends only on N and p.

The previous estimate necessitates 1<q2. This limitation can be bypassed in some cases using the Bernstein approach.

Lemma 2.4.

Assume that p,q>1 and M>0. If uC2(B¯2R) is a nonnegative solution of (1.1) in B2R, there holds

(2.6) | u ( x ) | c 9 ( | x | - 1 q - 1 + max | z - x ] | x | 2 u p q ( z ) ) for all  x B R 2 ,

where c9>0 depends on N, p, q and M.

Proof.

Set z=|u|2. Then by a classical computation and the use of Schwarz inequality,

- Δ | u | 2 + 1 N ( Δ u ) 2 + Δ u , u 0 .

Replacing Δu by its expression from p,qMu=0, we obtain

- Δ z + 2 N ( u 2 p + M 2 z q - 2 M u p z q 2 ) + 2 p u p - 1 z q M z q 2 - 1 z , u .

We notice that

q M z q 2 - 1 z , u q M z q 2 - 1 | z | z = q M z q 2 | z | z M 2 z q 2 N + 2 N q 2 M 2 | z | 2 z

and

4 M N u p z q 2 M 2 z q 2 N + 8 u 2 p N M 2 ,

thus

- Δ z + M 2 z q N 2 N q 2 M 2 | z | 2 z + 2 N ( 4 M 2 - 1 ) u 2 p .

For simplicity we set

A = M 2 N ,
B = 2 N q 2 M 2 ,
C = 2 N ( 4 M 2 - 1 ) + max | z - x | | x | 2 u 2 p ( z ) .

Then z satisfies

- Δ z + A z q B | z | 2 z + C in  B R 2 ( x )

and we obtain by [5, Lemma 3.1] (see also a simpler approach in [6, Lemma 2.2])

z ( x ) c 10 ( | x | - 2 q - 1 + C 1 q ) ,

where c10>0 depends on N, p, q and M. This yields (2.6). ∎

Remark.

The constants c9 and c10 can be expressed in terms of M, but their stability when M0 is not clear since in the limit case of the equation pu=0 the estimate of the gradient obtained by a very different and much simpler method combining the Keller–Osserman estimate and scaling methods.

Using Corollary 2.2, we obtain the new estimate:

Corollary 2.5.

Assume that 1<q<p and M>0. Then any nonnegative solution uC2(B2R) of (1.1) satisfies

(2.7) | u ( x ) | c 11 ( | x | - 1 q - 1 + max { M p q ( p - q | x | - p p - q , | x | - 2 p q ( p - 1 ) } ) for all  x B R 2 ,

where c11>0 depends on N, p, q and M.

Then we can combine this estimate with Proposition 2.1 to complete the cases not treated in Proposition 2.3 and obtain the following:.

Proposition 2.6.

Let 1<q<p. For any M>0 there exists a constant c12=c12(N,p,q,M)>0 such that:

  1. If 2 p p + 1 q < p , any positive solution u of ( 1.1 ) in B 2 R { 0 } with 0 < R 1 satisfies

    (2.8) | u ( x ) | c 12 max { M p q ( p - q ) | x | - p p - q , | x | - 2 p q ( p - 1 ) } for all  x B R { 0 } .

  2. If 1 < q 2 p p + 1 , any positive solution u of ( 1.1 ) in N B ¯ R 2 with R 1 satisfies

    (2.9) | u ( x ) | c 12 max { M p q ( p - q ) | x | - p p - q , | x | - 1 q - 1 } for all  x N B R .

Proof.

We can compare the different exponents of |x| which appear in the expressions (2.3) and (2.7)

(2.10)

p p - q < p + 1 p - 1 < 2 p q ( p - 1 ) < 1 q - 1 if   1 < q < 2 p p + 1 ,
1 q - 1 < 2 p q ( p - 1 ) < p + 1 p - 1 < p p - q if  2 p p + 1 < q < p ,

with equality if q=2pp+1. If 1<q2pp+1 (respectively 2pp+12), estimate (2.3) is better than (2.7) in BR{0} (respectively NB2R). Then (2.8) and (2.9) follow from (2.7) and (2.10). ∎

In the case M<0 an upper estimate on a solution is obtained by combining a result of Lions and the method of Keller and Osserman.

Proposition 2.7.

Let GRN be an open subset, M0 and p,q>1. If uC1(G) is a nonnegative solution of Lp,qMu=0, there exists c6=c6(N,p)>0, c7=c7(N,q)>0 and δ=δ(G)>0 such that there holds for all xG and 0<δδ(G),

(2.11) u ( x ) min { c 6 ( d G ( x ) ) - 2 p - 1 , c 7 | M | - 1 q - 1 ( d G ( x ) ) - 2 - q q - 1 + max d G ( z ) = δ u ( z ) } .

Proof.

This estimate follows from the fact that the solutions of p,qMu=0 are subsolutions of pu=0 and qMu=0. The estimate

u ( x ) c 6 ( d G ( x ) ) - 2 p - 1

corresponds to the Keller–Osserman estimate for solutions of pu=0. The second estimate corresponds to the fact that if u is a positive solution of qMu=0 in G, there holds (see [18, Theorem IV-1])

| u ( x ) | c 7 | M | - 1 q - 1 ( d G ( x ) ) - 1 q - 1 .

Integrating this inequality yields the second part of the inequality. ∎

Remark.

This estimate can be transformed into the universal estimate

u ( x ) min { c 6 ( d G ( x ) ) - 2 p - 1 , c 7 | M | - 1 q - 1 ( d G ( x ) ) - 2 - q q - 1 + c 6 δ - 2 p - 1 } ,

since maxdG(z)=δu(z)c6δ-2p-1 by (2.11).

The gradient estimates are due to Nguyen [20, Proposition 1.1]. Below we recall his result proved by the Bernstein method in a more general framework but which can also be obtained by scaling techniques in the present case.

Proposition 2.8.

Let p>1 and 1<q<2. For any M<0 and R>0 there exists a constant c12>0, depending on N, p, q, M, such that if u is a positive solution of (1.1) in B2R{0}, there holds

u ( x ) + | x | | u ( x ) | c 12 max { | x | - 2 p - 1 , | x | - 2 - q q - 1 } for all  x B R { 0 } .

Remark.

There are many estimates of positive solutions of (1.1) (or even with up replaced by f(u)) in a domain which tends to infinity on the boundary (large solutions) or of solutions in N (ground states). Many of these estimates are obtained by comparison with one-dimensional problems and they can be found in [2, 3, 14].

2.2 Proof of Theorem 1.1

Without loss of generality we can assume that uC2(Ω¯{0} and B¯2R0Ω with 2R01. If M0, then u is a nonnegative subsolution of -Δu+up=0, hence it is bounded in Ω¯ by [13].

Step 1.

We assume M>0 and we prove first that under condition (i) or (ii), |u|qL1(Ω), uLp(Ω), and then

Ω ( - u Δ ζ + u p ζ - M | u | q ζ ) 𝑑 x = 0 for all  ζ W 2 , ( Ω ) C c 1 ( Ω ¯ ) .

By Proposition 2.3

| u | q c | x | - ( p + 1 ) q p - 1 in  B R 0 ,

since q2pp+1, and where c depends also on M. By (i) or (ii), (p+1)qp-1<N. Hence uLlocq(Ω). For any ϵ>0 small enough denote by ρϵ a nonnegative C-function such that supp(ρϵ)B¯ϵ, 0ρϵ1, |ρϵ|2ϵ-1χB¯ϵ and we set ηϵ=1-ρϵ. Then

- B 2 R 0 u , ρ ϵ 𝑑 x + B 2 R 0 u p η ϵ 𝑑 x + B 2 R 0 u 𝐧 𝑑 S = M B 2 R 0 | u | q η ϵ 𝑑 x .

Next

(2.12) | B 2 R 0 u , ρ ϵ 𝑑 x | 2 c N ϵ N q - 1 ( B ϵ | u | q 𝑑 x ) 1 q 0 as  ϵ 0 ,

since 1<qNN-1. Since |u|qL1(B2R0), we deduce by monotone convergence that upL1(B2R0). Finally, if ζC0(Ω) and ζϵ=ζηϵ, we have

Ω u , ζ ϵ 𝑑 x + Ω u p ζ ϵ 𝑑 x - M Ω | u | q ζ ϵ 𝑑 x = 0 .

Letting ϵ0 and using (2.12), we infer that u satisfies

Ω u , ζ 𝑑 x + Ω u p ζ 𝑑 x - M Ω | u | q ζ 𝑑 x = 0 .

Hence it is a weak solution of (1.4) in Ω.

Step 2.

Let us assume that p>NN-2. If u is nonnegative and not identically zero, it is positive in Ω{0} by the maximum principle. We set u=vb with 0<b1. Then

- Δ v - ( b - 1 ) | v | 2 v + 1 b v 1 + ( p - 1 ) b - M b q - 1 v ( b - 1 ) ( q - 1 ) | v | q = 0 .

For ϵ>0,

v ( b - 1 ) ( q - 1 ) | v | q q ϵ 2 q 2 | v | 2 v + 2 - q 2 ϵ 2 2 - q v 1 + 2 b ( q - 1 ) 2 - q .

Therefore

(2.13) - Δ v + ( 1 - b - M q b q - 1 ϵ 2 q 2 ) | v | 2 v + 1 b v 1 + b ( p - 1 ) - M b q - 1 2 - q 2 ϵ 2 2 - q v 1 + 2 b ( q - 1 ) 2 - q = 0 .

We notice that 1+2b(q-1)2-q=1+b(p-1)-a with a=b2p-(p+1)q2-q0. We fix b as follows:

(2.14) ( p - 1 ) b + 1 = N N - 2 b = 2 ( N - 2 ) ( p - 1 ) ,

hence p>NN-2 if and only if 0<b<1. Next we impose

1 - b - M q b q - 1 ϵ 2 q 2 = 0 ϵ = ( 2 ( 1 - b ) M q b q - 1 ) q 2 = ( 2 ( ( N - 2 ) p - N ) M q b q - 1 ( N - 1 ) ( p - 1 ) ) q 2 .

This transforms (2.13) into

(2.15) - Δ v + ( N - 2 ) ( p - 1 ) 2 v N N - 2 - ( 2 - q ) b q - 1 2 ( q 2 ( 1 - b ) ) q 2 - q M 2 2 - q v N N - 2 - a 0 .

We first assume that 0<q<2pp+1. Then a>0, hence there exists A>0, depending on M, such that

- Δ v + ( N - 2 ) ( p - 1 ) 4 v N N - 2 A .

Set v~=(v-cAN-2N)+NN-2 with c=(4(N-2)(p-1))NN-2 satisfies

- Δ v ~ + ( N - 2 ) ( p - 1 ) 4 v ~ N N - 2 0 .

By [13], v~maxΩv~ which implies vcAN-2N+maxΩv and therefore u(x)B for some B0 in Ω. Furthermore, |u|q-1Lqq-1(Ω) since uLq(Ω), and qq-1>N as we assume q<NN-1. Writing (1.4) under the form

- Δ u + u p - M C ( x ) | u | = 0

with C(x)=|u(x)|q-1, it follows from Serrin’s theorem [25, Theorem 10] that the singularity at 0 is removable and u can be extended as a regular solution of (1.4) in Ω. Hence uC2(Ω). Then we assume that q=2pp+1. By the choice of b in (2.14), inequality (2.13) becomes

(2.16) - Δ v + ( 1 - b - M p b p - 1 p + 1 ϵ p + 1 p p + 1 ) | v | 2 v + ( 1 b - M b p - 1 p + 1 ( p + 1 ) ϵ p + 1 ) v N N - 2 0 .

Notice that

(2.17) 1 b - M b p - 1 p + 1 ( p + 1 ) ϵ p + 1 = 0 ϵ = ( M p + 1 ) 1 p + 1 b 2 p ( p + 1 ) 2 ,

and therefore

(2.18) 1 - b - M p b p - 1 p + 1 ϵ p + 1 p p + 1 = 1 - b - p b ( M p + 1 ) p + 1 p .

This coefficient vanishes if

p ( M p + 1 ) p + 1 p = p ( N - 1 ) - ( N + 1 ) 2 .

Therefore, if M satisfies

(2.19) p ( M p + 1 ) p + 1 p = p ( N - 2 ) - N 2 ,

we can choose ϵ>0 so that the coefficient of v(p-1)b+1 in (2.19) is equal to some τ>0. Therefore v satisfies

- Δ v + τ v N N - 2 0 in  Ω { 0 } .

It follows from [13] that vmaxΩv and the same type of uniform estimate holds for u. This ends the case p>NN-2.

Step 3.

Finally, we assume p=NN-2 and 1<q<2pp+1=NN-1. From (2.6),

M | u ( x ) | q c 9 | x | - q p + 1 p - 1 = c 9 | x | - q ( N - 1 ) := Q ( x )

and QL1(B2R0). Let {σn}C0(N) such that 0σn1

σ n ( x ) = { 1 if  2 n | x | R 0 , 0 if  | x | [ 0 , 1 n ] [ 2 R 0 , ) ,

and

| Δ σ n | 2 N n 2 χ B 2 n B 1 n + ϕ ,

where ϕ is a smooth nonnegative function with support in B2R0BR0. Then

(2.20) - { 1 n | x | 2 n } u Δ σ n 𝑑 x - { R 0 | x | 2 R 0 } u Δ σ n 𝑑 x + 1 n | x | u p σ n 𝑑 x = M 1 n | x | | u | q σ n 𝑑 x .

The right-hand side of (2.20) is bounded since |u|Lq(B2R0), the second term on the left is also uniformly bounded. Using the fact that |x|N-2u(x) is bounded by (2.1), we get

| { 1 n | x | 2 n } u Δ σ n 𝑑 x | C ,

for some C>0 independent of n. Letting n, we infer that uLlocp(Ω). By the maximum principle

u ( x ) u 1 ( x ) = C 𝐆 B 2 R 0 [ Q ] ( x ) + max | z | = 2 R 0 u ( z ) ,

where 𝐆B2R0 denotes the Green kernel in B2R0. Since Q(x)=C|x|-q(N-1), a direct computation shows that u1(x)cNC|x|2-q(N-1)=cNC|x|2-N+ϵ for some ϵ>0. We can write (1.1) under the form

- Δ u + c ( x ) u + d ( x ) | u | = 0 in  Ω { 0 } ,

with c(x)=u2N-2 and d(x)=|u|q-1. Then we have cLN2+ϵ1(B2R0) and dLN+ϵ2(B2R0); with ϵ1,ϵ2>0. It follows from [25, Theorem 10] that 0 is a removable singularity for u in the sense that it can be extended as a C2-solution in Ω. Theorem 1.1 is proved. ∎

When the conditions of the theorem are not fulfilled, there exist singular solutions. However, these singular solutions may exhibit different types of behavior according 1<q<2pp+1, 2pp+1<q<2 and q=2pp+1. In this case there may exist radial separable solutions of (1.4) under the form uX(r)=Xr-2p-1. By setting α=2p-1, it follows that X satisfies

(2.21) Φ p ( X ) := X p - 1 - M α 2 p p + 1 X p - 1 p + 1 + α ( N - 2 - α ) = 0

The following result is easy to prove by a standard analysis of the function Φp.

Proposition 2.9.

Let p>1 and MR.

  1. If M is arbitrary and 1 < p < N N - 2 , or M > 0 and p = N N - 2 , there exists one and only one positive solution X 1 to ( 2.21 ).

  2. If p > N N - 2 and M > m * , there exist two positive solutions X 1 < X 2 to ( 2.21 ).

  3. If p > N N - 2 and M = m * there exists one positive solution X 1 to ( 2.21 ).

  4. If p > N N - 2 and 0 < M < m * , or M 0 and p N N - 2 , there exists no positive solution to ( 2.21 ).

When q2pp+1, then the existence of singular solutions is much involved. It is developed in the subsequent paper [8].

Remark.

It is noticeable that in the case q=2pp+1, p>NN-2 and Mm*, the equation exhibits a phenomenon which is characteristic of Lane–Emden-type equations

- Δ u = u p in  B 1 { 0 } .

If u is nonnegative, then there exists α0 such that

- Δ u = u p + α δ 0 in  𝒟 ( B 1 ) .

If 1<p<NN-2, then α can be positive, but if pNN-2, then α=0. This means that the singularity cannot be seen in the sense of distributions, however there truly exist singular solutions, e.g., if p>NN-2,

u s ( x ) = c N , p | x | - 2 p - 1 .

Here also for q=2pp+1, p>NN-2, the isolated singularities are not seen in the sense of distributions.

2.3 Proof of Theorem 1.2

As in the proof of Theorem 1.1, we distinguish according to whether 1<q<2pp+1 or q=2pp+1 holds. Without loss of generality we can suppose that u>0. We perform the same change of unknown as in the previous theorem by putting u=vb, but now we choose b as follows:

( p - 1 ) b + 1 = r b = r - 1 p - 1 ,

and we first assume that

(2.22) 1 - b - M q b q - 1 ϵ 2 q 2 = 0 ϵ = ( 2 ( 1 - b ) M q b q - 1 ) q 2 = ( 2 ( p - r ) M q ( p - 1 ) b q - 1 ) q 2 .

Hence (2.15) becomes

(2.23) - Δ v + p - 1 r - 1 v r - ( 2 - q ) b q - 1 2 ( q 2 ( 1 - b ) ) q 2 - q M 2 2 - q v ( 2 r - p - 1 ) q + 2 ( p - r ) ( p - 1 ) ( 2 - q ) 0 .

Then

r ( 2 r - p - 1 ) q + 2 ( p - r ) ( p - 1 ) ( 2 - q ) 2 p - q ( p + 1 ) r ( 2 p - q ( p + 1 ) ) ,

since 1<r<p.

Assuming first that q<2pp+1, we obtain from (2.23)

- Δ v + p - 1 2 ( r - 1 ) v r A

for some constant A0. Since cap2,rN(K)=0, the function v is bounded from [4] and vcA1r+maxΩv for some c>0, hence u is also uniformly upper bounded in Ω by some constant a.

Next we have to show that uLq(Ω). Let {ρn} be a sequence of nonnegative C0(Ω)-functions such that 0ρn1, ρn=1 in a small enough neighborhood of K and ρnW2,r0 when n, and set ηn=1-ρn. Since

Ω u Δ ρ n 𝑑 x - Ω u 𝐧 𝑑 S + Ω u p η n 𝑑 x = M Ω | u | q η n 𝑑 x

and

| Ω u Δ ρ n 𝑑 x | c u L ρ n W 2 , r 0 as  n ,

we get

Ω u p 𝑑 x - Ω u 𝐧 𝑑 S = M Ω | u | q 𝑑 x .

Hence uLq(Ω). If ζC0(Ω) and ζn=ζηn, there holds

- Ω η n u Δ ζ 𝑑 x + Ω ζ u Δ ρ n 𝑑 x + Ω u p ζ n 𝑑 x = M Ω | u | q ζ n 𝑑 x .

Since the second term on the left-hand side tends to 0 and ζnζ when n, we obtain that

- Ω u Δ ζ 𝑑 x + Ω u p ζ 𝑑 x = M Ω | u | q ζ 𝑑 x .

Hence u is a solution in the sense of distribution in Ω.

Next we show that uL2(Ω). Multiplying (1.4) by uηn and integrating, we obtain

Ω | u | 2 η n 𝑑 x - Ω u u , ρ n 𝑑 x - Ω u u 𝐧 𝑑 S + Ω u p + 1 η n 𝑑 x = M Ω u | u | q η n 𝑑 x .

As

Ω u u , ρ n 𝑑 x = 1 2 Ω u 2 , ρ n 𝑑 x = - 1 2 Ω u 2 Δ ρ n 𝑑 x

and

| Ω u 2 Δ ρ n 𝑑 x | c u L 2 ρ n W 2 , r = o ( 1 ) as  n ,

we infer that

Ω | u | 2 𝑑 x - Ω u u 𝐧 𝑑 S + Ω u p + 1 𝑑 x = M Ω u | u | q 𝑑 x .

Finally, if ζC0(Ω) and ζn=ζηn, then

Ω η n u , ζ 𝑑 x - Ω ζ u , ρ n 𝑑 x + Ω u p ζ n 𝑑 x = M Ω | u | q ζ n 𝑑 x .

Since r2NN-2, there holds

ρ n W 1 , 2 c ρ n W 2 , r ρ n W 1 , 2 0 as  n .

Using the fact that uL2(Ω) and Hölder’s inequality, we derive

Ω ζ u , ρ n 𝑑 x 0 as  n .

Hence

Ω u , ζ 𝑑 x + Ω u p ζ 𝑑 x = M Ω | u | q ζ 𝑑 x .

This implies that u is a weak solution of (1.4) and it is therefore C2 in Ω.

Next we assume q=2pp+1. We choose b=r-1p-1 and (2.16) becomes

- Δ v + ( 1 - b - M p b p - 1 p + 1 ϵ p + 1 p p + 1 ) | v | 2 v + ( 1 b - M b p - 1 p + 1 ( p + 1 ) ϵ p + 1 ) v r 0 .

If (2.17) holds with this choice of b, (2.18) becomes

1 - b - M p b p - 1 p + 1 ϵ p + 1 p p + 1 = 1 - b - p b ( M p + 1 ) p + 1 p
= 1 p - 1 ( p - r - p ( r - 1 ) ( M p + 1 ) p + 1 p ) .

If M<mr* defined by (1.5), we can choose ϵ such that

1 - b - M p b p - 1 p + 1 ϵ p + 1 p p + 1 = 0

and

1 b - M b p - 1 p + 1 ( p + 1 ) ϵ p + 1 = τ := τ ( ϵ ) > 0 .

Then v satisfies

- Δ v + τ v r 0 in  Ω K .

Since cap2,rN(K)=0, it follows from [4] that vmaxxΩv(x). Hence u is bounded. The different steps of the proof in the first case applies without any modification: first uLq(Ω) and the equation holds in the sense of distributions in Ω, then uL2(Ω) and since r2NN-2 we infer that u is a weak solution and thus a strong one. ∎

3 Measure Data

Let ΩN be a bounded smooth domain with diameter smaller than 2R. Also any Radon measure in Ω is extended by 0 in Ωc with the same notation.

3.1 Proof of Theorem 1.3: The Case 1<q<NN-1

If 1<q<NN-1, then assumption (1.6) with μ0 reduces to

μ ( K ) C cap 2 , p N ( K ) for all compact set  K Ω .

The construction of solutions is based upon the following result due to Boccardo, Murat and Puel [10]. It is concerned with a general quasilinear equation in a domain GN:

𝒬 ( u ) := - Δ u + B ( , u , u ) = 0 in  𝒟 ( G ) ,

where BC(G××N) satisfies

| B ( x , r , ξ ) | Γ ( | r | ) ( 1 + | ξ | 2 ) for all  ( x , r , ξ ) G × × N ,

for some continuous increasing function Γ from + to +.

Theorem 3.1.

Let G be a bounded domain in RN. If there exists a supersolution ϕ and a subsolution ψ of the equation Qv=0 belonging to W1,(G) and such that ψϕ, then for any χW1,(G) satisfying ψχϕ there exists a function uW1,2(G) solution of Qu=0 such that ψuϕ and u-χW01,2(G).

The sub- and super-solutions are linked to the two problems in which p and q are bigger than 1, and μ and ω are Radon measures

(3.1)

- Δ v + | v | p - 1 v = μ in  Ω ,
v = 0 in  Ω

and

(3.2)

- Δ w - M | w | q = ω in  Ω ,
w = 0 in  Ω .

It is proved in [4, Theorem 4.1] that problem (3.1) admits a solution, vLp(Ω), necessarily unique, if and only if μ is absolutely continuous with respect to the Bessel capacity cap2,pN, that is:

For any compact set  E Ω cap 2 , p N ( E ) = 0 | μ | ( E ) = 0 .

Concerning (3.2), from [21, Theorem 1.9] a sufficient condition for solvability is the following estimate:

(3.3) For any compact set  E Ω | ω | ( E ) C cap 1 , q N ( E ) , for some  C > 0 .

When μ is nonnegative and has compact support in Ω, this condition turns out to be necessary. If (3.3) is satisfied, there exists ϵ0>0 such that (3.2) admits a solution with ω replaced by ϵω with 0<ϵϵ0. Furthermore, wLq(Ω) and the following estimates hold [9, Theorem 1.2]:

(3.4) | w ( x ) | c 13 ϵ 𝐈 1 2 R [ | ω | ] ( x ) ,

at least if ω has compact support or is a smooth function, and, with no such conditions on μ,

(3.5) | w ( x ) | c 14 ϵ 𝐆 Ω [ | ω | ] ( x ) ,

with c13,c14 depending on N and q, where 𝐈12R is the truncated Riesz potential in N defined for any measure μ by

𝐈 1 2 R [ μ ] ( x ) = 0 2 R μ ( B ρ ( x ) ) ρ N - 1 d ρ ρ for all  x N ,

and 𝐆Ω the Green potential in Ω. If R=, we denote by 𝐈1:=𝐈1 the classical Riesz potential; if Ω=N, the role of 𝐆Ω is played by the Newtonian potential 𝐈2. We start with the following easy result.

Lemma 3.2.

Let r>1, kN* and μM+(Ω). If μW-k,r(Ω) is nonnegative, then there exists C>0 such that

μ ( E ) C ( cap k , r Ω ( E ) ) 1 r for any compact set  E Ω .

where r=rr-1. Conversely, when k=1,2 and μ satisfies

(3.6) μ ( E ) C cap k , r Ω ( E ) for all compact set  E Ω ,

for some C>0, then:

  1. if k = 2 , then μ W - 2 , r ( Ω ) ,

  2. if k = 1 , then μ W - 1 , r ( Ω ) .

Proof.

Assume first that μ𝔐+(Ω)W-k,r(Ω). Let ζC0(Ω) such that 0ζ1 and ζχK. Then

μ ( E ) Ω ζ 𝑑 μ = μ , ζ ζ W 0 k , r μ W - k , r .

By the definition of capacity,

μ ( K ) μ W - k , r ( cap k , r p Ω ( K ) ) 1 r .

Conversely, if (3.6) holds with k=2, there exists ϵ0>0 such that for every ϵ(0,ϵ0] there exists zLr(Ω) satisfying

- Δ z = z r + ϵ μ in  Ω ,
z = 0 on  Ω

(see [24, Theorem 2.10, Remark 2.11]). Since z𝐆Ω[ϵμ], it follows that 𝐆Ω[μ]Lr(Ω) and therefore we have μW-2,r(Ω). Since 𝐆Ω is an isomorphism from Lr(Ω) into W2,r(Ω)W01,r(Ω), we infer by duality that 𝐆Ω is an isomorphism from W-2,r(Ω) into Lr(Ω). Hence μW-2,r(Ω).

Finally, if (3.6) holds with k=1, then there exists ϵ0>0 such that for every ϵ(0,ϵ0] there exists zW1,r(Ω) satisfying

- Δ z = | z | r + ϵ μ in  Ω ,
z = 0 on  Ω .

Then z satisfies z𝐆Ω[ϵμ]. Since zLr*(Ω) by the Sobolev imbedding theorem, we have 𝐆Ω[μ]Lr*(Ω), which implies the claim. ∎

Proof of Theorem 3.1.

We put μn=μηn, where {ηn}C0(N) is a sequence of mollifiers with the property that supp(ηn)B1n, and we denote by vn the solution of

- Δ v + v p = ϵ μ n χ Ω in  Ω ,
v = 0 on  Ω .

Since μ satisfies (3.6), so does μn with the same constant C. Hence μnW-2,p(Ω) and μnμ in W-2,p(Ω) as n. We also denote by zn a nonnegative solution of

- Δ z = z p + ϵ μ n χ Ω in  Ω ,
z = 0 on  Ω

and by wn a nonnegative solution of

- Δ w = M | w | q + ϵ μ n χ Ω in  Ω ,
w = 0 on  Ω .

Since wn is C2, it is unique by the strong maximum principle. Then there holds by (3.4) and (3.5),

v n ϵ 𝐆 Ω [ μ n ] w n c 14 ϵ 𝐆 Ω [ μ n ] c 14 ϵ 𝐈 2 [ μ n ] Ω ,
| w n | c 13 ϵ 𝐈 1 2 R [ μ n ] .

Since vn and wn are respectively a subsolution and a supersolution of

(3.7)

- Δ u + u p = M | u | q + ϵ μ n in  Ω ,
u = 0 on  Ω ,

it follows by Theorem 3.1 that there exists u=unW01,(Ω) satisfying (3.7) in the sense that for any ζCc2(Ω¯) there holds

(3.8) - Ω u n Δ ζ 𝑑 x + Ω ( u n p - M | u n | q ) ζ 𝑑 x = ϵ Ω ζ 𝑑 μ n .

It is unique by the strong maximum principle and it satisfies

(3.9) v n u n w n c 14 ϵ 𝐈 2 [ μ n ] Ω .

Since 𝐈2[μn]Ω is uniformly bounded in Lp(Ω), the sequence of functions {un} shares this property. If η=𝐆Ω[1], there holds

Ω ( u n + η u n p ) 𝑑 x = M Ω | u n | q η 𝑑 x + ϵ Ω η 𝑑 μ n .

Hence |un| is uniformly bounded in LdΩq(Ω), where dΩ(x)=dist(x,Ω). By (3.4),

(3.10) | u n | c 13 ϵ 𝐈 1 2 R [ | μ n - u n p | ] c 13 ( ϵ 𝐈 1 2 R [ μ n ] + 𝐈 1 2 R [ u n p ] ) c 13 ( ϵ 𝐈 1 2 R [ μ n ] + c 14 p ϵ p 𝐈 1 2 R [ ( 𝐈 2 [ μ n Ω ] ) p ] ) .

Using [16, Lemma 4.2], we have equivalence between

(3.11) 𝐈 1 [ ( 𝐈 2 [ μ n Ω ] ) p ] c 15 𝐈 1 [ μ n Ω ] ,

and

(3.12) 𝐈 2 [ ( 𝐈 2 [ μ n Ω ] ) p ] c 17 𝐈 2 [ μ n Ω ] ,

and c17c15C(N,p)c17. Moreover, since diam(Ω)<2R, it follows that

(3.13) ( 𝐈 1 2 R [ ( 𝐈 2 [ μ ] ) p ] ) q c 21 q ( 𝐈 1 [ μ ] ) q c q c 21 q ( 𝐈 1 2 R [ μ ] ) q ,

for some c=c(N,R)>0. Next, it is quoted in [16, Theorem 1.1] that inequality (3.12) is equivalent to the main assumption of Theorem 1.3,

(3.14) μ n Ω ( E ) C cap 2 , p N ( E ) for all compact set  E Ω ,

for some C>0. Actually this equivalence is proved in [19]. By [1, Theorem 3.14 (a)] there exists A=A(N)>0 such that

| { x N : | 𝐈 1 [ μ n Ω ] ( x ) | > λ } | A λ - N N - 1 μ n Ω L 1 N N - 1 .

Clearly, the above inequality holds if 𝐈1 is replaced by 𝐈12R and N by Ω. This is an estimate of 𝐈12R[μnΩ] in the Lorentz space LNN-1,(Ω) (or Marcinkiewicz space). Clearly,

μ n Ω L 1 μ 𝔐 .

Therefore (3.10) implies that |un| is bounded in LNN-1,(Ω), hence equi-integrable in Lq(Ω) since q<NN-1. By Lemma 3.2 (ii) and classical harmonic analysis results, 𝐈2[μnΩ𝐈2[μΩ in Lp(Ω) (see, e.g., [26]). It follows from (3.9) that un is equi-integrable in