Measure data problems for a class of elliptic equations with mixed absorption-reaction

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Measure data problems for a class of elliptic equations with mixed absorption-reaction

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron

To cite this version:

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Measure data problems for

a class of elliptic equations with mixed absorption-reaction. Advanced Nonlinear Studies, Walter de

Gruyter GmbH, 2021, 21 (2), pp.261-280. �hal-03147340v2�

Measure data problems for a class of elliptic equations with mixed absorption-reaction

Marie-Fran¸ coise Bidaut-V´ eron

, Marta Garcia-Huidobro

Laurent V´ eron

Abstract

We study the existence of nonnegative solutions to the Dirichlet problem L^Mp,qu:=−∆u+u^p−M|∇u|^q =µin a domain Ω⊂R^N whereµis a nonnegative Radon measure, whenp >1,q >1 andM ≥0. We also give conditions under which nonnegative solutions ofL^Mp,qu= 0 in Ω\KwhereKis a compact subset of Ω can be extended as a solution of the same equation in Ω.

2010 Mathematics Subject Classification: 35J62-35J66-31C15-28A12

Keywords: Elliptic equations, singularities, Bessel capacities, Riesz potential, maximal functions.

Contents

1 Introduction 2

2 Removable singularities 5

2.1 A prioriestimates . . . 5 2.2 Proof of Theorem 1.1 . . . 10 2.3 Proof of Theorem 1.2 . . . 14

3 Measure data 16

3.1 Proof of Theorem 1.3: the case 1< q < _N−1^N . . . 16 3.2 Proof of Theorem 1.3: the general case . . . 20 3.3 Proof of the Corollaries . . . 23

∗Laboratoire de Math´ematiques et Physique Th´eorique, Universit´e de Tours, 37200 Tours, France. E-mail: veronmf@univ-tours.fr

†Departamento de Matem´aticas, Pontifica Universidad Cat´olica de Chile, Casilla 307, Correo 2, Santiago de Chile. E-mail: mgarcia@mat.uc.cl

‡Laboratoire de Math´ematiques et Physique Th´eorique, Universit´e de Tours, 37200 Tours, France. E-mail: veronl@univ-tours.fr

1 Introduction

Let Ω be a bounded domain ofR^N,N ≥2, andL^Mp,q be the operator

u7→ L^Mp,qu:=−∆u+|u|^p−1u−M|∇u|^q for allu∈C²(Ω) (1.1) whereM ≥0 andp, q >1. We first provide an a prioriestimate 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

L^Mp,qu= 0 in Ω\K, (1.2)

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

L^Mp,qu=µ in Ω

u= 0 in∂Ω, (1.3)

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

IfM = 0,L^Mp,q reduces to the Emden-Fowler operator

u7→ L_pu:=−∆u+|u|^p−1u. (1.4) Singularity problems for solutions ofLpu= 0 have been investigated since fourty years, starting with the work of Brezis and V´eron [13] who gave conditions for the removability of an isolated singularity. Later on Baras and Pierre [3]

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 cap^R_2,p^N0 (p⁰ = _p−1^p ) 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 L^Mp,q is the Riccati operator

u7→ R^Mq u:=−∆u−M|∇u|^q. (1.5) The Dirichlet problem with measure data

−∆u−M|∇u|^q=µ in Ω

u= 0 on∂Ω (1.6)

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

WhenM >0 there is a balance between the absorption term|u|^p−1u and the source termM|∇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 _p+1^2p . This is due to the fact that ifq= _p+1^2p , (1.1) is equivariant with respect to the scaling transformationT` defined for` >0 by

T<sub>`</sub>[u](x) =`^p−12 u(`x) =`^2−qq−1u(`x). (1.7)

Ifq < _p+1^2p , the absorption term is dominant and the behaviour of the singular solutions is modelled by the equationLpu= 0 studied in [28]. Ifq > _p+1^2p , the diffusion is negligible and the behaviour of the singular solutions is modelled by positive separable solutions ofEp,q^Mu= 0 whereEp,q^M is an eikonal type operator defined by

Ep,q^Mu=u^p−M|∇u|^q. (1.8) This problem is studied in the forthcoming article [8]. Ifq= _p+1^2p , 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 whenq6= _p+1^2p since by an homothetyM can be assumed to be equal to 1.

Brezis and V´eron proved in [13] that isolated singularities of solutions of L_pu = 0 are removable when p≥ _N^N₋₂. The removability property has been extended to more general sets using a capacity framework in [3]. Using a change of variable inspired by [6] where boundary singularities of solutions of (1.8) are considered we prove a series of removability results for solutions of

L^Mp,qu= 0. (1.9)

Theorem 1.1 Assume 0∈Ω⊂R^N,N ≥3, M >0,p≥ _N^N₋₂, 1< q ≤ _p+1^2p and (p, q) 6= (_N−2^N ,_N−1^N ). Then any nonnegative solution u∈ C²(Ω\ {0}) of (1.9)inΩ\ {0}belongs toW_loc^1,q(Ω)∩L^p_loc(Ω), and it can be extended as a weak solution of(1.9)in Ω.

Furthermore, if we assume either (i)p≥ _N−2^N and1< q < _p+1^2p , or (ii) p > _N^N₋₂,q= _p+1^2p and

M < m^∗:= (p+ 1)

(N−2)p−N 2p

_p+1^p

, (1.10)

thenu∈C²(Ω).

The existence of radial singular solutions when (p, q) = (_N^N₋₂,_N^N₋₁) and M > 0, or when p > _N^N₋₂, q = _p+1^2p and M ≥ m^∗ shows the optimality of the statements (see [8]). A series of pointwise a priori estimates concerningu and∇uare 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= _N−2^N 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 [3] yields a more general removability result. For such a task we denote bycap^R_k,b^N the Bessel capacity relative toR^N with orderk >0 and power b ∈(1,∞). Ifk∈ N^∗ it coincides with the Sobolev capacity associated to the spaceW^k,b(R^N) by Calderon’s theorem (see e.g. [1] for a detailed presentation).

Theorem 1.2 Let p > _N^N₋₂ and _N^N₋₂ < r < p. Suppose that one of the following conditions is verified:

(i) either q= _p+1^2p and

0< M < m^∗(r) := (p+ 1)

p−r p(r−1)

_p+1^p

, (1.11)

(ii) or1< q < _p+1^2p andM >0.

Then, ifK is a compact subset ofΩsuch thatcap^R_2,r^N0(K) = 0, any nonnegative solution u∈C²(Ω\K) of (1.9) in Ω\K can be extended to Ω as a solution still denoted by uin the sense of distributions in Ω. Furthermore, if r≤_N^2N₋₂, thenu∈C²(Ω).

Next we obtain sufficient conditions on a positive measure in Ω in order (1.3) be solvable. In the sequel we assume that Ω⊂R^N,N ≥2, is a bounded smooth domain. We denote byM(Ω) (resp. M^b(Ω)) the set of Radon measures (resp.

bounded Radon measures) in Ω and byM+(Ω) (resp. M^b₊(Ω)) its positive cone.

The total variation norm of a bounded measureµiskµk_M.

Since for anyµ∈M^b₊(Ω) the nonnegative solutions ofLpv=µandR^M_q w= µare respectively a subsolution and a supersolution of equation (1.3) and they satisfy 0≤v≤w, the construction ofvandwis the key-stone for solving (1.3).

It is known that these two problems can be solved when the measureµsatisfies some continuity properties with respect to some specific capacities.

Theorem 1.3 Assume p >1,1< q <2. Letµ∈M^b₊(Ω). If µsatisfies µ(E)≤Cminn

cap^R_2,p^N0(E), cap^R_1,q^N0(E)o

for all Borel sets E⊂Ω, (1.12) there is a constant c0>0 such that for any0≤c≤c0 there exists a function u∈W₀^1,q(Ω)∩L^p(Ω),u≥0, satisfying

− Z

Ω

u∆ζdx+ Z

Ω

(u^p−M|∇u|^q)ζdx=c Z

Ω

ζdµ for allζ∈C_c²(Ω). (1.13) The condition on the measure is satisfied ifW^−1,q(Ω),→W^−2,p(Ω), and we prove the following:

Corollary 1.4 Let _N^{N p}_+p ≤q <2 andµ∈M^b₊(Ω) be such that

µ(E)≤Ccap^R_1,q^N0(E) for all Borel setE⊂Ω, (1.14) for someC >0, then there exists a constantc1>0such that for any0≤c≤c1

there exists a nonnegative functionu∈W₀^1,q(Ω)∩L^p(Ω) satisfying (1.13).

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

Corollary 1.5 Let _N^N₋₁ ≤q≤ _p+1^2p . Ifµ∈M^b₊(Ω) satisfies,

µ(E)≤Ccap^R_2,p^N0(E) for all Borel setE⊂Ω, (1.15) for some C > 0, then there exists c2 > 0 such that for any 0 ≤ c ≤c2 there exists a nonnegative function u∈W₀^1,q(Ω)∩L^p(Ω) satisfying(1.13).

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

Corollary 1.6 Let p >1,1< q <2 andµ∈M^b₊(Ω). There exists a function u ∈ W₀^1,q(Ω)∩L^p(Ω) solution of (1.13) if one of the following conditions is satisfied:

(i) When p < _N−2^N andq < _N−1^N , if kµk_M≤c₃ for some c₃>0.

(ii) Whenp < _N−2^N and _N^N₋₁ ≤q <2, if µsatisfies(1.14)for someC >0. In that case there exists c4 >0 such that there must hold0 < c < c4 in problem (1.12).

(iii) When p ≥ _N^N₋₂ and q < _N^N₋₁, if kµk_M ≤ c^∗₄M^−q−11 for some c^∗₄ = c^∗₄(N, q,Ω)>0 which can be estimated, and if

µ(E) = 0 for all Borel set⊂Ωsuch that cap^R_2,p^N0(E) = 0. (1.16) In the case (i) we show in a forthcoming article [8] and by a completely different method that there is no restriction onc ifµ=cδa for somea∈Ω. In the above mentioned article we construct many types of local or global singular solutions using methods inherited from dynamical systems.

Acknowledgements This article has been prepared with the support of the FONDECYT grants 1210241 and 1190102 for the three authors.

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 byc_j, (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 ofM. IfGis an open subset ofR^N we set d_G(x) = dist (x, ∂G) Proposition 2.1 Let G⊂R^N be an open subset, M >0 and 1 < q < p. If u∈C¹(G)is a nonnegative solution of(1.1), there holds,

u(x)≤c5maxn

M^p−q1 (d_G(x))^−p−qq ,(d_G(x))^−p−12 o

for all x∈G, (2.1) for somec₅=c₅(N, p, q)>0.

Proof. Following the method of Keller [17] and Osserman [22], we fix x ∈ G and 0< a < d_G(x) , and introduceU(z) =λ(a²− |z−x|²)^−b for someb >0.

Then puttingr=|x−z|and ˜U(r) =U(z), we have inBa(x) LU˜ =−U˜⁰⁰−N−1

r

U˜⁰−M|U˜⁰|^q+ ˜U^p

=λ(a²−r²)^−2−b

λ^p−1(a²−r²)^2−b(p−1)+ 2b(N−2(b+ 1))r²−2N ba²

−M2^qb^qλ^q−1r^q(a²−r²)^2+b−q(b+1) .

IfM >0, the two necessary conditions onb >0 to be fulfilled is order that ˜U is a supersolution inB_|a|(a) are

(i) 2−b(p−1)≤0⇐⇒b(p−1)≥2,

(ii) 2 +b−q(b+ 1)≥2−b(p−1)⇐⇒b(p−q)≥q.

The above inequalities are satisfied if b= max

2 p−1, q

p−q

. (2.2)

Ifq > _p+1^2p thenb=_p−q^q and LU˜ ≥λ a²−r^2−2p−qp−q

λ^q−1 λ^p−q−M2^qb^qρ^q

a²−r²

2p−q(p+1)

p−q −(3b+ 1)N a²

.

There exists c¹₅>0 depending onN, pandqsuch that if we choose λ=c¹₅maxn

M^p−q1 a^p−qq , a

2p(q−1) (p−1)(p−q)

o , there holds

LU˜ ≥0 inBa(x). (2.3)

Since ˜U(z)→ ∞whenr→a, we derive by the maximum principle thatu≤U˜ in B_a(x). In particular

u(x)≤U˜(x) =λa^−p−q2q =c¹₅maxn

M^p−q1 a^−p−qq , a^−p−12 o

. (2.4)

Ifq≤ _p+1^2p thenb=_p−1² and LU˜ ≥λ |a|²−ρ²−_p−1^2p

λ^p−1+ 2 p−1

N−2(p+ 1) p−1

ρ²− 2N p−1|a|²

−M2^q 2

p−1 q

λ^q−1ρ^q |a|²−ρ^22p−q(p+1)_p−1

≥λ |a|²−ρ²−_p−1^2p h

λ^p−1−c₂|a|²−c₃λ^q−1M|a|^{4p−q(p+3)p−1} i .

Hence, ifq=_p+1^2p , (2.3) holds if for somec²₅>0 depending on N, p, q, λ=c²₅maxn

M^p(p−1)p+1 ,1o

|a|^p−12 ,

which yields

u(x)≤U˜(x) =λa^−p−14 =c²₅maxn

M^p(p−1)p+1 ,1o

a^−p−12 . (2.5) While ifq < _p+1^2p , we choose

λ=c³₅maxn M^p−q1 a

4p−q(p+3)

(p−1)(p−q), a^p−12 o ,

wherec³₅>0 =c³₅(N, p, q), which implies u(x)≤U˜(x) =λa^−p−14 =c³₅maxn

M^p−q1 a^−p−qq , a^−p−12 o

. (2.6)

By letting a ↑ d_G(x) we derive (2.1) with a constant c5 = c³₅, depending on

N, p, q.

Corollary 2.2 Under the assumptions of Proposition 2.1 withG=B2R\ {0}, there holds forx∈B_R\ {0},

u(x)≤c5maxn

M^p−q1 |x|^−p−qq ,|x|^−p−12 o

. (2.7)

We infer from Proposition 2.1 an estimate of the gradient of a positive solution when M >0. We set σ = 2p−q(p+ 1), then σ >0 (resp. σ <0) according 2p > q(p+ 1) (resp. 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, M0R^p−1σ )such that, for0< M ≤M0 there holds:

(i) Ifq≤ _p+1^2p (thenσ≥0), any positive solutionuof(1.1)inB_2R\{0}satisfies

|∇u(x)| ≤c8maxn

M^p−q1 |x|^−p−qp ,|x|^−p+1p−1o

, (2.8)

for allx∈B_R\ {0}.

(ii) If _p+1^2p ≤q ≤2 (then σ≤0), any positive solutionu of (1.1) in R^N\BR 2

satisfies(2.1)for all x∈R^N \BR. Proof. (i) For 0< r <2Rwe set

u(x) =r^−p−12 u_r(^x_r) =r^−p−12 u_r(y) with y= ^x_r. If ^r₂ <|x|<2r, then ¹₂ <|y|<2 andu_r>0 satisfies

−∆ur+u^p_r−M r^{2p−q(p+1)p−1} |∇ur|^q = 0 in B₂\B1

2. (2.9)

Since 0< M r^p−1σ ≤M(2R)^p−1σ ≤M0(2R)^p−1σ as σ≥0, it follows that max

|∇ur(z)|: ²₃<|z|< ³₂ ≤cmax

|ur(z)|: ¹₂ <|z|<2 , (2.10) where c depends on N, p, q and R^p−1σ M₀ (see e.g. [14, Chapter 13]). From Proposition 2.1 there holds

max

|ur(z)|: ¹₂<|z| ≤2 ≤2^p−12 c5maxn M^p−q1 r

2p−q(p+1) (p−1)(p−q),1o by (2.1). Therefore

max

|∇u(y)|: ^r₂ <|z|<2r ≤2^p−12 cc5r^−p+1p−1maxn M^p−q1 r

2p−q(p+1) (p−1)(p−q),1o

≤c8maxn

M^p−q1 |x|^−p−qp ,|x|^−p+1p−1o ,

(2.11) which implies (2.8).

(ii) Forr > Rwe defineu_r as in (i). It satisfies (2.9) and sinceσ≤0, we have again 0< M r^p−1σ ≤M R^p−1σ ≤M₀R^p−1σ ifr≥R. Since 1< q <2, (2.10) holds

and we derive (2.8).

Remark. Ifq=_p+1^2p the constantc8depends only on N andp.

The previous estimate necessitates 1 < q ≤ 2. This limitation can be by passed in some cases using the Bernstein approach.

Lemma 2.4 Assume p, q >1 and M > 0. If u ∈C²(B2R) is a nonnegative solution of(1.1)in B2R, there holds

|∇u(x)| ≤c₉ |x|^−q−11 + max

|z−x]≤^|x|₂

u^pq(z)

!

for all x∈BR

2, (2.12) c₉>0 depends onN,p,q andM.

Proof. Setz=|∇u|², then by a classical computation and the use of Schwarz inequality,

−∆|∇u|²+ 1

N(∆u)²+h∇∆u,∇ui ≤0.

Replacing ∆uby its expression fromL^Mp,qu= 0, we obtain

−∆z+ 2 N

u^2p+M²z^q−2M u^pz^q2

+ 2pu^p−1z≤qM z^q2−1h∇z,∇ui.

We notice that

qM z^q2−1h∇z,∇ui ≤qM z^q2−1|∇z|√

z=qM z^q2|∇z|

√z ≤M²z^q

2N +2N q² M²

|∇z|² z , and

4M

N u^pz^q2 ≤ M²z^q

2N + 8u^2p N M², thus

−∆z+M²z^q

N ≤2N q² M²

|∇z|² z + 2

N 4

M²−1

u^2p.

For simplicity we set A= M²

N , B= 2N q²

M² and C= 2 N

4 M² −1

+

max

|z−x|≤^|x|₂

u^2p(z) Thenz satisfies

−∆z+Az^q≤B|∇z|²

z +C inBR 2(x)

and obtain by [5, Lemma 3.1] (see also a simpler approach in [7, Lemma 2.2]), z(x)≤c₁₀

|x|^−q−12 +C^1q

(2.13) wherec₁₀>0 depends onN,p,qandM. This yields (2.12).

Remark. The constants c9 and c10 can be expressed in terms of M, but their stability whenM →0 is not clear since in the limit case of the equationLpu= 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 Assume1< q < p andM >0. Then any nonnegative solution u∈C²(B_2R) of(1.1)satisfies

|∇u(x)| ≤c11

|x|^−q−11 + maxn

M^q(p−qp |x|^−p−qp ,|x|^{−q(p−1)2p} o

for all x∈BR 2, (2.14) wherec11>0 depends onN,p,q andM.

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

Proposition 2.6 Let 1 < q < p. For any M > 0 there exists a constant c12=c12(N, p, q, M)>0 such that:

(i) If _p+1^2p ≤q < p, any positive solutionuof(1.1)inB2R\ {0}with0< R≤1 satisfies,

|∇u(x)| ≤c₁₂maxn

M^q(p−q)p |x|^−p−qp ,|x|^{−q(p−1)2p} o

for allx∈B_R\ {0}. (2.15) (ii) If 1 < q ≤ _p+1^2p , any positive solution u of (1.1) in R^N \BR

2 with R ≥1 satisfies,

|∇u(x)| ≤c12maxn

M^q(p−q)p |x|^−p−qp ,|x|^−q−11 o

for allx∈R^N \BR. (2.16) Proof. We can compare the different exponents of|x|which appear in the ex- pressions (2.8) and (2.14)

(i) p

p−q <p+ 1

p−1 < 2p

q(p−1) < 1

q−1 if 1< q < 2p p+ 1,

(ii) 1

q−1 < 2p

q(p−1) < p+ 1 p−1 < p

p−q if 2p

p+ 1 < q < p,

(2.17)

with equality if q= _p+1^2p . If 1< q ≤ _p+1^2p (resp. _p+1^2p ≤≤2), estimate (2.8) is better than (2.14) inBR\ {0}(resp. R^N\B2R). Then (2.15) and (2.16) follow

from (2.14) and (2.17).

In the caseM <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 G ⊂ R^N be an open subset, M ≤ 0 and p, q > 1. If u∈C¹(G)is a nonnegative solution ofL^Mp,qu= 0, there existsc6=c6(N, p)>0, c7 =c7(N, q)> 0 and δ = δ(G) >0 such that there holds for all x∈ G and Ø< δ≤δ(G),

u(x)≤min

c6(d_G(x))^−p−12 , c7|M|^−q−11 (d_G(x))^{−2−qq−1} + max

d_G(z)=δu(z),

. (2.18) Proof. This estimates follows from the fact that the solutions ofL^Mp,qu= 0 are subsolutions of Lpu = 0 andR^Mq u= 0. The estimate u(x) ≤c6(d_G(x))^−p−12 corresponds to the Keller-Osserman estimate for solutions of Lpu = 0. The second estimate corresponds to the fact that ifuis a positive solution ofR^Mq u= 0 inGthere holds (see [18, Theorem IV-1])

|∇u(x)| ≤c⁰₇|M|^−q−11 (d_G(x))^−q−11 .

Integrating this inequality yields the second part of the inequality.

Remark. This estimate can be transformed into the universal estimate u(x)≤minn

c6(d_G(x))^−p−12 , c7|M|^−q−11 (d_G(x))^{−2−qq−1} +c6δ^−p−12 ,o

, (2.19) since max

dG(z)=δu(z)≤c6δ^−p−12 by (2.18).

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 constantc⁰₁₂=c⁰₁₂(N, p, q, M,)>0 such that: ifuis a positive solution of (1.1) inB2R\ {0}, there holds

u(x) +|x||∇u(x)| ≤c⁰₁₂maxn

|x|^−p−12 ,|x|^{−2−qq−1}o

for allx∈BR\ {0}. (2.20) Remark. There are many estimates of positive solutions of (1.1) (or even with u^preplaced byf(u)) in a domain which tends to infinity on the boundary (large solutions) or of solutions in R^N (ground states). Many of these estimates are obtained by comparison with one dimensional problems and they can be found in [2], [4], [15].

2.2 Proof of Theorem 1.1

Without loss of generality we can assume that u∈C²(Ω\ {0} and B2R₀ ⊂Ω with 2R0≤1. IfM ≤0,uis a nonnegative subsolution of−∆u+u^p= 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|^q ∈L¹(Ω),u∈L^p(Ω), and then Z

Ω

(−u∆ζ+u^pζ−M|∇u|^qζ)dx= 0 ∀ζ∈W^2,∞(Ω)∩C_c¹(Ω). (2.21) By Proposition 2.3

|∇u|^q ≤c|x|^{−(p+1)qp−1} in B_R0,

since q ≤ _p+1^2p , and where c depends also on M. By (i) or (ii), ^(p+1)q_p−1 < N.

Hence ∇u∈L^q_loc(Ω).

For any > 0 small enough we denote by ρ a nonnegative C^∞-function such that supp(ρ)⊂B, 0≤ρ≤1,|∇ρ| ≤2⁻¹χ

B and we setη= 1−ρ. Then

− Z

B2R0

h∇u,∇ρidx+ Z

B2R0

u^pηdx+ Z

∂B2R0

∂u

∂ndS=M Z

B2R0

|∇u|^qηdx.

(2.22) Next

Z

B2R0

h∇u,∇ρidx

≤2cN^Nq0−1 Z

B

|∇u|^qdx ¹_q

→0 as →0, (2.23)

since 1< q ≤ _N−1^N . Since |∇u|^q ∈L¹(B_2R0) we deduce by monotone conver- gence thatu^p∈L¹(B_2R0). Finally, ifζ∈C₀^∞(Ω) andζ=ζη, we have

Z

Ω

h∇u,∇ζidx+ Z

Ω

u^pζdx−M Z

Ω

|∇u|^qζdx= 0.

Letting→0 and using (2.23), we infer thatusatisfies Z

Ω

h∇u,∇ζidx+ Z

Ω

u^pζdx−M Z

Ω

|∇u|^qζdx= 0.

Hence it is a weak solution of (1.9) in Ω.

Step 2. Let us assume that p > _N−2^N . Ifuis nonnegative and not identically zero, it is positive in Ω\ {0} by the maximum principle. We set u=v^b with 0< b≤1. Then

−∆v−(b−1)|∇v|² v +1

bv^1+(p−1)b−M b^q−1v^{(b−1)(q−1)}|∇v|^q= 0. (2.24) For >0,

v^{(b−1)(q−1)}|∇v|^q ≤q^2q 2

|∇v|²

v + 2−q 2^2−q2

v^{1+2b(q−1)2−q} .

Therefore

−∆v+ 1−b−Mqb^q−12q 2

!|∇v|² v +1

bv^1+b(p−1)−M b^q−12−q 2^2−q2

v^{1+2b(q−1)2−q} = 0.

(2.25) We notice that 1 +^2b(q−1)_2−q = 1 +b(p−1)−awitha=b^2p−(p+1)q_2−q ≥0. We fix b as follows,

(p−1)b+ 1 = N

N−2 ⇐⇒b= 2

(N−2)(p−1), (2.26) hencep > _N^N₋₂ if and only if 0< b <1. Next we impose

1−b−Mqb^q−12q

2 = 0⇐⇒=

2(1−b) M qb^q−1

^q₂

=

2((N−2)p−N) M qb^q−1(N−1)(p−1)

^q₂ . (2.27) This transforms (2.25) into

−∆v+(N−2)(p−1)

2 v^N−2N −(2−q)b^q−1 2

q 2(1−b)

_2−q^q

M^2−q2 v^{N−2N −a}≤0.

(2.28) We first assume that 0 < q < _p+1^2p . Then a > 0, hence there exists A > 0, depending onM, such that

−∆v+(N−2)(p−1)

4 v^N−2N ≤A. (2.29)

Set ˜v= (v−cA^N−2N )

N N−2

+ withc=

4 (N−2)(p−1)

_N−2^N

satisfies

−∆˜v+(N−2)(p−1)

4 ˜v^N−2N ≤0. (2.30)

By [13], ˜v≤max

∂Ω v˜which implies v≤cA^N−2N + max

∂Ω v and therefore u(x)≤B for someB ≥0 in Ω. Furthermore|∇u|^q−1∈L^q−1q (Ω) since ∇u∈L^q(Ω), and

q

q−1 > N as we assumeq < _N^N₋₁. Writing (1.9) under the form

−∆u+u^p−M C(x)|∇u|= 0,

withC(x) =|∇u(x)|^q−1, it follows from Serrin’s theorem [25, Theorem 10] that the singularity at 0 is removable anducan be extended as a regular solution of (1.9) in Ω. Henceu∈C²(Ω).

Then we assume that q= _p+1^2p . By the choice ofb in (2.26), inequality (2.25) becomes

−∆v+ 1−b−M pb^p−1p+1p+1p p+ 1

!|∇v|²

v + 1

b − M b^p−1p+1 (p+ 1)^p+1

!

v^N−2N ≤0. (2.31) Notice that

1

b − M b^p−1p+1

(p+ 1)^p+1 = 0⇐⇒= M

p+ 1 _p+1¹

b

2p

(p+1)2, (2.32)

and therefore

1−b−M pb^p−1p+1p+1p

p+ 1 = 1−b−pb M

p+ 1 ^p+1_p

. (2.33)

This coefficient vanishes if p

M p+ 1

^p+1_p

= p(N−1)−(N+ 1)

2 .

Therefore, ifM satisfies p

M p+ 1

^p+1_p

=p(N−2)−N

2 , (2.34)

we can choose > 0 so that the coefficient of v^(p−1)b+1 in (2.34) is equal to some τ >0. Thereforev satisfies

−∆v+τ v^N−2N ≤0 in Ω\ {0}. (2.35) It follows by [13], v ≤max

∂Ω v and the same type of uniform estimate holds for u. This ends the casep > _N^N₋₂.

Step 3. Finally we assumep= _N−2^N and 1< q < _p+1^2p = _N−1^N . From (2.12), M|∇u(x)|^q≤c9|x|^−qp+1p−1 =c9|x|^−q(N−1):=Q(x).

andQ∈L¹(B_2R0). Let{σ_n} ⊂C₀^∞(R^N) such that 0≤σ_n≤1 σn(x) =

( 1 if _n² ≤ |x| ≤R0

0 if|x| ∈[0,_n¹]∪[2R₀,∞),

and

|∆σ_n| ≤2N n²χ_B

2 n

\B1 n

+φ,

whereφis a smooth nonnegative function with support inB_2R0\B_R0. Then

− Z

{_n¹≤|x|≤_n²}

u∆σndx− Z

{R0≤|x|≤2R0}

u∆σndx+ Z

1 n≤|x|

u^pσndx=M Z

1 n≤|x|

|∇u|^qσndx.

(2.36) The right-hand side of (2.36) is bounded since |∇u| ∈ L^q(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

Z

{_n¹≤|x|≤_n²}

u∆σndx

≤C,

for some C >0 independent of n. Lettingn → ∞we infer that u∈L^p_loc(Ω).

By the maximum principle

u(x)≤u₁(x) =CG^B2R0[Q](x) + max

|z|=2R0

u(z), (2.37)

where G^B2R0 denotes the Green kernel in B2R0. Since Q(x) = C|x|^−q(N−1), a direct computation shows thatu1(x)≤cNC|x|^2−q(N−1)=cNC|x|^2−N+ for some >0. We can write (1.1) under the form

−∆u+c(x)u+d(x)|∇u|= 0 in Ω\ {0},

with c(x) = u^N−22 and d(x) = |∇u|^q−1. Then c ∈ L^N2+1(B2R₀) and d ∈ L^N+2(B2R₀); with 1, 2 > 0. It follows from [25, Theorem 10] that 0 is a removable singularity foruin the sense that it can be extended as aC²solution

in Ω.

When the conditions of the theorem are not fulfilled there exist singular solutions. However these singular solutions may exhibit different types of be- haviour according 1< q < _p+1^2p , _p+1^2p < q <2 andq= _p+1^2p . In this case there may exist radial separable solutions of (1.9) under the form uX(r) =Xr^−p−12 . Setting α= _p−1² , thenX satisfies

Φp(X) :=X^p−1−M α^p+12p X^p−1p+1 +α(N−2−α) = 0 (2.38) The following result is easy to prove by a standard analysis of the function Φ_p. Proposition 2.9 Let p >1 andM ∈R.

(i) If M is arbitrary and 1 < p < _N^N₋₂, orM > 0 and p= _N−2^N , there exists one and only one positive solution X1 to(2.38).

(ii) If p > _N^N₋₂ and M > m^∗, there exist two positive solutions X1 < X2 to (2.38).

(iii) Ifp > _N^N₋₂ andM =m^∗ there exists one positive solution X₁ to(2.38).

(iv) If p > _N^N₋₂ and 0 < M < m^∗, or M ≤0 and p≥ _N^N₋₂, there exists no positive solution to(2.38).

Whenq 6= _p+1^2p the existence of singular solutions is much involved. It is developed in the subsequent paper [8].

Remark. It is noticeable that in the caseq= _p+1^2p ,p > _N^N₋₂ andM ≥m^∗, the equation exhibits a phenomenon which is characteristic of Lane-Emden type equations

−∆u=u^p in B1\ {0}. (2.39) Ifuis nonnegative then there existsα≥0 such that

−∆u=u^p+αδ0 in D⁰(B1). (2.40) If 1< p < _N−2^N thenαcan be positive, but ifp≥ _N−2^N , thenα= 0. This means that the singularity cannot be seen in the sense of distributions, however there truly exist singular solutions, e.g. ifp > _N−2^N ,

us(x) =cN,p|x|^−p−12 . (2.41) Here also for q = _p+1^2p , p > _N−2^N , 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 1 < q < _p+1^2p or q = _p+1^2p . Without loss of generality we can suppose that u >0. We perform the same change of unknown as in the previous theorem putting u=v^b, but now we choosebas follows

(p−1)b+ 1 =r⇐⇒b=r−1

p−1, (2.42)

and we first assume that 1−b−Mqb^q−12q

2 = 0⇐⇒=

2(1−b) M qb^q−1

^q₂

=

2(p−r) M q(p−1)b^q−1

^q₂

. (2.43) Hence (2.28) becomes

−∆v+p−1

r−1v^r−(2−q)b^q−1 2

q 2(1−b)

2−q^q

M^2−q2 v

(2r−p−1)q+2(p−r)

(p−1)(2−q) ≤0. (2.44) Conditionr≥(2r−p−1)q+2(p−r)

(p−1)(2−q) is equivalent to 2p−q(p+ 1)≤r(2p−q(p+ 1)), since 1< r < p.

Assuming first thatq < _p+1^2p , we obtain from (2.44)

−∆v+ p−1

2(r−1)v^r≤A. (2.45)

for some constant A ≥ 0. Since cap^R_2,r^N0(K) = 0 the function v is bounded from [3] and v≤cA^1r + max

∂Ω v for somec >0, hence uis also uniformly upper bounded in Ω by some constanta.

Next we have to show that∇u∈L^q(Ω). Let{ρ_n}be a sequence ofC₀^∞(Ω) non- negative functions such that 0≤ρ_n ≤1,ρ_n= 1 in a small enough neighborhood ofK andkρ_nk_W2,r0 →0 when n→ ∞, and setη_n= 1−ρ_n. Since

Z

Ω

u∆ρndx− Z

∂Ω

∂u

∂ndS+ Z

Ω

u^pηndx=M Z

Ω

|∇u|^qηndx,

and

Z

Ω

u∆ρ_ndx

≤ckuk_L∞kρ_nk_W2,r0 →0 asn→ ∞, we get

Z

Ω

u^pdx− Z

∂Ω

∂u

∂ndS=M Z

Ω

|∇u|^qdx.

Hence ∇u∈L^q(Ω). Ifζ∈C₀^∞(Ω) and ζ_n=ζη_n, there holds

− Z

Ω

ηnu∆ζdx+ Z

Ω

ζu∆ρndx+ Z

Ω

u^pζndx=M Z

Ω

|∇u|^qζndx.

Since the second term on the left-hand side tends to 0 andζn→ζwhenn→ ∞, we obtain that

− Z

Ω

u∆ζdx+ Z

Ω

u^pζdx=M Z

Ω

|∇u|^qζdx.

Hence uis a solution in the sense of distribution in Ω.

Next we show that∇u∈L²(Ω). Multiplying (1.9) by uη_n and integrating, we obtain

Z

Ω

|∇u|²ηndx−

Z

Ω

uh∇u,∇ρnidx−

Z

∂Ω

u∂u

∂ndS+

Z

Ω

u^p+1ηndx=M Z

Ω

u|∇u|^qηndx.

As Z

Ω

uh∇u,∇ρ_nidx= 1 2 Z

Ω

h∇u²,∇ρ_nidx

=−1 2 Z

Ω

u²∆ρndx,

and

Z

Ω

u²∆ρndx

≤ckuk²_L∞kρnk_W2,r0 =o(1) as n→ ∞, we infer that

Z

Ω

|∇u|²dx− Z

∂Ω

u∂u

∂ndS+ Z

Ω

u^p+1dx=M Z

Ω

u|∇u|^qdx.

Finally ifζ∈C₀^∞(Ω) andζn=ζηn, then Z

Ω

η_nh∇u,∇ζidx− Z

Ω

ζh∇u,∇ρ_nidx+ Z

Ω

u^pζ_ndx=M Z

Ω

|∇u|^qζ_ndx.

Sincer≤ _N^2N₋₂ there holds

kρ_nk_W1,2 ≤ckρ_nk_W2,r0 =⇒ kρ_nk_W1,2 →0 as n→ ∞.

Using the fact that∇u∈L²(Ω) and H¨older’s inequality, we derive Z

Ω

ζh∇u,∇ρnidx→0 as n→ ∞.

Hence

Z

Ω

h∇u,∇ζidx+ Z

Ω

u^pζdx=M Z

Ω

|∇u|^qζdx.

This implies thatuis a weak solution of (1.9) and it is thereforeC²in Ω.

Next we assumeq=_p+1^2p . We chooseb=_p−1^r−1 and (2.31) becomes

−∆v+ 1−b−M pb^p−1p+1p+1p p+ 1

!|∇v|²

v + 1

b − M b^p−1p+1 (p+ 1)^p+1

!

v^r≤0. (2.46) If (2.32) holds with this choice ofb, (2.33) becomes

1−b−M pb^p−1p+1p+1p

p+ 1 = 1−b−pb M

p+ 1 ^p+1_p

= 1

p−1 p−r−p(r−1) M

p+ 1 ^p+1_p !

.

(2.47)

IfM < m^∗_r defined by (1.10), we can choose such that 1−b−M pb^p−1p+1p+1p

p+ 1 = 0, and

1

b − M b^p−1p+1

(p+ 1)^p+1 =τ:=τ()>0.

Thenv satisfies

−∆v+τ v^r≤0 in Ω\K.

Sincecap^R_2,r^N0(K) = 0 it follows from [3] thatv≤max

x∈∂Ωv(x). Henceuis bounded.

The different steps of the proof in the first case applies without any modification:

first∇u∈L^q(Ω) and the equation holds in the sense of distributions in Ω, then

∇u∈L²(Ω) and sincer≤ _N−2^2N we infer thatuis a weak solution and thus a

strong one.

3 Measure data

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

3.1 Proof of Theorem 1.3: the case 1 < q <

If 1< q <_N^N₋₁ assumption (1.12) withµ≥0 reduces to

µ(K)≤Ccap^R_2,p^N0(K) for all compact set K⊂Ω. (3.1)

The construction of solutions is based upon the following result due to Boccardo- Murat-Puel [10]. It is concerned with a general quasilinear equation in a domain G⊂R^N

Q(u) :=−∆u+B(., u,∇u) = 0 inD⁰(G), (3.2) whereB ∈C(G×R×R^N) satisfies

|B(x, r, ξ)| ≤Γ(|r|)(1 +|ξ|²) for all (x, r, ξ)∈G×R×R^N, (3.3) for some continuous increasing function Γ fromR⁺ toR⁺.

Theorem 3.1 Let G be a bounded domain in R^N. If there exists a superso- lution φ and a subsolution ψ of the equation Qv = 0 belonging to W^1,∞(G) and such that ψ ≤φ, then for any χ∈ W^1,∞(G) satisfying ψ ≤χ ≤φ there exists a function u∈ W^1,2(G) solution of Qu= 0 such that ψ ≤ u≤ φ and u−χ∈W₀^1,2(G).

The sub and super solutions are linked to the two problems in whichpand qare bigger than 1, and µandω are Radon measures

−∆v+|v|^p−1v=µ in Ω

v= 0 in ∂Ω, (3.4)

and

−∆w−M|∇w|^q =ω in Ω

w= 0 in∂Ω. (3.5)

It is proved in [3, Theorem 4.1] that Problem (3.4) admits a solution,v∈L^p(Ω), necessarily unique, if and only ifµis absolutely continuous with respect to the Bessel capacitycap^R_2,p^N0, that is

For any compact setE⊂Ω, cap^R_2,p^N0(E) = 0 =⇒ |µ|(E) = 0. (3.6) Concerning (3.5), from [21, Theorem 1.9] a sufficient condition for solvability is the estimate

For any compact setE⊂Ω, |ω|(E)≤Ccap^R_1,q^N0(E), (3.7) for some C > 0. When µis nonnegative and has compact support in Ω, this condition turns out to be necessary. If (3.7) is satisfied there exists₀>0 such that (3.5) admits a solution withωreplaced byωwith 0< ≤₀. Furthermore

∇w∈L^q(Ω) and the following estimates hold [9, Theorem 1.2],

|∇w(x)| ≤c13I^2R₁ [|ω|](x), (3.8) at least if ω has compact support or is a smooth function, and, with no such conditions onµ,

|w(x)| ≤c14G^Ω[|ω|](x), (3.9) withc13, c14depending onN andq, whereI^2R₁ is the truncated Riesz potential in R^N defined for any measureµby

I^2R₁ [µ](x) = Z 2R

0

µ(Bρ(x)) ρ^N−1

dρ

ρ for all x∈R^N, (3.10) andG^Ωthe Green potential in Ω. IfR=∞we denote byI1:=I^∞₁ the classical Riesz potential and if Ω = R^N the role of G^Ω is played by the Newtownian potentialI₂. We start with the following easy result:

Lemma 3.2 Let r >1,k∈N^∗ andµ∈M₊(Ω). Ifµ∈W^−k,r(Ω) is nonnega- tive, then there exists C >0 such that

µ(E)≤C

cap^Ω_k,r0(E)_r¹0

for any compact setE⊂Ω. (3.11) wherer⁰=_r−1^r . Conversely when k= 1,2andµ satisfies

µ(E)≤Ccap^Ω_k,r0(E) for all compact setE ⊂Ω, (3.12) for someC >0, then,

(i) if k= 2 thenµ∈W^−2,r(Ω), (ii) ifk= 1then µ∈W^−1,r(Ω).

Proof. Assume first that µ ∈M₊(Ω)∩W^−k,r(Ω). Let ζ ∈C₀^∞(Ω) such that 0≤ζ≤1 andζ≥χ_K. Then

µ(E)≤ Z

Ω

ζdµ=hµ, ζi ≤ kζk_Wk,r0 0

kµk_W−k,r.

By the definition of capacity

µ(K)≤ kµk_W−k,r cap^Ω_k,rp0(K)_r¹0

.

Conversely if (3.12) holds with k = 2 there exists 0 > 0 such that for every ∈(0, 0], there existsz∈L^r(Ω) satisfying

−∆z=z^r+µ in Ω

z= 0 on∂Ω, (3.13)

(see [24, Theorem 2.10, Remark 2.11]). Since z ≥ G^Ω[µ], it follows that G^Ω[µ]∈L^r(Ω) and thereforeµ∈W^−2,r(Ω). SinceG^Ωis an isomorphism from L^r0(Ω) intoW^2,r0(Ω)∩W₀^1,r0(Ω), we infer by duality thatG^Ωis an isomorphism fromW^−2,r(Ω) intoL^r(Ω). Henceµ∈W^−2,r(Ω).

Finally, if (3.12) holds with k = 1, then there exists 0 >0 such that for every ∈(0, 0] there existsz∈W^1,r(Ω) satisfying

−∆z=|∇z|^r+µ in Ω

z= 0 on∂Ω. (3.14)

Thenz satisfiesz≥G^Ω[µ]. Sincez∈L^r∗(Ω) by Sobolev imbedding theorem, we have thatG^Ω[µ]∈L^r∗(Ω), which implies the claim.

Proof of the theorem. We putµn =µ∗ηn where{ηn} ⊂C₀^∞(R^N) is a sequence of mollifiers with supp(ηn)⊂B1

n, and we denote by vn the solution of

−∆v+v^p=µ_nχ_Ω in Ω

v= 0 on∂Ω. (3.15)

Since µ satisfies (3.12) so does µ_n with the same constant C. Hence µ_n ∈ W^−2,p(Ω) and µ_n → µ in W^−2,p(Ω) as n → ∞. We also denote by z_n a nonnegative solution of

−∆z=z^p+µ_nχ_Ω in Ω

z= 0 on∂Ω, (3.16)