Nonlinear elliptic equations with measure valued absorption potentials

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Nonlinear elliptic equations with measure valued absorption potentials

Nicolas Saintier, Laurent Veron

To cite this version:

Nicolas Saintier, Laurent Veron. Nonlinear elliptic equations with measure valued absorption potentials. Annali della Scuola Normale Superiore di Pisa, inPress. �hal-02368308�

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Nonlinear elliptic equations with measure valued absorption potentials

Nicolas Saintier^∗ Laurent V´eron^†

AbstractWe study the semilinear elliptic equation−∆u+g(u)σ=µwith Dirichlet boundary conditions in a smooth bounded domain where σ is a nonnegative Radon measure, µ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that σ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) =|r|^q−1r, with q > 1 and µ satisfies an absolute continuity condition expressed in terms of some capacities involvingσ.

2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05.

Key words: Radon measures; Morrey class; capacities; potential estimates;θ-regular measures.

1 Introduction

Let Ω⊂R^N be a bounded domain with aC²boundary,σa nonnegative Radon measure in Ω andg:R→Ra continuous function satisfying, for somer0≥0,

rg(r)≥0 for allr∈(−∞,−r0]∪[r0,∞). (1.1) In this article we consider the following problem

−∆u+g(u)σ=µ in Ω

u= 0 in ∂Ω, (1.2)

whereµis a Radon measure defined in Ω. By a solution we mean a functionu∈L¹(Ω) such thatρg(u)∈L¹_σ(Ω), whereρ(x) = dist (x, ∂Ω) andL¹_σ(Ω) is the Lebesgue space of functions integrable with respect toσ, satisfying

− Z

Ω

u∆ζdx+ Z

Ω

g(u)ζdσ= Z

Ω

ζdµ, (1.3)

for allζ ∈W₀^1,∞(Ω) such that ∆ζ ∈L^∞(Ω). In the sequel, such a solution is called avery weak solution. A measureµsuch that the problem admits a solution is called agood measure.

We emphasize on the particular cases where g(r) =|r|^q−1r with q >0, org(r) =e^αr−1 withα >0 andN = 2.

Whenσis a measure with constant positive density with respect to the Lebesgue measure in R^N, this problem has been initiated by Brezis and Benilan [4], [5] who gave a general existence result for any bounded measureµunder an integrability condition ofgat infinity;

their proof is based upon an a priori estimate of approximate solutionsun in Lorentz spaces L^q,∞(Ω), yielding the uniform integrability ofg(un) and hence the pre-compactness inL¹(Ω).

Ifg(r) =|r|^q−1r, integrability condition is fufilled if and only if 0< q < _N−2^N (anyq >0 if N = 2). In the 2-dim case the integrability condition have been replaced by the exponential order of growth ofgin [27]. Wheng(u) =|u|^q−1uwithq≥ _N−2^N not any bounded measure is eligible for solving (1.2). In fact Baras and Pierre [3] proved that whenN >2 andq >1, a bounded Radon measureµis eligible if and only if it vanishes on Borel sets withc2,q⁰-capacity zero, where q⁰ = _q−1^q is the conjugate exponent of q. Contrary to the previous subcritical case, the method for proving the necessity of this condition is based upon a duality-convexity argument, while the sufficiency uses the fact that any positive Radon measure absolutely continuous with respect to the c2,q⁰-capacity can be approximated from below by an nondecreasing sequence of positive measures in W^−2,q(Ω) (see [13]). Furthermore they also

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gave a necessary and sufficient condition for a compact subset K⊂Ω to be removable for equation

−∆u+|u|^q−1u= 0 in Ω\K, (1.4)

namely that c2,q⁰(K) = 0.

The aim of this paper is to extend the previous constructions of Benilan-Brezis, Baras- Pierre and Vazquez to the case where σ is a general measure. In order to be able to deal with the convergence of approximate solutions we assume thatσbelongs to the Morrey class M⁺N

N−θ

(Ω) for someθ∈[0, N] which means

|B_r(x)|_σ :=

Z

Br(x)

dσ≤cr^θ for all (x, r)∈Ω×(0,∞), (1.5) for some c >0. Note that we extend σ by 0 in R^N\Ω and slightly abuse notation putting _N^N_−θ =∞ when θ=N.

Our first result is the following:

Theorem A Assume σ ∈ M⁺_N

N−θ

(Ω) for some θ ∈ (N −2, N] and that g satisfies (1.1). Then, for anyµ∈L¹_ρ(Ω), there exists a very weak solutionuof problem(1.3).

If we assume moreover that g is nondecreasing and if u⁰ is a very weak solution of (1.3) with right-hand side µ⁰ ∈L¹_ρ(Ω), then the following estimates hold

− Z

Ω

|u−u⁰|∆ζdx+ Z

Ω

|g(u)−g(u⁰)|ζdσ≤ Z

Ω

|µ−µ⁰|dx, (1.6) and

− Z

Ω

(u−u⁰)+∆ζdx+ Z

Ω

(g(u)−g(u⁰))+ζdσ ≤ Z

Ω

(µ−µ⁰)+dx (1.7) for all ζ ∈W₀^1,∞(Ω) such that∆ζ ∈L^∞(Ω)and ζ ≥0.

Note that (1.6) implies the uniqueness of the solution of (1.3), that we denote by u_µ, and (1.7) the monotonicity of the mappingµ7→u_µ.

The next result extends Benilan-Brezis unconditional existence result for measures.

Theorem B Let N > 2 and σ ∈ M⁺_N

N−θ

(Ω) with N ≥ θ > N − _N−1^N . Assume that g satisfies (1.1) and |g(r)| ≤ ˜g(|r|) for all |r| ≥ r0 where g˜ is a continuous nondecreasing function on [r₀,∞) verifying

Z ∞ r0

˜

g(t)t⁻¹⁻^N−2^θ dt <∞. (1.8)

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Then, for any bounded Radon measure µ, there exists a very weak solution u of problem (1.3) which moreover belongs to L¹_σ(Ω). Moreover, if we assume that g is nondecreasing then the solution is unique.

Note that we recover Benilan-Brezis result whenσ is the Lebesgue measure (so that θ = N). Note also that when g(r) = |r|^q−1r, the integrability condition (1.8) is fullfilled if and only if 0< q < _N−2^θ .

In the 2-dimensional case the condition on θ is 2 ≥ θ > 0 but (1.8) has to be modified. If f :R7→R+ is nondecreasing we define its exponential order of growth at∞ (see [27]) by

a∞(f) = inf

α≥0 : Z ∞

0

f(s)e^−αsds <∞

. (1.9)

Similarly, if h:R7→R− is nondecreasing its exponential order of growth at−∞ is a-∞(h) = sup

α≤0 :

Z 0

−∞

h(s)e^αsds >−∞

. (1.10)

If g : R 7→ R satisfies (1.1) but is not necessarily nondecreasing, we define the monotone nondecreasing hull g^∗ ofg by

g^∗(r) =







sup{g(s) :s≤r} for all r≥r₀ 0 for all r∈(−r₀, r0) inf{g(s) :s≥r} for all r≤ −r₀.

(1.11) We set

a∞(g) =a∞(g₊^∗) and a-∞(g) =a-∞(g^∗₋). (1.12) Theorem C Let σ ∈ M⁺₂

2−θ

(Ω)with 2≥θ >0 and g:R7→R satisfies(1.1).

(I) If a∞(g) = 0 = a-∞(g), then for any µ ∈M_b(Ω), problem (1.3) admits a very weak solution.

(II) If 0 < a∞(g) < ∞ and −∞ < a-∞(g) < 0 there exists δ > 0 such that if µ∈M_b(Ω) satisfieskµk_M

b ≤δ problem (1.3) admits a very weak solution.

In thesupercritical case, that is when (1.8) is not satisfied, all the measures are not eligible for solving (1.3). Following [16], [28, Th 4.2 ] we can give a sufficient existence condition involving the Green function of the Laplacian. LetG(., .) be the Green kernel defined in Ω×Ω andG[.] the corresponding potential operator acting on bounded measures ν namely G[ν](x) = R

ΩG(x, y)dν(y). We have the following result:

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Theorem D Let σ ∈ M⁺_N

N−θ

(Ω) with N ≥ θ > N − _N−1^N and assume that g is nondecreasing and vanishes at 0.

(I) If µ∈M_b(Ω) satisfies

ρg(G[|µ|])∈L¹_σ(Ω), (1.13)

then problem (1.3)admits a unique very weak solution.

(II) Let µ=µ_r+µ_s where µ_r is absolutely continuous with respect to the Lebesgue measure and µ_s is singular. Assume that g satisfies the ∆₂ condition, namely that

|g(r+r⁰)| ≤a(|g(r)|+|g(r⁰)|) +b for all r, r⁰ ∈R, (1.14) for some a >1 and b≥0. Then the previous assertion holds if(1.13)is replaced by ρg(G[|µ_s|])∈L¹_σ(Ω). (1.15)

Notice that (1.13) holds if either (i)σandµhave disjoint support, or (ii)µ∈ M_p(Ω) for somep > ^N₂. Indeed if (i) holds then G[|µ|] is bounded pointwise on the support ofσ, and if (ii) holds then by Lemma 2.2G[|µ|] is bounded pointwise in Ω. Obviously the same comment holds in the setting of II.

In order to make more explicit conditions (1.13), (1.15), we introduce the following growth assumption on g:

|g(r)| ≤c(1 +|r|^q) for all r∈R, (1.16) for some q > 1. Notice that ˜g(r) = 1 +r^q satisfies (1.8) if and only if q < _N−2^θ . When σ is the Lebesgue measure and g(r) = |r|^q−1r, Baras and Pierre [3] gave a necessary and sufficient condition for the existence of a solution to (1.2) involving certain capacities associated to the Bessel potential spaces H^s,p(R^N) where s ∈ R and p∈[1,∞]. Let us recall that

H^s,p(R^N) =

f :f =Gs∗h, h∈L^p(R^N) , (1.17) where G_s is the Bessel kernel of orders. By extensionG₀ =δ₀, hence H^s,p(R^N) = L^p(R^N). When s is a positive integer, it is proved by Calder´on [2, Theorem 1.2.3]

that H^s,p(R^N) is the standard Sobolev space W^s,p(R^N). Ifs >0, we denote byc_s,p the associated capacity, called the Bessel capacity. It is defined for any compact set K ⊂R^N by

c_s,p(K) = inf{kφk^p_Hs,p : φ∈ S(R^N), φ≥1 onK}. (1.18) The definition of c_s,p is then extended first to open sets and then to arbitrary sets.

We refer to [2] for general properties of Bessel spaces and their associated capacities

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c_s,p. We say that a measureµ∈M_b(Ω) is absolutely continuous with respect to the cs,p-capacity if for any Borel subsetE ⊂R^N,

cs,p(E) = 0 =⇒ |µ|(E) = 0.

Baras and Pierre’s result states that equation (1.2), withσstanding for the Lebesgue measure and g(r) =|r|^q−1r, has a solution if and only if µis absolutely continuous with respect to the c_2,q⁰-capacity. The next result generalizes the ”if” part to the case where σ belongs to some Morrey space.

Theorem E Let σ ∈ M⁺_N

N−θ

(Ω) with N ≥ θ > N − _N^N₋₁ and assume that g is nondecreasing and satisfies (1.1) and (1.16). Let p > 1 and s ≥ 0 such that N >

sp > N−θand _N−sp^θp ≥q. Ifµ∈M_b(Ω)is absolutely continuous with respect to the c2−s,p⁰-capacity, then (1.2) admits a unique very weak solution.

As a particular case, we take p = q and obtain that if µ is absolutely continuous with respect to thec₂₋^N−θ

q ,q⁰-capacity, then (1.3) admits a unique solution. We thus recover Baras-Pierre’s sufficient condition [3] when θ=N.

We give an explicit condition on the measure µin terms of Morrey spaces implying that it satisfies the conditions of Theorem E.

Proposition 1.1 Under the assumptions onσandgof Theorem E, ifµ∈ M N N−θ∗(Ω) for some θ^∗ > ^{(N−2)q−θ}_q−1 , then (1.3)admits a unique very weak solution.

Notice that the condition onµgiven in Proposition 1.1 is weaker than the one given after Theorem D.

When g(r) = |r|^q−1r with q > 1, one can find a necessary conditions for the existence of a solution of (1.3) in the supercritical case under additional regularity assumptions on σ. By [2, Def 2.3.3, Prop. 2.3.5], the following expression

c^σ_q(E) = inf Z

Ω

|v|^q⁰dσ:v∈L^q_σ⁰(Ω), v≥0,G[vσ]≥1 onE

, (1.19)

whereE is any subset of Ω defines an outer capacity. The measure is calledθ-regular

if 1

cr^θ≤ Z

Br(x)

dσ≤cr^θ for all (x, r)∈Ω×(0,1],

The next result gives a necessary condition for a measure to be a good measure.

Theorem F Let q > 1 and σ ∈ M⁺_N

N−θ

(Ω) be θ-regular with N ≥ θ > N −2.

If µ ∈ M⁺_b(Ω) is such that problem (1.3) with g(r) = |r|^q−1r admits a very weak solution, then µvanishes on any Borel set E such that c^σ_q(E) = 0.

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Furthermore the c^σ_q- capacity admits the following representation in terms of Besov capacities. If Γ⊂Ω is the support ofσ, we denote byB²⁻

N−θ q ,Γ

q⁰,∞ (Ω) the closed subspace of distributions ζ ∈B²⁻

N−θ q

q⁰,∞ (Ω) such that the support of the distribution

∆ζ is a subset of Γ. Then c^σ_q(K)∼c²⁻

N−θ q ,Γ

q⁰,∞ (K) := inf





 kζk^q⁰

B

2−N−θ q q0,∞

:ζ ∈B²⁻

N−θ q ,Γ

q⁰,∞ (Ω), ζ ≥χ_K







, (1.20) for all compact subset K⊂Ω.

Finally a complete characterization of removable sets can be obtained under a much stronger assumption onσ, namely thatdσ=wdxwithω:=w⁻^q−1¹ ∈L¹_loc(Ω).

If K⊂Ω is compact, we set

c^ω_q(K) = inf Z

Ω

|∆ζ|^q⁰ωdx:ζ ∈C₀^∞(Ω),0≤ζ ≤1, ζ = 1 in a neighborhood ofK

. (1.21) This defines a capacity on Borel sets of Ω.

Theorem G.Assumeq >1and there exists a nonnegative Borel functionwin Ωin the Muckenhoupt class A_q(Ω)such that dσ=wdx. IfK ⊂Ωis compact, a function u∈L¹_loc(Ω\K) such that |u|^qw∈L¹_loc(Ω\K) which satisfies

−∆u+w|u|^q−1u= 0, (1.22)

in the sense of distributions in Ω\K can be extended as a solution of the same equation in the whole Ω if and only if cq,w(K) = 0.

The assumption w ∈ Aq(Ω) can be weakened and replaced by ω =w^1−q¹ is q⁰- admissible in the sense of [15, Chap 1], a condition which implies in particular the validity of the Gagliardo-Nirenberg and the Poincar´e inequalities.

2 Preliminaries

In the whole paper c denotes a generic positive constant whose value can change from one ocurrence to another even within a single string of estimates. Sometimes, in order to avoid ambiguity, we are led to introduce other notations for constant, for examplec⁰.

We denote byM_b(Ω) the space of outer regular bounded Borel measures on Ω equipped with the total variation norm, and byM⁺_b (Ω) its positive cone. Since Ω is

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bounded we can identify bounded Radon measures in Ω with measures µin Ω such that |µ|(∂Ω) = 0. All the measures are extended by 0 in R^N\Ω.

Let G(., .) be the Green kernel defined in Ω×Ω and G[.] the corresponding potential operator acting on bounded measuresνnamelyG[ν](x) =R

ΩG(x, y)dν(y).

We denote L^p,∞(Ω) the usual weak L^p space. The next result is classical and valid in a much more general setting (see e.g. [6], [11]).

Lemma 2.1 Let µ∈M_b(Ω)and v=G[µ] be the (very weak) solution of

−∆v =µ in Ω

v = 0 in ∂Ω. (2.1)

I- If N ≥2, then v∈L^N−2^N ^,∞(Ω), ∇v∈L^N−1^N ^,∞(Ω)and kvk

L^N−2^N ^,∞+k∇vk

L^N−1^N ^,∞ ≤ckµk_M

b. (2.2)

II- If N = 2, then v∈BM O(Ω), ∇v∈L^2,∞(Ω) and kvk_{BM O}+k∇vk_L2,∞ ≤ckµk_M

b. (2.3)

This result can be refined when more information is available on the degree of concentration ofµ. This leads to the definition of Morrey spaces of measures.

2.1 Morrey spaces of measures

If 1≤p≤ ∞we define the Morrey spaceM_p(Ω) as the set of bounded outer regular Borel measures µdefined in Ω and extended by 0 in Ω^c, satisfying

|B_r(x)|_µ:=

Z

Br(x)

d|µ| ≤cr^N⁽¹⁻¹^p⁾ for all (x, r)∈Ω×R+, (2.4) for somec >0. In particularµ∈ M N

N−θ(Ω),θ∈[0, N], if Z

Br(x)

d|µ| ≤cr^θ for all (x, r)∈Ω×R+.

We refer to [19] for a detailed study ofM_p(Ω) and full proofs of the various results we will recall now. Endowed with the norm

kµk_M

p= sup

(x,r)∈Ω×R+

r^N⁽¹^p⁻¹⁾|B_r(x)|_µ, (2.5)

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M_p(Ω) is a Banach space and M⁺_p(Ω) = M_p(Ω)∩M⁺_b (Ω) is its positive cone.

We also set Mp(Ω) = M_p(Ω)∩L¹_loc(Ω); it is a closed subspace of M_p(Ω) and, if 1< p <∞, the following imbedding holds

L^p(Ω),→L^p,∞(Ω),→Mp(Ω). (2.6) Note that since Ω is bounded and any measure in Ω is extended toR^N by 0, it is easily seen that if 1≤q ≤p≤ ∞we have a continuous embeddingM_p(Ω),→ M_q(Ω) with

kvk_M

q ≤(diam(Ω))^N^q⁻^N^p kvk_M

p for all v∈ M_p(Ω). (2.7) Indeed for any x∈Ω the ball centered atx with radius diam(Ω) contains Ω so that it is enough to consider r ≤diam(Ω). We have

r^−N^(1−1/q)|B_r(x)|_µ≤r^{−N(1−1/q)}kµk_M

pr^N^(1−1/p)≤(diam(Ω))^N^q⁻^N^p kµk_M

p. The following imbedding inequalities holds.

Lemma 2.2 Let µ∈ M_p(Ω) andv be the solution of(2.1).

I- If 1< p < ^N₂, then v∈M_q(Ω)with ¹_q = ¹_p −_N² and there holds kvk_M

q ≤ckµk_M

p. (2.8)

II- If p > ^N₂, then v is bounded pointwise and

(i) v(x)≤ckµk_M

p for all x∈Ω,

(ii) sup

x6=y

|v(x)−v(y)|

|x−y|^α ≤ckµk_M

p with α= 2−N

p if N > p > N 2, (iii) sup

x6=y

|v(x)−v(y)|

|x−y|^α ≤ckµk_M

p with α∈(0,1) if N =p,

(iv) sup

x

|∇v(x)| ≤ckµk_M

p if N < p.

(2.9) Remark. The previous regularity results are proved in [19, Prop. 3.1, 3.5] when v = I_α ∗ µ where I_α is the Riesz potential. However it is easily seen that the proof in [19] can be adapted to our setting. In particular for (2.8) we need that G(x, y)≤c|x−y|^2−N, for (i) we use (2.7).

Remark. If we assume thatµ∈M_ρ(Ω)∩ M_p,loc(Ω), the previous estimates acquire a local aspect and remain valid provided the supremum in the norms on the left-hand sides are taken on compact subsets of Ω.

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2.2 Trace embeddings

Some applications of Morrey spaces to imbedding theorems (also called trace inequalities) can be found in Adams-Hedberg’s book [2]. For the sake of completeness, we quote here the main result therein we will use in the sequel. If 0< α < N we recall that I_α (resp. G_α) is the Riesz potential (resp. the Bessel potential) of orderα in R^N. The next result is [2, Th 7.2.2, 7.3.2 ] (recall that thec_Iα_,p-Riesz capacity of a ball Br(x) is proportional tor^N^−αp- see [2, Prop. 5.1.2].)

Proposition 2.3 Let σ be a nonnegative Radon measure in R^N, N > αpand 1< p < q < _N−αp^{N p} .

(I)- The following assertions are equivalent:

kI_α∗fk_L^q

σ(R^N)≤c₁kfk_Lp(R^N) for all f ∈L^p(R^N), (2.10) for some c1 =c1(N, α, p, q)>0, and

σ ∈ M_r(R^N) with 1 r =q

1 q −1

p + α N

. (2.11)

(II)- The mapping f 7→G_α∗f is continuous from L^p(R^N) toL^q_σ(R^N) if and only if σ(K)¹^q ≤c₂(c_α,p(K))¹^p for all K⊂R^N, (2.12) where cα,p denotes the Bessel capacity of orderα defined in(1.18). In fact this holds if and only if

σ(B_r(x))≤c₃(c_α,p(B_r(x)))^q/p for all x∈R^N,0< r≤1. (2.13) (III)- A necessary and sufficient condition in order the mapping f 7→ Gα ∗f be compact from L^p(R^N) to L^qσ(R^N) is

(i) lim

δ→0 sup

x∈R^N, r≤δ

σ(B_r(x)) (c_α,p(B_r(x)))^q^p

= 0 (ii) lim

|x|→∞sup

r≤1

σ(B_r(x)) (c_α,p(B_r(x)))^q^p

= 0.

(2.14)

If R^N is replaced by a smooth bounded set Ω, we extend any bounded Radon measure in Ω by zero in Ω^c. In view of [2, 5.6.1] the c_Iα_,p-Riesz capacity andc_α,p- Bessel capacity of balls Br(x) withx∈Ω and r≤1 are then equivalent. It follows that c_α,p(B_r(x))'r^N−αp. Then, it follows from II and III above, the definition of H^α,p(R^N) and the existence of an extension operatorH^α,p(Ω),→H^α,p(R^N) that the following holds,

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Proposition 2.4 Under the assumptions of Proposition 2.3, the embeddingH^α,p(Ω),→ L^qσ(Ω)is:

(I)- continuous if and only if (σ(K))¹^q ≤c₂(c_α,p(K))¹^p for all K ⊂R^N, i.e. if and only if σ∈ M⁺_r(R^N) with ¹_r =q

1

q −¹_p +_N^α . (II)- compact if and only if

r→0limsup

x∈Ω

σ(Br(x)) r

(N−αp)q p

= 0. (2.15)

As an immediate corollary, Proposition 2.5 Let σ ∈ M⁺_N

N−θ

(Ω), i.e. σ(B_r(x)) ≤ cr^θ, N > αp and 1 < p <

q < _N−αp^{N p} . Then the embedding

H^α,p(Ω),→L^q_σ(Ω), (2.16)

is continuous iff σ(K) ≤ c1(cα,p(K))

q

p for all K ⊂ R^N which holds iff q ≤ _N−αp^θp . And the embedding (2.16) is compact iff q < _N−αp^θp .

Other trace inequalities can be found in [21]. In the caseN =αpthe following estimate holds, see e.g. [1], [20, Corollary 8.6.2], [31].

Proposition 2.6 Let σ be a nonnegative Radon measure in R^N with compact support and N = αp, p > 1. Then there exists a constant b = b(N, α, p) > 0 such that

sup

kfk_Lp≤1

Z

R^N

exp

b|G_α∗f|^p⁰

dσ <∞ (2.17)

if and only if σ ∈ M⁺_τ(R^N) for some τ ∈(1,∞).

Whenp= 1 the next result is proved in [20, Sec 1.4.3]

Proposition 2.7 Let σ be a nonnegative bounded Radon measure in R^N, α be an integer such that 1≤α≤N and q≥1. Then the following estimate holds

kfk_L^q

σ ≤c2

X

|β|=α

kD^αfk₁ for allf ∈C₀^∞(R^N), (2.18)

for some c2 =c2(N, p, q, α)>0 if and only if σ∈ M⁺ _N

N−q(N−α)

(R^N).

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3 The subcritical case

3.1 The variational construction

We prove in this section that if µ∈W^−1,2(Ω) then, under some assumptions on g and σ, equation (1.2) has a variational solution.

We assume that g ∈ C(R) satisfies (1.1), and set G(r) :=

Z r 0

g(s)ds. We will find a solution to (1.2) minimizing the functional

J(v) := 1 2 Z

Ω

|∇v|²dx+ Z

Ω

G(v)dσ− hµ, vi, (3.1) over the set

X_G(Ω) :={v∈W₀^1,2(Ω) :G(v)∈L¹_σ(Ω)}. (3.2) The next proposition is a variant of a result in [8].

Proposition 3.1 Assume σ ∈ M⁺_N

N−θ

(Ω) with N ≥ θ > ^N₂ −1. If µ ∈W^−1,2(Ω) there exists u ∈ XG(Ω) which minimizes J in XG(Ω). Furthermore u is a weak solution of (1.2) in the sense that

Z

Ω

∇u.∇ζdx+ Z

Ω

g(u)ζdσ=hµ, ζi for all ζ ∈C₀^∞(Ω). (3.3) If g is nondecreasing this solution is unique and denoted by u_µ, and the mapping µ7→uµ is nonnecreasing.

Proof. Step 1: Existence of a minimizer. If N >2 we apply (2.16) withα = 1 and p= 2, recalling that by Fourier transformH^1,2(Ω) =W^1,2(Ω) (it is a special case of Calder´on’s theorem), to obtain that

W₀^1,2(Ω),→L

2θ N−2

σ (Ω). (3.4)

If N = 2 withp= 2 we take anyα <1 and obtain kfk

L

θ σ1−α

≤c1kfk_Wα,2 ≤c⁰₁kfk_W1,2. (3.5) According to Proposition 2.5 the imbedding of W₀^1,2(Ω) into L^pσ(Ω) is compact for any p∈[1,_N^2θ₋₂) ifN >2 and 1≤p <∞ifN = 2.

Let us first assume thatgis bounded. Then|G(v)| ≤m|v|. Sincegis continuous, G(v) ∈ L¹_σ(Ω) for any v ∈ W₀^1,2(Ω) and the functional J is well defined and is of class C¹ inW₀^1,2(Ω). Furthermore

kvklim

W1,2→∞J(v) = +∞. (3.6)

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Let {u_n} be a minimizing sequence. By (3.6), {u_n} is bounded in W₀^1,2(Ω) and thus relatively compact in L¹_σ(Ω) and in L²(Ω). Hence there exist u ∈ L²(Ω) and v ∈L¹_σ(Ω) such that, up to a subsequence, un→ v inL¹_σ(Ω), and un → u strongly in L²(Ω) and weakly in W₀^1,2(Ω). We can also assume that u_n → u c_1,2-quasi almost everywhere in the sense that there exists E ⊂ Ω with c1,2(E) = 0 such that un(x)→u(x) for anyx ∈Ω\E. According to Proposition 2.5, σ is absolutely continuous with respect to thec_1,2-capacity. It follows thatσ(E) = 0 so thatu_n→u σ-almost everywhere and thus u = v σ-almost everywhere. Thus we have that un→u inL²(Ω), in L¹_σ(Ω), σ-almost everywhere and weakly in W₀^1,2(Ω). Then we have that hµ, u_ni → hµ, ui. By the dominated convergence theorem we have also that G(un)→G(u) in L¹_σ(Ω). Therefore

J(u)≤lim inf

n→∞ J(u_n), (3.7)

which implies that u is a minimizer ofJ inW₀^1,2(Ω).

Ifgis unbounded, we writeg=g1+g2whereg1=gχ_(−r₀_,r₀₎,g2 =gχ_(−∞−r₀_]∪[r₀_,∞), wherer₀ is defined in (1.1). HenceG(r) =G₁(r) +G₂(r) where |G₁(r)| ≤m|r|and G₂(r) is nonnegative. Using again (2.14) we obtain that (3.6) holds. A minimizing sequence {u_n} inherits the same property as above, hence un →u σ-almost everywhere in Ω and in L¹_σ(Ω), this implies thatG₁(u_n)→G₁(u) inL¹_σ(Ω) and G₂(u) is σ-measurable. By Fatou’s lemma

Z

G2(u)dσ≤lim inf

n→∞

Z

G2(un)dσ,

which implies that (3.7) holds. Notice that, among the consequences, X_G is closed subset of W₀^1,2(Ω). Henceu in a minimizer ofJ in X_G(Ω).

Uniqueness holds if g is nondecreasing since it implies that J is stricly convex and actually XG is a closed convex set.

Step 2: The minimizer is a weak solution. Fork > r₀ we defineg_k by g_k(r) =







g(r) if |r| ≤k g(k) if r > k g(−k) if r <−k

Then g_k is continuous and bounded and the minimizeru_k∈W₀^1,2(Ω) of J_k(v) = 1

2 Z

Ω

|∇v|²dx+ Z

Ω

G_k(v)dσ− hµ, vi where G_k(r) = Z s

0

g_k(s)ds, is a weak solution (i.e. in the sense given by (3.3)) of

−∆u+gk(u)σ=µ in Ω

u= 0 on∂Ω. (3.8)

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The following energy estimate holds Z

Ω

|∇u_k|²dx+ Z

Ω

u_kg_k(u_k)dσ=hµ, u_ki ≤ kµk_W−1,2ku_kk_W1,2, (3.9) and it implies

Z

Ω

|∇u_k|²dx+ Z

Ω

|u_kg_k(u_k)|dσ≤ kµk²_W−1,2 +mσ(Ω) =M, (3.10) for some m = m(r0) > 0. Up to a subsequence, {u_k}_k converges to some u as k → ∞, weakly in W₀^1,2(Ω), strongly in L²(Ω), and almost everywhere in Ω. By Proposition 2.4 the imbedding of W^1,2(Ω) in L^qσ(Ω) is compact for any q < _N−2^2θ . Hence the subsequence can be taken such that u_k → u, σ-almost everywhere as k → ∞, and consequently g_k(u_k) → g(u) σ-almost everywhere. Let E ⊂ Ω be a Borel set, then for any λ > r₀,

M ≥ Z

E

|g_k(uk)uk|dσ

= Z

E∩{|u_k|>λ}

|g_k(u_k)u_k|dσ+ Z

E∩{|u_k|≤λ}

|g_k(u_k)u_k|dσ

≥λ Z

E∩{|uk|>λ}

|g_k(u_k)|dσ+ Z

E∩{|uk|≤λ}

|g_k(u_k)u_k|dσ.

Therefore Z

E

|g_k(u_k)|dσ= Z

E∩{|u_k|>λ}

|g_k(u_k)|dσ+ Z

E∩{|u_k|≤λ}

|g_k(u_k)|dσ

≤ M

λ + max{|g(r)|: |r| ≤λ}σ(E)

For >0 we first chooseλsuch that ^M_λ ≤ ₂ and thenσ(E)≤ 1+2 max{|g(r)|≤λ} . This implies the uniform integrability of {g_k(u_k)}_k in L¹_σ(Ω). Hence g_k(u_k) → g(u) in L¹_σ(Ω) by Vitali’s convergence theorem. Since uk is a weak solution of (3.8), there holds for any ζ ∈C₀^∞(Ω),

Z

Ω

∇u_k.∇ζdx+ Z

Ω

g_k(u_k)ζdσ =hµ, ζi. (3.11) Letting k→ ∞ we obtain, using the above convergence results,

− Z

Ω

∇u.∇ζdx+ Z

Ω

g(u)ζdσ=hµ, ζi. (3.12) Hence u is a weak solution. If g is monotone, uniqueness is also a consequence of the weak formulation. Furthermore ifµ, µ⁰ belong toW^−1,2(Ω) are such thatµ−µ⁰ is a nonnegative measure, thenhµ⁰−µ,(u⁰_µ−u_µ)₊i ≤0. Taking (u⁰_µ−u_µ)₊for test function in the weak formulation yields (u⁰_µ−uµ)+= 0.

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3.2 The L¹ case In the sequel we set

X(Ω) ={ζ ∈C¹(Ω), ζ = 0 on∂Ω and ∆ζ ∈L^∞(Ω)}, (3.13) and X+(Ω) = X(Ω)∩ {ζ ∈ C¹(Ω) : ζ ≥ 0 in Ω}. We recall (see e.g. [29]) that if f ∈L¹_ρ(Ω) and u∈L¹(Ω) is a very weak solution of

−∆u=f in Ω, (3.14)

there holds

− Z

Ω

|u|∆ζdx≤ Z

Ω

fsign(u)ζdx for all ζ∈X+(Ω), (3.15) and

− Z

Ω

u⁺∆ζdx≤ Z

Ω

fsign₊(u)ζdx for all ζ ∈X+(Ω). (3.16) Proposition 3.2 Assume N ≥2, σ∈ M⁺_N

N−θ

(Ω)withN ≥θ > N−2 and g:R7→

Ris a continuous nondecreasing function vanishing at0. Ifµ∈L¹_ρ(Ω)there exists a uniqueu:=uµ∈L¹(Ω)very weak solution of(1.2). Furthermore, ifuµ, uµ⁰ ∈L¹(Ω) are the very weak solutions of (1.2) with right-hand sides µ, µ⁰∈L¹_ρ(Ω), then

− Z

Ω

u_µ−u_µ⁰

∆ζdx+ Z

Ω

g(u_µ)−g(u_µ⁰) ζdσ ≤

Z

Ω

(µ−µ⁰)sign(u_µ−u_µ⁰)ζdx, (3.17) and

− Z

Ω

(u_µ−u_µ⁰)₊∆ζdx+ Z

Ω

(g(u_µ)−g(u_µ⁰))₊ζdσ ≤ Z

Ω

(µ−µ⁰)sign₊(u_µ−u_µ⁰)ζdx (3.18) for any ζ ∈X+(Ω). In particular the mapping µ→u_µ is nondecreasing.

The following result will be used several time in the sequel. Its proof is standard but we present it for the sake of completeness.

Lemma 3.3 Assume N > q ≥ 1 and σ ∈ M⁺_N

N−θ

with N ≥ θ > N −q. Then σ vanishes on any Borel set with c1,q-capacity zero.

Proof. It suffices to prove the result whenEis compact. We define the Λ_θHausdorff measure of a set E by

Λθ(E) = lim

κ→0Λ^κ_θ(E) := lim

κ→0 inf







∞

X

j=1

r^θ_j : 0< rj ≤κ≤ ∞, E ⊂

∞

[

j=1

Brj(aj)





 .

(3.19)

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Note that Λ^∞_θ (E) is the Hausdorff content of E and it is smaller than (diam (E))^θ. For any covering ofE by ballsBrj(aj),j ≥1, we have

σ(E)≤

∞

X

j=1

σ(Brj(aj))≤ kσk N N−θ

∞

X

j=1

r^θ_j.

It follows that

σ(E)≤ kσk N

N−θΛ_θ(E).

Next, ifc1,q(E) = 0 then Λ_θ(E) = 0 according to [2, Th. 5.1.13], and thusσ(E) = 0

by the previous inequality.

We introduce the flow coordinates near∂Ω defined by

Π(x) = (ρ(x), τ(x))∈[0, ₀]×∂Ω where τ(x) =proj_∂Ω(x).

It is well-known that for 0 small enough, Π is a C¹-diffeomorphism from Ω0 :=

{x∈Ω :ρ(x)≤₀} to [0, ₀]×∂Ω. With this diffeomorphism we can assimilate the surface measure dS on Σ = {x ∈ Ω : ρ(x) = } with the surface measure dS on Σ0 =∂Ω by setting

Z

Σ

v(x)dS(x) = Z

Σ0

v(, τ)dS(τ).

Lemma 3.4 Assume N ≥2 and µ∈M(Ω)satisfies Z

Ω

ρd|µ|<∞. (3.20)

Then u=G[µ] satisfies

→0lim Z

Σ0

|u|(, τ)dS(τ) = 0. (3.21) Proof. If u=G[µ], it is the unique weak solution of −∆u =µ in Ω, u= 0 on ∂Ω.

Hence u=u1−u2 whereu1=G[µ⁺] and u2=G[µ⁻]. Sinceµ+ and µ− satisfy the integrability condition (3.20) both u₁ and u₂ have a zero measure boundary trace (M-boundary trace in the sense of [18, Sec 1.3]). Hence, taking for test function the functionζ = 1,

→0lim Z

Σ0

uj(, τ)dS(τ) = 0, (3.22)

which implies (3.20).

This result allows us to obtain the uniqueness of the solution even if the right- hand side is a measure.

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Lemma 3.5 Assume N ≥ 2, σ ∈ M⁺_N

N−θ

(Ω) with N ≥θ > N −2 and g :R 7→ R is a continuous nondecreasing function. If µ∈M(Ω)there exists at most one very weak solution of (1.2).

Proof. By Lemma 3.3 withα= 1,p= 2,σ is absolutely continuous with respect to thec_1,2capacity (it is diffuse in the terminology of [9]), and ifh∈L¹_σ(Ω) the measure h₊σ, which is the increasing limit of inf{n, h₊}σ is also diffuse. Similarly h−σ is diffuse and so is hσ. Next we assume that u and u⁰ are two very weak solutions of (1.2) and set w=u−u⁰. Hence

−∆w+ (g(u)−g(u⁰))σ= 0.

Since ρ(g(u)−g(u⁰))∈L¹_σ(Ω), it follows from Lemma 3.4 that

→0lim Z

Σ

|w|(, τ)dS(τ) = 0

We use Kato inequality for measures as in [10, Th 1.1]: Sincew∈L¹(Ω), ∆w⁺ is a diffuse measure and

∆w⁺≥χ_{w≥0}∆w=χ_{w≥0}(g(u)−g(u⁰))σ≥0 in Ω

Since w⁺has a M-boundary trace by Lemma 3.4, we can apply [18, Lemmma 1.5.8]

withµ=−χ_{w≥0}(g(u)−g(u⁰))σwhich is a measure inM_ρ(Ω) :={ν ∈M(Ω) :ρν ∈ M_b(Ω)}. Then there exists τ ∈M⁺_ρ(Ω) such that

−∆w⁺=µ−τ.

Equivalently

−∆w⁺+χ_{w≥0}(g(u)−g(u⁰))σ=−τ.

Since the M-boundary trace of w⁺ is zero, it follows that w⁺ =−G[χ_{w≥0}(g(u)− g(u⁰))σ+τ]. Hencew⁺ = 0 andu≤u⁰. Similarlyu⁰ ≤u.

The following variant will be useful in the sequel.

Lemma 3.6 Assume N ≥ 2, σ ∈ M⁺_N

N−θ

(Ω) with N ≥θ > N −2 and g :R 7→ R is a continuous nondecreasing function. If u, u⁰ ∈ L¹(Ω) are such that ρg(u) and ρg(u⁰) belong to L¹_σ(Ω)and satisfy

− Z

Ω

(u−u⁰)∆ζdx+ Z

Ω

(g(u)−g(u⁰))ζdσ= Z

Ω

ζdν for all ζ ∈X+(Ω) (3.23) for some ν ∈M₊(Ω) diffuse with respect to the c_1,2-capacity, then u≥u⁰ c_1,2-quasi everywhere in Ω.

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Proof. We use Kato’s inequality, Lemma 3.4 and [18, Lemma 1.5.8] in the same way as in the proof of Lemma 3.5 since the measures (g(u)−g(u⁰))dσ andν are diffuse,

∆(u⁰−u) is diffuse, hence

∆(u⁰−u)₊≥χ_{u0≥u}∆(u⁰−u) = (g(⁰)−g(u))χ_{u0≥u}+χ_{_u⁰_≥u}ν ≥0

Since u⁰−u∈W₀^1,q(Ω) for any 1< q < _N^N₋₁, we conclude that (u⁰−u)+= 0 almost everywhere and c1,2-quasi everywhere by [2, Th 6.1.4].

The next result and the corollary which follows are the key-stone for the proof of Proposition 3.2.

Lemma 3.7 Let σ ∈ M⁺_N

N−θ

(Ω) with N ≥θ > N−2, h ∈L^∞_σ (Ω), f ∈L^s(Ω) with s > ^N₂ andw∈L¹(Ω)be the very weak solution of

−∆w+hσ=f in Ω

w= 0 in ∂Ω. (3.24)

Then w is continuous in Ωand for any nondecreasing bounded function γ ∈C²(R) vanishing at 0, there holds

− Z

Ω

j(w)∆ζdx+ Z

Ω

γ(w)hζdσ≤ Z

Ω

γ(w)ζf dx for all ζ ∈X+(Ω), (3.25) where j(r) =

Z r 0

γ(s)ds.

Proof. The solution is unique and expressed by w = G[f −hσ]. Since _N−θ^N > ^N₂, w ∈ C^α(Ω) for some α ∈ (0,1) by Lemma 2.2. Hence γ(w) is continuous and therefore measurable. We extendσ by zero in Ω^cand denoteσn=σ∗ηnwhere{η_n} is a sequence of mollifiers. Then σn →σ in the narrow topology of Ω. Forn∈N^∗, let w_n be the solution of

−∆w_n+hσ_n=T_n(f) in Ω

wn= 0 in ∂Ω, (3.26)

whereTn(f) = min{|f|, n}sgn(f). Thenwn∈W^2,s(Ω)∩W₀^1,∞(Ω) for all 1< s <∞.

By Green’s formula

− Z

Ω

j(w_n)∆ζdx+ Z

Ω

γ(w_n)hζdσ≤ Z

Ω

γ(w_n)ζf dx for all ζ ∈X+(Ω). (3.27)

Since wn→wuniformly in Ω, (3.25) follows.

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Corollary 3.8 Under the assumptions of Lemma 3.7, there holds

− Z

Ω

|w|∆ζdx+ Z

Ω

sign0(w)hζdσ≤ Z

Ω

sign0(w)ζf dx, (3.28) and

− Z

Ω

w+∆ζdx+ Z

Ω

sign+(w)ζhdσ≤ Z

Ω

sign+(w)ζf dx, (3.29) for any ζ ∈ X+(Ω). Moreover there exists a constant C >0 depending only on Ω

such that Z

Ω

sign₀(w)hdσ≤C Z

Ω

|f|dx. (3.30)

Proof. For proving (3.28) we consider a sequence {γ_k} of odd nondecreasing functions such that

γ_k(r) =







1 if r≥2k⁻¹

0 if −k⁻¹≤r≤k⁻¹

−1 if r≤ −2k⁻¹

and such that {rγ_k(r)} is nondecreasing for any r. Usingγ_k in place ofγ in (3.25) we obtain

− Z

Ω

j_k(w)∆ζdx+ Z

Ω

γ_k(w)ζhdσ≤ Z

Ω

γ_k(w)ζf dx for all ζ ∈X+(Ω), (3.31) where j_k(r) =

Z r 0

γ_k(s)ds. Since γ_k(w) ↑ w on Ω+ := {x ∈ Ω : w(x) > 0}, there holds by the monotone convergence theorem,

Z

Ω+

γ_k(w)ζ|h|dσ↑ Z

Ω+

wζ|h|dσ as k→ ∞.

Since Z

Ω+

(w−γ_k(w))ζhdσ

≤ Z

Ω+

|(w−γ_k(w))ζh|dσ= Z

Ω+

(w−γ_k(w))ζ|h|dσ, we obtain

Z

Ω+

γ_k(w)hζdσ→ Z

Ω+

whζdσ as k→ ∞.

Similarly, γ_k(w)↓w on Ω− :={x∈Ω :w(x)<0}so that Z

Ω−

γk(w)hζdσ→ Z

Ω−

whζdσ as k→ ∞.

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Combining these two results yields Z

Ω

γk(w)ζhdσ→ Z

Ω+

wζhdσ− Z

Ω−

wζhdσ= Z

Ω

sign0(w)ζhdσ.

Usiing dominated convergence theorem there holds Z

Ω

γk(w)∆ζdx→ Z

Ω

sign0(w)∆ζdx,

and Z

Ω

γ_k(w)ζf dx→ Z

Ω

sign₀(w)ζf dx.

This implies (3.28). The proof of (3.17) is similar.

Eventually we prove (3.30). Let η₁ be the solution of

−∆η₁ = 1 in Ω

η1 = 0 in∂Ω. (3.32)

Then η₁ = G[1] ∈ X+(Ω) and there exists c, c⁰ > 0 depending only on Ω such that cρ ≤ η₁ ≤ c⁰ρ. Given α ∈ (0,1], let j(r) = (r +)^α −^α, r ≥ 0, and ζ =j(η1). Note thatζ ∈C²(Ω), 0≤ζ ≤η^α, ζ = 0 on ∂Ω, j⁰ >0,j⁰⁰ <0, so that

−∆ζ =j⁰(η₁)−j⁰⁰(η₁)|∇η₁|²≥0. We deduce from (3.28) that Z

Ω

sign₀(w)(η+)^αhdσ≤ Z

Ω

sign₀(w)η^α|f|dx+^α Z

Ω

sign₀(w)hdσ.

We obtain Z

Ω

sign0(w)ρ^αhdσ ≤C Z

Ω

ρ^α|f|dx+^α|˜σ(Ω)|

Letting →0 and then α→0 we infer the result by dominated convergence.

We are now in position to prove Proposition 3.2.

Proof of Proposition 3.2. We divide the proof into several steps.

Step 1: We assume that µ ∈ L^∞(Ω). Let {η_n} be a sequence of molifiers and σn=σ∗ηn. Ifµ∈L^∞(Ω), the solution un=un,µ of

−∆u_n+g(un)σn=µ in Ω

un= 0 in∂Ω, (3.33)

is continuous in Ω. Since

−G[µ⁻]≤ −u⁻_n ≤0≤u⁺_n ≤G[µ⁺] (3.34)

Nonlinear elliptic equations with measure valued absorption potentials

HAL Id: hal-02368308

https://hal.archives-ouvertes.fr/hal-02368308

Nonlinear elliptic equations with measure valued absorption potentials

Nicolas Saintier, Laurent Veron

To cite this version:

Nonlinear elliptic equations with measure valued absorption potentials

Contents

1 Introduction

2 Preliminaries

3 The subcritical case