On singular quasilinear elliptic equations with data measures

Theorem 2.1

Let p > 1, γ > 0 and λ > 0. We suppose that a ∈ L¹(Ω)⁺ and there exists a ball B₀ in Ω such that, b(x) ≥ C₀ > 0 a.e x ∈ B₀ and ∫B0f>0. Then there exists 0 < λ^* < ∞ such that (P_λ) does not have any solution for λ > λ^*.

Furthermore, when (P_λ) has a solution, then

where Cp=p−1ppp−1C01p−1.

Proof

See Theorem 2.1 [Alaa-Pierre, [Theorem 2.1, [6]]] for a similar detailed proof.□

Remark 2.2

The condition (2.1) is at the same time a size and regularity condition on f. It is similar to the results obtained for quasilinear elliptic equations and multidimensional Riccati equations. In other words,

1. a regularity condition is required on f as soon as p > 1;

2. moreover, a size condition is also required if p > 2.

For various discussions on the meaning of (2.1) and its relationship with nonlinear capacities, we refer the reader to [6] and [19].

Proposition 2.3

[Alaa-Pierre, [Proposition 2.2, [6]]] Under the hypothesis of Theorem 2.1, if Problem (P_λ) has a solution for some λ > 0, then the measure f does not charge the sets of W^1,p′-capacity zero.

Remark 2.4

We recall that a compact set K ⊂ Ω is of W^1,p′-capacity zero if there exists a sequence of C0∞- functions φ_n greater than 1 on K and converging to 0 in W^1,p′. The above statement in Proposition 2.3 implies that

Obviously, this is not true for any measure f as soon as N > p′ or p > NN−1. See [11] for more details.

Proof

of Proposition 2.3 See [6] for a similar detailed proof.□

In the following section, we restrict our attention to the existence of a weak solution to (P_λ) for p = 1. To this aim, we proceed by an approximation argument. The main step is to get a priori estimates on the approximate solution sequences, for any value of γ > 0.

Theorem 3.1

Let a ∈ L¹(Ω)⁺ and b ∈ L^N+η(Ω)⁺. Then, for all γ > 0, λ > 0 and for all f ∈ Mb+ (Ω), Problem (P_ε) has a nonnegative weak solution u in W01,q (Ω) for 1 ≤ q < NN−1.

The main tool in the proof of this theorem is the isoperimetric inequality that we will use under the following form [25].

Lemma 3.2

Let u ∈ W01,1 (Ω). Then

where ω_N is the Lebesgue measure of the unit ball of ℝ^N, and

Proof

of Theorem 3.1

1. Let us approximate our Problem (P_ε). For this purpose, we define the truncated function T_k as follows

   Tk(r)=max(−k,min(r,k)). (3.4)

   Now, we truncate the functions a, b and f by considering the three sequences a_n, b_n and f_n which are defined by

   let n∈N,an(x)=min(a(x),n),bn(x)=min(b(x),n), (3.5)

   and

   fn∈C0∞(Ω), such that fn≥0,||fn||L1(Ω)≤||f||Mb(Ω) and fn→f in Mb(Ω). (3.6)

   Let us now consider the following approximated problem

   un∈W01,∞(Ω),1nun−Δun=an(x)(un+ε)γ+bn(x)|∇un|1+1nbn(x)|∇un|+λfninΩ. (3.7)

   The constant M¯=max((2n||an||∞)1γ+1−1n+2λn||fn||∞) is a supersolution of (3.7) and M = 0 is a subsolution. Then by applying the classical theory (see p.34 of [7] and the Main theorem of [2]), we obtain the existence of u_n solution of (3.7).

2. At this level, we will prove the existence of a constant C independent of n such that

   ∫Ω|∇un|q≤M,1≤q<NN−1. (3.8)

   First of all, we introduce the following function

   pt,h(r)=0ifr≤t,r−thift≤r≤t+h,1ifr>t+h. (3.9)

   We multiply (3.7) by p_t,h(u_n) and then we integrate on Ω to obtain

   ∫Ω1nun−Δunpt,h(un)=∫Ωan(un+ε)γ+bn|∇un|1+1nbn|∇un|+λfnpt,h(un). (3.10)

   We observe that

   bn|∇un|1+1nbn|∇un|≤b|∇un|, (3.11)

   thus,

   1h∫[t≤un≤t+h]|∇un|2≤∫[t≤un≤t+h]an(un+ε)γun−th+∫[un≥t+h]an(un+ε)γ+∫[un≥t]b|∇un|+λ||fn||L1(Ω). (3.12)

   Since 0≤un−th≤1 on the set [t ≤ u_n ≤ t + h] and ∣∣f_n∣∣_L¹(Ω) ≤ ∣∣f∣∣_Mb(Ω), we obtain

   1h∫[t≤un≤t+h]|∇un|2≤∫[un≥t]an(un+ε)γ+||b||LN+η(Ω)∫[un≥t]|∇un|q1q+λ||f||Mb(Ω), (3.13)

   where q=N+η′=NN−1−ε(η),ε(η)>0.

   Hence

   1h∫[t≤un≤t+h]|∇un|2≤∫[un≥t]an(t+ε)γ+||b||LN+η(Ω)∫[un≥t]|∇un|q1q+λ||f||Mb(Ω). (3.14)

   Finally, we get

   1h∫[t≤un≤t+h]|∇un|2≤C1εγ+Cq∫[un≥t]|∇un|q1q+Cλ, (3.15)

   where C₁ = ∣∣a∣∣_L¹(Ω), C_q = ∣∣b∣∣_L^N+η(Ω) and C_λ = λ ∣∣f∣∣_Mb(Ω). Now, we assume that N ≥ 2 so that q < 2, and we use the following two inequalities

   1h∫[t≤un≤t+h]|∇un|q≤1h∫[t≤un≤t+h]|∇un|2q2(μ(t)−μ(t+h)h)2−q2 (3.16)

   and

   1h∫[t≤un≤t+h]|∇un|≤1h∫[t≤un≤t+h]|∇un|q1q(μ(t)−μ(t+h)h)q−1q. (3.17)

   Next, we take the q^th power of (3.17) and we multiply it by the square of (3.16) to find

   1h∫[t≤un≤t+h]|∇un|q1h∫[t≤un≤t+h]|∇un|q≤1h∫[t≤un≤t+h]|∇un|2q(μ(t)−μ(t+h)h). (3.18)

   Now, we plug the inequality (3.26) into the previous inequality, and we let h tend to zero, to obtain a differential inequality satisfied by σn(t)=∫[un≥t]|∇un|q and defined in the following sense

   −ddt∫[un≥t]|∇un|q−σn′(t)≤C1εγ+Cqσn(t)1q+Cλq. (3.19)

   On the other hand, according to the isoperimetric inequality (3.2), we get

   NqωnqNμn(t)q(1−1N)(−σn′(t))≤C1εγ+Cqσn(t)1q+Cλq−μn′(t). (3.20)

   Using Young’s inequality on the right hand side term leads to

   −σn′(t)≤N−qωn−qND1εγq+Dqσn(t)+Dλμn(t)q(1N−1)(−μn′(t))), (3.21)

   where D₁ = C₁^q, D_q = C_q^q and D_λ = C_λ^q. This implies that

   −σn′(t)≤D1^εγq+Dq^σn(t)+Dλ^μn(t)q(1N−1)(−μn′(t))), (3.22)

   where D1^=N−qωn−qND1,Dq^=N−qωn−qNDq and D^λ=N−qωn−qNDλ.

   This can be rewritten as

   −ddte−kμn(t)ασn(t)≤ddte−kμn(t)α1Dq^D1^εγq+Dλ^, (3.23)

   where α=1−qN−1N and kα=Dq^.

   Then by integrating from t = 0 to t = ∣∣u_n∣∣_∞, and knowing σ_n(∣∣u_n∣∣_∞) = 0 and μ_n(∣∣u_n∣∣_∞) = 0, we obtain

   e−kμn(0)ασn(0)≤1Dq^D1^εγq+Dλ^. (3.24)

   Since μ_n(0) ≤ ∣Ω∣, we get

   ∫Ω|∇un|q≤C, (3.25)

   where C=ek|Ω|α1Dq^D1^εγq+Dλ^.

3. We have

   an(un+ε)γL1(Ω)≤||a||L1(Ω)εγ, (3.26)

   and then from (3.7) and (3.25), we deduce

   ||Δun||L1(Ω)≤Cand||un||W01,q(Ω)≤C. (3.27)

   This yields to the compactness of u_n in W01,q(Ω) for 1≤q<NN−1. Then there exists a function u such that (up to not relabeled sub-sequences), the sequence u_n converges to u strongly in W01,q (Ω), and (u_n, ∇ u_n) converges to (u, ∇ u) a.e in Ω.

   Moreover, by compact embedding, we obtain that u_n converges strongly to u in L¹(Ω).

   Thus, taking φ in Cc1 (Ω), we have that

   anφun+εγ≤||φ||L∞(Ω)εγa. (3.28)

   Applying Lebesgue’s dominated convergence theorem, we obtain

   limn→∞∫Ωanφun+εγ=∫Ωaφuγ. (3.29)

   Finally, since b ∈ L^N+η(Ω), then b ∣∇ u_n∣ converges strongly to b ∣∇ u∣ in L¹(Ω). This concludes the proof since it is straightforward to pass to the limit in the last term containing f_n.□

Theorem 3.3

Let 0 < γ < 1, a ∈ L^∞(Ω)⁺ and b ∈ L^N+η(Ω)⁺. Then for all λ > 0 and all f ∈ Mb+ (Ω), Problem (P_λ) has a solution u in W01,q (Ω) for every 1 ≤ q < NN−1.

The proof of Theorem 3.3 strictly follows the main steps of the previous proof of Theorem 3.1. We will then sketch it by enlightening the main differences. Estimates will mainly be based on the isoperimetric inequality, and so they will be formally very similar to the previous proof. The main challenge in this case will be to control the singular term 1unγ, for which we will show that u_n is bounded from below on the compact subsets of Ω.

Proof

1. Let us now consider the following approximated problem

   un∈W01,q(Ω),1≤q<NN−1,−Δun=a(x)un+1nγ+b(x)|∇un|+λfinΩ. (3.30)

   The existence of a solution for (3.30) is ensured by Theorem 3.1 by letting ε = 1n in Problem (P_ε).

2. Here, we show that u_n is bounded from below on the compact subsets of Ω. In particular, we check that the sequence u_n is such that for every ω ⊂ ⊂ Ω, there exists a constant c_ω > 0 such that

   un(x)≥cωinω,for everyn∈N. (3.31)

   In fact, we have

   −Δun≥λf. (3.32)

   Hence, using the uniform Hopf principle as formulated in [3] and [16], there exists a constant C only depending on Ω such that

   Gf≥C(Ω)∫Ωfϕ1ϕ1, (3.33)

   where ϕ₁ denotes the first eigenfunction of −Δ with Dirichlet homogeneous boundary conditions, and G denotes the inverse in L¹(Ω) of the operator −Δ under homogeneous Dirichlet conditions. Therefore we have

   un≥λGf≥λC(Ω)∫Ωfϕ1ϕ1. (3.34)

   Thus, for all compact subset ω of Ω, there exists a constant c_ω (not depending on n) such that u_n ≥ c_ω, in which c_ω can be taken as c_ω = λ C̃(Ω) min {ϕ₁}(x), x ∈ ω}, where C~(Ω)=C(Ω)∫Ωfϕ1.

3. Let us take φ = p_t,h(u_n) as a test function in the weak formulation (3.30), where p_t,h is given by (3.9). Applying (3.34), we have 1unγ≤Cϕ1γ, in which C = (λ C̃(Ω))^γ. Using the fact that u_n + 1n ≥ u_n, we obtain

   ∫Ω∇un∇pt,h(un)≤C∫Ωa(x)pt,h(un)ϕ1γ+∫Ωb(x)|∇un|pt,h(un)+λ∫Ωfpt,h(un),

   which implies that

   1h∫[t≤un≤t+h]|∇un|2≤C∫[t≤un≤t+h]a(x)ϕ1γ(un−t)h+C∫[un≥t+h]a(x)ϕ1γ+∫[un≥t]b(x)|∇un|+λ||f||Mb(Ω).

   Since 0≤un−th≤1 on the set [t ≤ u_n ≤ t + h], we obtain

   1h∫[t≤un≤t+h]|∇un|2≤C∫Ωa(x)ϕ1γ+∫[un≥t]b(x)|∇un|+λ||f||Mb(Ω).

   Therefore

   1h∫[t≤un≤t+h]|∇un|2≤C∫Ωa(x)ϕ1γ+||b||LN+η(Ω)∫[un≥t]|∇un|q1q+λ||f||Mb(Ω),

   in which q=N+η′=NN−1−ε(η),ε(η)>0.

   Thus

   1h∫[t≤un≤t+h]|∇un|2≤C1∫Ω1ϕ1γ+Cq∫[un≥t]|∇un|q1q+Cλ,

   where C₁ = C ∣∣a∣∣_L^∞(Ω), C_q = ∣∣b∣∣_L^N+η(Ω) and C_λ = λ ∣∣f∣∣_Mb(Ω).

   Analogously to the proof of the previous theorem, we finally get

   ∫Ω|∇un|q≤C. (3.35)

4. Similarly to the passage to the limit in (3.7), we may assume that u_n converges strongly to u in L¹(Ω) and a.e. in Ω.

   Thus, taking φ in Cc1 (Ω), we have that

   ∣anφ(un+1n)γ∣≤||φ||L∞(Ω)ϕ1γa. (3.36)

   Finally, applying Lebesgue’s dominated convergence theorem, we obtain

   limn→∞∫Ωanφ(un+1n)γ=∫Ωaφuγ. (3.37)

   By a straightforward re-adaptation of the previous theorem, u is a solution to (P_λ).□

Theorem 3.4

Let γ ≥ 1, a ∈ L¹(Ω)⁺ and b ∈ L^N+η(Ω)⁺. Then for all f ∈ Mb+ (Ω) and λ > 0, (P_λ) has a solution u in Wloc1,q (Ω) for every 1 ≤ q < NN−1. Furthermore, Tk(u)γ+12∈H01(Ω).

Proof

Analogously to Step 1 and Step 2 in the proof of Theorem 3.3, we obtain the existence of a solution u_n for the approximated Problem (3.30) for γ ≥ 1, such that for all ω ⊂ ⊂ Ω, there exists a constant c_ω > 0 such that

In what follows, we show that Tk(u)γ+12∈H01(Ω).

To this aim, let H be a function in C¹(ℝ) defined by

We introduce the test function

Next, we multiply (3.30) (with γ ≥ 1) by ϕ and we integrate on Ω to obtain

Hence

By using the definition of H given by (3.39), the second term of the right-hand side of the inequality (3.42) vanishes and for the first term, we have

Since unγ(un+1n)γ≤1, then

Using Young’s inequality in the third term of (3.42), we obtain

Concerning the last term in (3.45), we have

Since H(unk)γeβun≤(unk)γeβun≤eβk on the set [u_n ≤ k], we get

Thus