On singular quasilinear elliptic equations with data measures

The study of nonlinear elliptic problems with singular nonlinearities is motivated by its various applications in many elds. For example, we can mention uid mechanics, newtonian uids, and glaciology [14]. They are also applicable to model problems arising from boundary layer phenomena for viscous uids and chemical heterogeneous catalysts. Furthermore, they can be regarded as mathematical models of electrostatic MEMS devices or Micro-Electro Mechanical systems [20].


Introduction
In this work, we restrict our attention to the study of a class of quasilinear elliptic problem with a singular nonlinearity and data measure namely in Ω, where Ω is an open bounded subset of R N for N ≥ , with smooth boundary ∂Ω and f ∈ M + b (Ω) is a given nite nonnegative Radon measure. We assume that a and b are nonnegative functions, γ > , λ > , p ≥ and |.| designates the euclidean norm in R N . We stress that the problem is singular as one asks to the solution to be zero on the boundary.
The study of nonlinear elliptic problems with singular nonlinearities is motivated by its various applications in many elds. For example, we can mention uid mechanics, newtonian uids, and glaciology [14]. They are also applicable to model problems arising from boundary layer phenomena for viscous uids and chemical heterogeneous catalysts. Furthermore, they can be regarded as mathematical models of electrostatic MEMS devices or Micro-Electro Mechanical systems [20].
In order to trace the objectives of our work, we will start by recalling some previous studies where three types of problems were treated: quasilinear equations with regular data, semilinear problems with singular nonlinearities and coupling of both problems in the regular case. _ Case where f is regular: • Case where b ≡ , the problem is simply written in the form in Ω, u = on ∂Ω. (1.1) The homogeneous case (i.e. λ = ) was considered in the pioneer works of [15,23] and references therein. The authors showed using the method of sub-and supersolutions, that if a(x) is a bounded smooth function, then (1.1) has a classical solution. The case where a(x) is only a function in L (Ω) was treated in [13] where the authors obtained some existence and regularity results for Problem (1.1) depending on the value of γ. In fact, they showed that if γ ≤ , Problem (1.1) has a weak solution u ∈ H (Ω). Otherwise if γ > , there exists a solution u ∈ H loc (Ω) such that u γ+ ∈ H (Ω). The nonhomogeneous case (i.e. λ > ) has also been treated in [17], where the authors proved the existence of bounded solutions to (1.1) in the case where a and f belong to L q (Ω) for q > N .
• Case where a ≡ , the problem writes in Ω, u = on ∂Ω. (1.2) This problem was considered in [8] in the case where < p ≤ , b ∈ L ∞ (Ω) and f is regular enough. The authors showed that if (1.2) has a subsolution u and a supersolution u in W ,q (q > N) with u ≤ u in Ω, then there exists a solution u to (1.2) such that u ≤ u ≤ u. This problem was also studied in [24], where the authors showed that if f ∈ W ,∞ (Ω) and (1.2) has a nonnegative supersolution in W ,q (Ω) for (q > N), then it has a solution no matter the value of p ( ≤ p < ∞). An important step in resolving such problems is to obtain an estimate on the norm of the gradient of the solution in L ∞ (Ω). The method used to get this estimate was originally introduced by Bernstein and later developed and systematized in [21,22,30,31].
_ Case where f is only integrable or a Radon measure: • Case where b ≡ was treated in [26], where two di erent cases γ ≤ and γ > were distinguished. For γ ≤ , using an approximation argument, the authors obtained the existence of a weak solution u ∈ W ,q (Ω) of (1.1) for ≤ q < N N− . For γ > , existence and uniqueness of the solution were obtained only in W ,q loc (Ω) for every ≤ q < N N− such that T k (u) γ+ ∈ H (Ω) (where T k (u) represents the truncated function of u). The use of the truncations of u was necessary since the presence of the measure f does not allow to conclude that u γ+ itself belongs to H (Ω).
• Case where a ≡ was studied in [6]. Since f is a nonnegative integrable function or, more generally a given nite nonnegative measure on Ω, it is not regular enough. Hence the usual techniques that lead to the W ,∞ -solutions can not be applied. This di culty was the main motivation behind the work [6], where the authors distinguished three cases such that, for di erent p values in (1.2), existence and nonexistence results are established. Firstly, considering a linear growth on the gradient, they proved the existence of a solution for (1.2) using the isoperimetric inequality. Secondly, they showed that if p > , the existence of a solution is obtained if λ is su ciently small and the measure f does not charge the sets of W ,p − capacity zero p + p = . Finally, if p = , assuming the existence of a supersolution in W , (Ω), the authors obtained the existence of a solution for Problem (1.2).
In [19], the authors studied the existence of weak solutions for the following generalized elliptic Riccati equation _ Case where λ ≡ , b(x) ≡ and < p ≤ was treated in [1], in which the model problem was given by The authors proved that if p = , (1.4) admits a distributional solution for all a ∈ L (Ω). The case where < p < was treated di erently depending on the function a. Indeed, if a(x) ∈ L ∞ (Ω), then the existence was obtained for every γ > . However, for the general case a(x) ∈ L (Ω), the existence of a solution to (1.4) was proved under the condition γ > γ , where the exact value of the constant γ was given.
We conclude this section by recalling some works on the parabolic version of our problem. Recently, the authors in [12] considered the following singular nonlinear parabolic equation in Ω × ( , T), where γ > , µ ≥ , r > and f ∈ M + b (Ω). If r > , the existence of a solution to (1.5) was established for suitable small data a and f . Otherwise, if < r < , there exists a solution for every data.
Closely related to Problem (1.5) is the following one given by (1.6), which has been considered for the p-laplacian operator in [27]  where p > − N + and γ > .
Other results concerning the well-posedness of the following triply nonlinear degenerate elliptic parabolic equation were obtained in [10], (1.7) Existence, uniqueness and continuous dependence on data u and f when [b + ψ](R) = R and ϕ • [b + ψ] − is continuous, were established.
The main new aspect of this paper is the fact that λ and the functions a and b are not identically zero. Our aim in this work is to prove the existence of a suitable weak solution to (P λ ). Here, as well as in the proof of other similar results, the rst step is to precise in which sense we want to solve our problem. On one hand, a solution to (P λ ) has to be understood in the weak distributional meaning. On the other hand, we have to take into account the singular nonlinearity at zero. For this purpose, we adopt the following de nitions: De nition 1.2. If γ > , then a weak solution to Problem (P λ ) is a function The rest of our paper is organized as follows. Section 2 is devoted to necessary conditions on the data to get existence of weak solutions in (P λ ). In section 3, we investigate the existence of a solution for Problem (P λ ), when p = . Three di erent cases will be treated separately depending on the value of γ : the non-singular sublinear problem for any γ > , the singular sublinear problem for γ < and the strongly singular problem for γ ≥ . Finally, in section 4, we show the uniqueness of a solution of (P λ ) when it exists, for every Now, in what follows, we give necessary conditions for existence. For this purpose, we prove that for su ciently large value of λ, the equation (P λ ) has no weak solution.

Necessary conditions for existence . Size condition
Theorem 2.1. Let p > , γ > and λ > . We suppose that a ∈ L (Ω) + and there exists a ball B in Ω such that, b(x) ≥ C > a.e x ∈ B and B f > . Then there exists < λ * < ∞ such that (P λ ) does not have any solution for λ > λ * . Furthermore, when (P λ ) has a solution, then 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, -a regularity condition is required on f as soon as p > ; -moreover, a size condition is also required if p > .
For various discussions on the meaning of (2.1) and its relationship with nonlinear capacities, we refer the reader to [6] and [19].
Obviously, this is not true for any measure f as soon as N > p or p > N N− . 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 = . 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 > .

Existence Results for any nite nonnegative Radon measure
In this section, we present existence results of which the proofs are based on the isoperimetric inequality [25], for linear growth on the gradient (p = ), and for any nite measure f ∈ M + b (Ω). Three di erent problems will be treated separately in each subsection: the non-singular sublinear problem for any γ > , the singular sublinear problem for γ < and the strongly singular problem for γ ≥ .

. Existence of solutions to the non-singular sublinear problem
Let us consider the following regularized problem in which we regularize the singular term a(x) u γ by a(x) (u + ε) γ where ε > , to become not singular at the origin. The problem then rewrites (3.1) The main tool in the proof of this theorem is the isoperimetric inequality that we will use under the following form [25].
where ω N is the Lebesgue measure of the unit ball of R N , and Proof. of Theorem 3.1 Step 1. Existence for the approximating problem. Let us approximate our Problem (Pε). For this purpose, we de ne the truncated function T k as follows T k (r) = max(−k, min(r, k)). (3.4) Now, we truncate the functions a, b and f by considering the three sequences an, bn and fn which are de ned by let n ∈ N, and Let us now consider the following approximated problem The constant M = max(( n||an||∞) γ+ − n + λn||fn||∞) is a supersolution of (3.7) and M = 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 un solution of (3.7).
Step 2. Estimates on the approximating solutions. At this level, we will prove the existence of a constant C independent of n such that First of all, we introduce the following function (3.14) Finally, we get where C = ||a|| L (Ω) , Cq = ||b|| L N+η (Ω) and C λ = λ||f || M b (Ω) . Now, we assume that N ≥ so that q < , and we use the following two inequalities (3.17) Next, we take the q th power of (3.17) and we multiply it by the square of (3.16) to nd Now, we plug the inequality (3.15) into the previous inequality, and we let h tend to zero, to obtain a di erential inequality satis ed by σn(t) = [un≥t] |∇un| q and de ned in the following sense On the other hand, according to the isoperimetric inequality (3.2), we get (3.20) Using Young's inequality on the right hand side term leads to −σ n (t) ≤ N −q ω −q N n D ε γq + Dq σn(t) + D λ µn(t) q( N − ) (−µ n (t))), (3.21) where D = C q , Dq = Cq q and D λ = C λ q . This implies that −σ n (t) ≤ D ε γq +Dq σn(t) +D λ µn(t) q( N − ) (−µ n (t))), (3.22) where α = − q N− N and kα =Dq. Then by integrating from t = to t = ||un||∞, and knowing σn(||un||∞) = and µn(||un||∞) = , we obtain Step 3. Passage to the limit. We have and then from (3.7) and (3.25), we deduce ||∆un|| L (Ω) ≤ C and ||un|| W ,q (Ω) ≤ C. (3.27) This yields to the compactness of un in W ,q (Ω) for ≤ q < N N − . Then there exists a function u such that (up to not relabeled sub-sequences), the sequence un converges to u strongly in W ,q (Ω), and (un , ∇un) converges to (u, ∇u) a.e in Ω. Moreover, by compact embedding, we obtain that un converges strongly to u in L (Ω). Thus, taking φ in C c (Ω), we have that Finally, since b ∈ L N+η (Ω), then b|∇un| converges strongly to b|∇u| in L (Ω). This concludes the proof since it is straightforward to pass to the limit in the last term containing fn.

. Existence of solutions to the singular sublinear problem and for every nonnegative Radon measure
Theorem 3.3. Let < γ < , a ∈ L ∞ (Ω) + and b ∈ L N+η (Ω) + . Then for all λ > and all f ∈ M + b (Ω), Problem (P λ ) has a solution u in W ,q (Ω) for every ≤ q < N N − .
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 di erences. 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 un γ , for which we will show that un is bounded from below on the compact subsets of Ω. Proof.
Step 1. Existence for the approximating problem.
Let us now consider the following approximated problem The existence of a solution for (3.30) is ensured by Theorem 3.1 by letting ε = n in Problem (Pε).
Step 2. Local uniform bound from below. Here, we show that un is bounded from below on the compact subsets of Ω. In particular, we check that the sequence un is such that for every ω ⊂⊂ Ω, there exists a constant cω > such that un(x) ≥ cω in ω, for every n ∈ N. (3.31) In fact, we have −∆un ≥ 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 where ϕ denotes the rst eigenfunction of −∆ with Dirichlet homogeneous boundary conditions, and G denotes the inverse in L (Ω) of the operator −∆ under homogeneous Dirichlet conditions. Therefore we have Step 3. Estimates on the approximating solutions.
Analogously to the proof of the previous theorem, we nally get Ω |∇un| q ≤ C. (3.35) Step 4. Passage to the limit Similarly to the passage to the limit in (3.7), we may assume that un converges strongly to u in L (Ω) and a.e. in Ω. Thus, taking φ in C c (Ω), we have that Proof. Analogously to Step 1 and Step 2 in the proof of Theorem 3.3, we obtain the existence of a solution un for the approximated Problem (3.30) for γ ≥ , such that for all ω ⊂⊂ Ω, there exists a constant cω > such that un(x) ≥ cω in ω. (3.38) In what follows, we show that T k (u) γ+ ∈ H (Ω). To this aim, let H be a function in C (R) de ned by Choosing β such that β − ε > leads to Consequently Even if we replace k by k, we obtain the desired result.
Next, we show the boundedness of un in W ,q loc (Ω) into two steps. For xed k > , we will make use of the two truncations functions T k (r) given by (3.4) and G k (r) de ned as G k (r) = (|r| − k) + sign(r).
Step 1: G (un) is bounded in W ,q (Ω) for all ≤ q < N N − .
In other words, we have to prove that there exists a constantC k depending only on k such that [un≥ ] |∇un| q ≤C k . (3.54) Analogously to the case γ < , we take ϕ = p t,h (un) as a test function in ( an (un + n ) γ + [un≥t] b|∇un| + λ||f || M b (Ω) .
Let us now plug the inequality (3.58) into the previous inequality (3.18). By tending h to zero, we obtain a di erential inequality satis ed by the function σn which is de ned in the following sense σn(t) = [un≥t] |∇un| q , On the other hand, according to the isoperimetric inequality (3.2), we get Using Young's inequality on the right-hand side term leads to where D = C q , Dq = Cq q and D λ = C λ q . This implies that −σ n (t) ≤ D t γq +Dq σn(t) +D λ µn(t) q( N − ) (−µ n (t))), (3.62) where α = − q N− N and kα =Dq.
Step 2: T (un) is bounded in H loc (Ω). We have to investigate the behavior of (un) for its small values (un ≤ ). To do so, we need to prove that ∀ω ⊂⊂ Ω, ω |∇T (un)| ≤ C .
Finally, we deduce that u is a solution to (P λ ) by a straightforward re-adaptation of the passage to the limit in the previous theorem. . Then for all γ > , λ > and f ∈ M + b (Ω), the solution of (P λ ) is unique if it exists.

Uniqueness of weak solutions
In order to prove this result, we start by recalling the following technical lemma Lemma 4.2. Let us consider j(r) = |r| p . The function j is convex and we have ∀r ∈ R n , ∃A ∈ ∂j(r) such that ∀r ∈ R n , j(r) − j(r) ≥ < A, r −r >, (4.1) where ∂j(r) is the sub-di erential of j(r) de ned as follows: (4.2) Consequently, we deduce that for u ∈ W ,p (Ω), there exists A(x) ∈ ∂j(∇u) such that ∀û ∈ W ,p (Ω), |∇u| p − |∇û| p ≥ < A, ∇(u −û) > . For p = , we may deduce from (4.2), that ||A(x)|| ≤ , for all x ∈ Ω. Hence A ∈ L ∞ (Ω) N . Thus, again we have A ∈ L N+η (Ω) N . Finally, the uniqueness result that we obtain is a consequence of the following two lemmas: Then ω ≤ . and since θ ≥ in Ω, then θ = in Ω.
Proof. of Theorem 4.1 Let u be a supersolution of (P λ ) andû a subsolution, and let w = u −û.
We take the di erence between the equations associated to u andû respectively, we obtain Therefore, thanks to Lemma 4.4, we get w = , which completes the proof.