An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term

In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions


Introduction
In this paper we study the following problem: where Ω is an open bounded set of R N , N ≥ 3, B, p > 0 and θ > 0. We assume that b : Ω → R is a measurable function such that for some positive constants α and β Moreover f is a positive function belonging to some Lebesgue space L m (Ω), with m ≥ 1. We point out three characteristics of this problem: the operator A(v) = −div b(x) (1 + |v|) p ∇v is defined on H 1 0 (Ω) but is not coercive on this space when v is large, as proved in [20]. The lower order term has a quadratic growth with respect to the gradient and is singular in the variable u. As we will see, existence and summability of solutions to problem (1) depend on these features.
It is known that the degenerate coercivity has in some sense a bad effect on the summability of the solutions to problem (3) −div (a(x, u)∇u) = f in Ω, u = 0 on ∂Ω, as proved in [9]. There f ∈ L m (Ω) was not assumed to be positive, a : Ω × R → R was a Carathéodory function such that α (1 + |s|) p ≤ a(x, s) ≤ β, for p ∈ (0, 1) and α, β > 0. Apart from the case where m > N 2 , the summability of the solutions is lower than the summability of the solutions to elliptic coercive problems.
Indeed, in [9] it is shown that if 2N N + 2 − p(N − 2) < m < N 2 there exists a , there exists a W 1,s 0 (Ω) distributional solution, with s = N m(1 − p) N − m(1 + p) . For p > 1 the authors prove a non-existence result for constant sources f . Note that a bad effect on the regularity of the solutions appears even when the right hand side of (3) is an element of H −1 (Ω), such as −div(F ), with F ∈ L 2 (Ω). As a matter of fact, in this case the solutions are in general not in H 1 0 (Ω) (see [16]). The presence of lower order terms can have a regularizing effect on the solutions. In [7] and [14] three kinds of lower order terms are considered for elliptic problems with degenerate coercivity, with no restriction on p. In the first paper the author analyses a lower order term defined by a Carathéodory function g : Ω × R × R N → R N with the following properties. There exists d ∈ L 1 (Ω), two positive constants µ 1 , µ 2 > 0 and a continuous increasing real function h such that g(x, s, ξ)s ≥ 0, µ 1 |ξ| 2 ≤ |g(x, s, ξ)| when |s| ≥ µ 2 and |g(x, s, ξ)| ≤ d(x)h(|s|)|ξ| 2 . It is proved that for a L 1 (Ω) source there exists a H 1 0 (Ω) distributional solution to −div (a(x, u)∇u) + g(x, u, ∇u) = f in Ω, u = 0 on ∂Ω .
This proves that the summability of the gradient of the solutions is much larger than that one of the solutions of problem (3). It is even larger than the summability of the gradient of the solutions to elliptic coercive problems with L 1 (Ω) sources, which is L s (Ω) for every s < N N −1 (see [10] for example). We remark moreover that the lower order term gives the existence of a solution for p ≥ 1; for these values of p, (3) has no solution.
In a previous article [14] we consider two kinds of lower order terms h(u). For h(u) = |u| q−1 u, with q > p + 1, we stablish the existence of a distributional solution u ∈ W 1,t 0 (Ω) ∩ L q (Ω) , t < (Ω) such that |u| qm ∈ L 1 (Ω). These results show that if q is sufficiently large, there exists a distributional solution for any source; this is not the case for problem (3). The second lower order term analysed in [14] is h(u), where h : [0, s 0 ) → R is a continuous, increasing function such that h(0) = 0 and lim s→s − 0 h(s) = +∞ for some s 0 > 0. The regularizing effects of this lower order term are even better than the previous one. Indeed for a positive L 1 (Ω) source, there exists a bounded H 1 0 (Ω) solution.
In the literature we find several papers about elliptic coercive problems with lower order terms having a quadratic growth with respect to the gradient (see [6,10,11,12,8] for example and the references therein), that is, for problem In these works it is assumed that M : Ω → R N 2 is a bounded elliptic Carathéodory map, so that there exists α > 0 such that α|ξ| 2 ≤ M (x)ξ · ξ for every ξ ∈ R N . Various assumptions are made on g. With no attempt of being exhaustive, we will describe some recent results where a singular g has been considered, namely 3,4], has been studied in [2,3,4,8,13,15]. From this body of literature one can deduce that for a then the solution u belongs to W 1,q 0 (Ω), with q = N m(2 − θ) N − mθ . The authors of [5] consider the general case θ < 2, assuming that f is a strictly positive function on every compactly contained subset of Ω. They prove that if f ∈ L 2N N +2 (Ω) there exists a positive H 1 0 (Ω) solution. Finally, in [15] the lower order term is taken to be λu + µ |∇u| 2 |u| θ χ {u>0} , where χ {u>0} denotes the characteristic function of the set {u > 0}, λ > 0 and µ ∈ R.
In this paper we consider the same lower order term as above in an elliptic problem defined by an operator with degenerate coercivity. We will see that if 0 < θ < 2, then |∇u| 2 |u| θ has a regularizing effect, even if it is singular in u. We are going to state our results. We will distinguish the cases 0 < θ < 1 and 1 ≤ θ < 2.
, and satisfies for every ω ⊂⊂ Ω. Then there exists a function u ∈ H 1 0 (Ω), strictly positive on Ω, such that |∇u| 2 u θ ∈ L 1 (Ω) and , and satisfies ess inf {f (x) : x ∈ ω} > 0 for every ω ⊂⊂ Ω. Then there exists a function u ∈ W 1,σ 0 (Ω), strictly positive on Ω, such that |∇u| 2 u θ ∈ L 1 (Ω) and We remark that if θ < N N − 1 we are able to prove the existence of solutions when the source f belongs to L 1 (Ω). We would like to point out the regularizing effects of the lower order term, in the case where p > θ−1 and 0 < θ < 2. Our results furnish H 1 0 (Ω) solutions for less summable sources than for problem (3), since .
Even in the case where the source f is less summable, we get a better regularity of solutions than for problem (3): In the case where 0 < p ≤ θ − 1, we are able to prove the existence of a solution to problem (1) with the same regularity as the solutions of problem (3). (1) If m > N 2 , then there exists a strictly positive H 1 0 (Ω) ∩ L ∞ (Ω) solution to problem (1).
, then there exists a strictly positive , then there exists a strictly Moreover |∇u| 2 u θ ∈ L 1 (Ω). In the case where θ ≥ 2, the situations changes. Indeed we will prove a nonexistence result of finite energy solutions. Let λ 1 (f ) denote the first positive eigenvalue of −∆u = λf u in Ω, where f in L q (Ω), with q > N 2 . Using a result of [5], it is quite easy to prove the following Theorem 1.6. Let f ≥ 0, f ≡ 0, be a L q (Ω) function, with q > N 2 . If either θ > 2, or θ = 2 and λ 1 (f ) > β Bα , then there is no H 1 0 (Ω) solution to problem (1).

A priori estimates
To prove the existence of solutions to problem (1) we use the following approximating problems: where, for n ∈ N and s ∈ R T n (s) = max{−n, min{n, s}} .
These problems are well-posed due to the following result proved in [6,11,12].
By Theorem 2.1 the solutions u n of the above approximating problems are bounded H 1 0 (Ω) non-negative functions, since f is assumed to be positive and the lower order term has the same sign as u n . This implies that u n satisfies We are now going to prove some a priori estimates. The next lemma gives a control of the lower order term.
Lemma 2.2. Let u n be the solutions to problems (6). Then it results Proof. Let us consider T h (u n ) h , h > 0, as a test function in (6). We have, dropping the non-negative operator term, It is now sufficient to pass to the limit as h → 0, using Fatou's lemma and the fact that We prove now two a priori estimates on u n , which are true for every p > 0 and θ ∈ (0, 2). In the sequel C will denote a positive constant independent of n; µ(E) will be the Lebesgue measure of a set E ⊂ R N .
. Then the solutions u n to problems (6) are uniformly bounded in H 1 0 (Ω). Thus there exists a function u ∈ H 1 0 (Ω) such that, up to a subsequence, u n → u weakly in H 1 0 (Ω) and a.e. in Ω. Proof. The assertion follows by proving that the solutions u n to problems (6) are uniformly bounded in H 1 0 (Ω). If we take (u n + 1) θ − 1 as a test function in problem (6) we obtain dropping the positive operator term. We can estimate the right hand side using (7) in order to get By working in {u n ≥ 1}, the previous inequality gives We use the Sobolev inequality in the left hand side and the Hölder inequality Since we are assuming θ < 2, we deduce that It follows from (8) that Let us search for the same kind of estimate in {u n < 1}. Taking T 1 (u n ) as a test function in problem (6), we get using hypothesis (2) and dropping the non-negative lower order term. As a consequence of estimates (9) and (10), u n is uniformly bounded in H 1 0 (Ω). By compactness, there exists a function u ∈ H 1 0 (Ω) such that, up to a subsequence, u n → u weakly in H 1 0 (Ω) and a.e. in Ω.
. Then the solutions u n to problems (6) are uniformly bounded in Thus there exists a function u ∈ W 1,σ 0 (Ω) such that, up to a subsequence, u n → u weakly in W 1,σ 0 (Ω) and a.e. in Ω.
Taking T 1 (u n ) as a test function in (6) and dropping the non-negative the lower order term, we get (2). This last estimate and (14) imply that u n is uniformly bounded in W 1,σ 0 (Ω). Since σ > 1, there exists a function u ∈ W 1,σ 0 (Ω) such that, up to a subsequence, u n → u weakly in W 1,σ 0 (Ω) and a.e. in Ω.
In the following lemma, we will assume some hypotheses on p. This will give, in some cases, some better estimates than Lemmata 2.3 and 2.4.
, the solutions of (6) are uniformly bounded , the solutions of (6) are uniformly bounded in W 1,s 0 (Ω). Thus there exists a function u ∈ W 1,s 0 (Ω) such that, up to a subsequence, u n → u weakly in W 1,s 0 (Ω) and a.e. in Ω.
Proof. In problems (6) consider as a test function the same test functions as in [9]. With this choice, the lower order term is non-negative and we can take into account only the term given by the operator. Therefore one can follow the same proofs as in [9] to get the above estimates.
Remark 1. Let p > θ − 1. Lemmata 2.3 and 2.4 give a further uniform estimate on u n than Lemma 2.5. Indeed, if one chooses u n as a test function in (6), then, by hypothesis (2) If p > θ − 1, the lower order term has a leading role in the left hand side of the previous inequality.
We are going to prove the a.e. convergence of the gradients of u n . We will follow the same technique as in [8]. Remark that a similar technique was used for elliptic degenerate problems in [1].
Lemma 2.6. Let u n be the solutions to problems (6) and u be the function found in Lemmata 2.3, or 2.4 or 2.5, according to the summability of f . Up to a subsequence, ∇u n converges to ∇u a.e. in Ω.
Proof. Let h, k > 0. In the sequel C will denote a constant independent of n, h, k. Let us consider T h (u n − T k (u)) as a test function in problems (6). Then By estimate (7) on the right hand side and by hypothesis (2) on the left one, we get Then we can write At the limit as n → ∞ one has lim sup We recall that u n is uniformly bounded in W 1,η 0 (Ω), where η equals 2 or σ or s, according to the statements of Lemmata 2.3, 2.4 and 2.5. Let q ∈ (1, η). We can write Using the Hölder inequality with exponent 2 q on the first term of the right hand side and exponent η q on the other ones, we have where we have used that u n is uniformly bounded in W 1,η 0 (Ω) to estimate the last two terms. By (15) the limit as n → ∞ gives The limit as h → 0 implies At the limit as k → +∞, µ({|u| > k}) converges to 0. Therefore ∇u n → ∇u in L q (Ω). Up to a subsequence, ∇u n → ∇u a.e. in Ω.

3.
Existence results in the case 0 < θ < 1 To prove the existence of solutions to problem (1), the key point is to prove that the function u found by compactness in the lemmata of Section 2 is strictly positive. In the case 0 < θ < 1, we use a technique similar to that in [8].
Proof. We define, for s ≥ 0, Observe that H is well-defined, since θ < 1. We choose e −BHn(un) φ, where φ is a positive C ∞ 0 (Ω) function, as a test function in (6). This gives φ by hypothesis (2). The last quantity is positive, due to the choice of H n and φ. As a consequence With these definitions, we remark that we have just proved that the inequality holds distributionally. Observe that for every n ∈ N, P n (u n ) ∈ H 1 0 (Ω), since P ′ n is bounded and u n ∈ H 1 0 (Ω). Let z n be the Then −div(b(x)∇(P n (u n ))) ≥ −div(b(x)∇z n ) .
Proof. We pass to the limit in (7). The a.e. convergence of u n to u (see Lemmata 2.3, 2.4 and 2.5), the a.e. convergence of ∇u n to ∇u (see Lemma 2.6) and Proposition 1 imply We are going to prove Theorem 1.1.
Proof. We are going to prove that the function u found in Lemma 2.3, and studied in Lemma 2.6, Proposition 1 and Corollary 1, is a weak solution to problem (1). We use the same technique as in [8].
We will prove that (4) holds true for every positive and bounded ϕ ∈ H 1 0 (Ω). The general case follows from the fact that every such function ϕ can be written as ϕ + − ϕ − with ϕ ± bounded, positive and belonging to H 1 0 (Ω). We pass to the limit as n → ∞ in where ϕ is a positive bounded H 1 0 (Ω) function. Regarding the first term we observe that b(x) (1 + T n (u n )) p ∇ϕ strongly converges to b(x) (1 + u) p ∇ϕ in L 2 (Ω) and ∇u n weakly converges to ∇u in L 2 (Ω). For the second one we use the a.e. convergence of ∇u n , proved in Lemma 2.6. Fatou's lemma implies The proof of the opposite inequality is more delicate. To this aim, we define, for n ∈ N and s ≥ 0, Note that by hypothesis (2) and inequality the sum of the last two terms is non-negative. At the limit as n → ∞ we have using the weak convergence of u n to u in H 1 0 (Ω) in the left hand side and Fatou's lemma in the right one. Now we pass to the limit as j → ∞, using that e −H0(u) e H 1 j (Tj (u)) ≤ 1 and Corollary 1. We obtain Inequalities (16) and (17) imply that for every positive and bounded ϕ ∈ H 1 0 (Ω). We are going to prove Theorem 1.2.
Proof. We are going to prove that the function u found in Lemma 2.4 and studied in Lemma 2.6, Proposition 1 and Corollary 1, is a weak solution to problem (1). We use the same technique as in [11,21].
We first prove (5) for every positive C 1 0 (Ω) function ϕ. With the same argument as in the previous theorem (i.e., using Fatou's lemma) one can prove that To prove the opposite inequality, we slightly modify the previous proof, since we no longer have uniform estimates of u n in H 1 0 (Ω). Observe that, however, T k (u n ) is uniformly bounded in H 1 0 (Ω). Indeed, it is sufficient to consider T k (u n ) as a test function in (6): we obtain by hypothesis (2). We will use, for k ∈ N and s ∈ R to define a test function. We set, for t ≥ 0, This is possible, since θ < 1. We consider where ϕ is a positive C 1 0 (Ω) function and j ∈ N , as a test function in (6). Then The sum of the last two terms is positive, since b(x) ≥ α by hypothesis (2) and by inequality Dropping the non-negative term at the limit as n → ∞ we have, by Fatou's lemma, the weak convergence of u n in W 1,σ 0 (Ω) and the weak convergence of T k (u n ) in H 1 0 (Ω), As in the previous proof, it is now sufficient to pass to the limit as j → ∞ first, using that e −H0(u) e H 1 j (Tj (u)) ≤ 1 and Corollary 1, and then to the limit as k → ∞, using that R k (u) tends to 1. We thus obtain Inequalities (18) and (19) imply that for every positive ϕ ∈ C 1 0 (Ω). Now, let ϕ any C 1 0 (Ω) function. We define ϕ ε ± = ρ ε * ϕ ± as the convolution of a mollifier ρ ε , for ε > 0, with ϕ ± . Then ϕ ε ± is a positive C 1 0 (Ω) function, for ε sufficiently small. By (20) we have Since ϕ ε − − ϕ ε − → ϕ uniformly in Ω and in W 1,q 0 (Ω) for every q ≥ 1, as ε → 0, the result follows.

4.
Existence results in the case 1 ≤ θ < 2 As in the above case, we need to prove that the function u found in Section 2 is not 0 in Ω. To this aim, we are going to prove that for every ω ⊂⊂ Ω there exists a positive constant c ω such that the solutions u n to problems (6) satisfy u n ≥ c ω in ω for every n ∈ N. We will follow a similar technique to that one in [5]. The following theorem, proved in [18] (and in [5]), will be useful to us. Theorem 4.1. Let B : Ω × R → R be a Carathéodory function such that for every ω ⊂⊂ Ω there exists m ω > 0 such that B(x, s) ≥ m ω l(s) for a.e. x ∈ Ω and for every s ≥ 0. Assume that l : R + → R + is a continuous increasing function such that l(s)/s is increasing for s sufficiently large and for some t 0 > 0 Then for every ω ⊂⊂ Ω there exists a constant C ω > 0 such that every sub- Remark 2. We recall that a sub-solution of −div(b(x)∇v) + l(v)g(x) = 0 is a W 1,1 loc (Ω) function such that for every C ∞ c (Ω) positive function φ. Remark 3. In the literature condition (21) is called the Keller-Osserman condition, due to the papers [17,19] on semilinear equations. Proposition 2. Let 1 ≤ θ < 2. Let u n be the solutions of (6). Then for every ω ⊂⊂ Ω there exists a strictly positive constant c ω such that u n ≥ c ω in ω for every n ∈ N.
Proof. Step 1. Let u n be a H 1 0 (Ω) ∩ L ∞ (Ω) solution to (6). We perform a change of variable in order to get a sub-solution of an elliptic semi-linear problem, as in Theorem 4.1.
We set a n (s) = 1 (1 + T n (s)) p . Then u n satisfies, distributionally, that is, Let k n (t) = t 1 B αr θ a n (r) dr and ψ n (s) = a ′ n (s) a n (s) − B αs θ a n (s) .
Step 3. We are going to prove that l satisfies the hypotheses of Theorem 4.1. We observe that l is continuous and increasing, since ψ −1 1 is decreasing and k 1 is increasing. We claim that l(s)/s is increasing for s sufficiently large. This is is decreasing for small positive t. Now, We remark that l ′ (ψ 1 (s)) = B αs θ a β α +1 1 (s) . Let w 0 ∈ (0, 1) be such that h(t) = is decreasing in (0, w 0 ]. Therefore We have proved that If t is sufficiently small, the last quantity is positive, since h is decreasing for small positive t. Therefore (27) holds.
We are going to study the last condition on l, that is, the existence of a positive t 0 such that

Using the change of variable
It is easy to see that e −k1(τ ) − 1 ≥ 1 2 e −k1(τ ) for τ ≤ τ 0 sufficiently small. Moreover a 1 (τ ) ≥ 1 2 , for τ ≤ 1. Therefore it suffices to find t 0 sufficiently large (t 0 > ψ 1 (τ 0 )) such that The last integral can be estimated, using the change w = ψ −1 1 (t) and the fact that a 1 (s) ≤ 1, in the following way: dτ where w 0 is chosen in such a way that k ′ 1 is decreasing in (0, w 0 ]. We observe that Hence it suffices to prove that there exists a strictly positive constant c such that Observe that e 2k1(w) → 0 as w → 0, since k 1 (w) = w 1 B αt θ a 1 (t) dt → −∞ as w → 0, by hypothesis θ ≥ 1. Therefore (28) is proved.
Proof. As in the proof of Corollary 1, we pass to the limit in (7) using the a.e. convergence of u n to u (see Lemmata 2.3, 2.4 and 2.5), the a.e. convergence of ∇u n to ∇u (see Lemma 2.6) and Proposition 2.
Corollary 3. For every ω ⊂⊂ Ω there exists a positive constantc ω such that u n (u n + 1 n ) 1+θ ≤c ω ∀ x ∈ ω . Proof. It is sufficient to observe that in every subset ω ⊂⊂ Ω u n (u n + 1 since u n ≥ c ω > 0 in ω by Proposition 2. As in [5] we prove the strong convergence of T k (u n ) in H 1 loc (Ω). This will be used to compute the limit of the lower order term in problems (6). for all positive φ ∈ C ∞ c (Ω). Let ϕ λ (s) = se λs 2 , λ > 0. As in [11], we will consider as a test function ϕ λ (T k (u n ) − T k (u))φ, where λ will be chosen later. In the sequel ε(n) will denote any quantity converging to 0, as n → ∞. From (6) we get (29) It is not difficult to prove that as n → ∞. Indeed for the first limit one can use the Lebesgue Theorem. For the second one it is sufficient to observe that ∇u n converges weakly in some Sobolev space given by the statements of Lemmata 2.3, 2.4 and 2.5 and b(x) (1 + T n (u n )) p ∇φ ϕ λ (T k (u n )− T k (u)) is uniformly bounded with respect to n.
We are going to treat the left hand side of (29). We choose ω φ ⊂⊂ Ω, with suppφ ⊂ ω φ . Then by Corollary 3. We deduce from (29) that We remark that Hence inequality (30) is equivalent to Remark that Adding the above quantity in both sides of (31) we get By hypothesis (2) on b, we obtain It is easy to prove that We deduce from (32) that the quantity (33) tends to 0. Now, ϕ λ has the following property: for every a, b > 0, Applying this inequality to the quantity (33), the statement of the theorem is proved.
We are now going to prove Theorems 1.3 and 1.4 in a unique proof. As we will see the only difference is the choice of the test functions ϕ. Theorem 1.5 can be proved with the same technique.
Proof. By Lemmata 2.3 and 2.4 the solutions u n to (6) are uniformly bounded in H 1 0 (Ω) and W 1,σ 0 (Ω) respectively; moreover ∇u n converges to ∇u a.e. in Ω up to a subsequence, by Lemma 2.6. The solutions u n satisfy For the proof of Theorem 1.3 we consider for ϕ a bounded H 1 0 (Ω) function. For the proof of Theorem 1.4, ϕ is a C 1 0 (Ω) function. To compute the limit of the first term in the case where u n weakly converges to u in H 1 0 (Ω) (Theorem 1.3) it is sufficient to use that b(x) (1 + T n (u n )) p ∇ϕ strongly converges to b(x) (1 + u) p ∇ϕ in (L 2 (Ω)) N for every ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω). In the case where u n weakly converges to u in W 1,σ 0 (Ω), with σ < 2 (Theorem 1.4), one uses that b(x) (1 + T n (u n )) p ∇ϕ strongly converges to b(x) (1 + u) p ∇ϕ in (L r (Ω)) N for every r ≥ 1 and for every ϕ ∈ C 1 0 (Ω).