Nonlinear degenerate elliptic equations with a convection term

We prove some existence and regularity results for the solutions of the Dirichlet problem associated with a nonlinear degenerate elliptic equation with a convection term.


Introduction
This paper deals with the homogeneous Dirichlet problem associated with the nonlinear second order elliptic equation where Ω is a bounded, open subset of ℝ N , with N > 2 , M ∶ Ω → ℝ N 2 is a matrix with measurable entries such that with , > 0 and and are two real numbers such that and > 0 . The terms E and f are a vector field and a function, respectively, satisfying suitable summability assumptions. Already in the case E = 0 , the main difficulty in dealing with Eq. (1) is due to the fact that the nonlinear operator though well defined from W 1,2 0 (Ω) into its dual, is not coercive when v is large. Nevertheless, some existence and regularity results of the solutions to the Dirichlet problem depending on the summability of the source term f have been proved in [1,3,4,6].
On the other hand, if E ≠ 0 , = 0 and = 1 the study of the boundary value problem goes back to the paper [16] by Stampacchia, where the existence and regularity of weak solution have been studied provided and If no smallness assumption on ‖�E�‖ L N (Ω) is assumed, the operator fails to be coercive and the existence and regularity of the solutions of problem (5) have been studied in [2] (see also [9][10][11][12][13][14][15]) by approximating the linear problem (5) with a sequence of nonlinear coercive Dirichlet problems. Dealing with the Eq. (1) and in order to follow the strategy used in the linear nondegenerate framework, the main tool is the achievement of a logarithmic estimate on the gradient of suitable approximating solutions, which we can obtain, at least formally, as follows. Assume that we can choose as test function in the weak formulation of the homogeneous Dirichlet problem associated with Eq. (1).
(1 + |s|) +2 ds 1 3 Nonlinear degenerate elliptic equations with a convection… Then, easy calculations lead to the estimate and if + ≤ 1 we obtain At last, by Sobolev imbedding, for any k > 0 the N-dimensional measure of the level set {x ∈ Ω ∶ |u(x)| > k} satisfies the inequality with c > 0 independent of u, which, in turn, is the starting point for obtaining suitable a priori estimates. Consequently, the most interesting choice of the parameters and is = 1 − and, up to now, we will concern with the problem Namely, we prove some existence and summability results of weak or distributional solutions, according to the summability of the right hand side.
The paper is organized as follows. In the next section we state the main theorems; in Sect. 3 we introduce a sequence of approximating problems and we prove a priori estimates. At last, Sect. 4 is devoted to the proofs of the main results.

Statements of the main results
The first theorem we state concerns with the existence of finite energy solutions of the problem (7). (2), (3), (6) hold and let f ∈ L m (Ω) with Then there exists u ∈ W 1,2 0 (Ω) ∩ L (1− )m * * (Ω) which is a weak solution of the problem (7), that is
As usually, if 1 < p < N we set p * = pN N − p .
When the summabilities of the source f and of the convection term E are higher then N 2 and N respectively, the weak solutions of the problem (7) are bounded according to the following Let u ∈ W 1,2 0 (Ω) be a weak solution of the problem (7).

Then
We point out that the number is bigger than the duality exponent , then we can't expect solutions with finite energy as the following example shows.

Example 1
Let Ω = B(0, 1) be the unit ball of ℝ N centered at the origin and We point out that, thanks to the choice of the parameters , and , , but Du ∉ L 2 (Ω).
However, if the summability of the datum is lower then we can prove the existence of distributional solutions of the problem (7), i.e. functions belonging to W 1,1 0 (Ω) (infinite energy solutions) such that the following integral identity holds for every ∈ C ∞ 0 (Ω). Actually, if the summability of f is sufficiently larger than 1, we can prove the existence of distributional solutions of the problem (7) belonging to W 1,q 0 (Ω) , with suitable 1 < q < 2 , as the following theorem states: Theorem 3 Assume that hypotheses (2), (3), (6) hold and let f ∈ L m (Ω) with , which is a distributional solution to the problem (7).

Remark 1
The results stated above coincide with the classical regularity results for uniformly elliptic equations if = 0 (see [2,16]), and with those obtained for degenerate elliptic equation without convection term if 0 < < 1 (see [1,3,6]). Now, we focus our attention on the existence of distributional solutions when the right-hand side has the borderline summability that is Namely, we will prove the following theorems.

Then there exists
, which is a distributional solution of the problem (7).
Then there exists u ∈ W 1,1 0 (Ω) which is a distributional solution of the problem (7).

Remark 2
If E = 0 the existence of W 1,1 0 -solutions was first studied in [5] in the case = 0 and in [4] in the degenerate framework (see also [7,8]). Our results retrieve the results already known in these cases.

A priori estimates
In this section we follow the approach used in [2,6] (see also [1,3]).
As usual, for any k > 0 we denote by T k (s) the standard truncation function defined by and we set For every n ∈ ℕ and for a.e. x ∈ Ω , let us introduce the bounded functions and let us consider the following approximating problems: .

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Nonlinear degenerate elliptic equations with a convection… Thanks to the Schauder's Theorem and using the existence and boundedness results of [6], for every n ∈ ℕ , there exists u n ∈ W 1,2 0 (Ω) ∩ L ∞ (Ω) which is a weak solution to the problem (12).
For every n ∈ ℕ and k > 0 we denote by the level set of u n and by |A n (k)| its N− dimensional measure. Next Lemma shows that, under minimal assumptions on the summabilities of E and f, the N− dimensional measure of the set A n (k) is small enough as k goes to infinity, uniformly with respect to n. (2), (3) hold and let |E| ∈ L 2 (Ω) and f ∈ L 1 (Ω) . Let u n ∈ W 1,2 0 (Ω) be a weak solution of the problem (12). Then, for any > 0 there exists k such that uniformly with respect to n. (12), we use the assumption (2) and Young's inequality and we get with > 0 to be choosen later. Here and in the sequel we will denote by c various positive constants, whose values may depend on , N, meas(Ω), , ‖�E�‖ L N (Ω) and on the L m (Ω) norm of the source f.

Lemma 1 Assume that hypotheses
Choosing a suitable 0 < < 1 , from the above relation it follows and using Sobolev's inequality in the left-hand side we get the following estimate and, consequently, the Lemma follows. ◻ Lemma 2 Assume that hypotheses (2), (3) hold and let |E| ∈ L 2 (Ω) and f ∈ L 1 (Ω) . Let u n ∈ W 1,2 0 (Ω) be a weak solution of the problem (12). Then, there exists a positive constant c, only depending on such that for any k > 0.
Proof Let k > 0 . We take v = T k (u n ) as test function in (12). By using Young's inequality we have Easy calculations lead to which implies estimate (17). ◻ Now, we state some a priori estimates on u n depending on the summability of the source term f.
Proof In the following, as usual, we set [s] + = max{0, s}, ∀s ∈ ℝ . We choose v = |u n | 1+ − k 1+ + sgn(u n ) as test function in (12). By using (2) and taking into account the estimate we have Let > 0 . Noting that |u n | ≤ |G k (u n )| + k and applying Hölder's and Young's inequalities from the previous estimate we deduce Thanks to the hypotheses |E| ∈ L N (Ω) and (19), and using Sobolev's and Young's inequalities, we derive .
Therefore, thanks to Lemma 1 we can choose suitable < 1 and k > k * such that hence From (22) and (17) Proof We take as test function in (12) Let > 0 . Applying Young's and Hölder's inequalities in the right-hand side of the above inequality we obtain

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Nonlinear degenerate elliptic equations with a convection… Choosing a suitable 0 < < 1 , we obtain By using Sobolev's inequality in the left-hand side of (25) we get Assumption |E| ∈ L N (Ω) and Lemma 1 imply that there exists k > 0 such that, if k ≥k , we have Therefore, .

Nonlinear degenerate elliptic equations with a convection…
The Hölder's inequality with exponent 2 q and 2 2−q together with the estimates (29) and (30) gives This estimate implies that the sequence u n is bounded in W 1,q 0 (Ω) . ◻

Then the sequence u n is bounded in
Proof We take as test function in (12) the following function where is a positive number that we will choose later in order to have 2 − 1 − > 0 . Using Young's and Sobolev's inequalities, we obtain Assumption |E| ∈ L N (Ω) and Lemma 1 imply that there exists k such that, if k ≥k , we have Therefore, Let 1 ≤ q < (1 − )N N − (1 + ) and = N(2−q) 2(N−q) (note that q < 2 and 2 − 1 − > 0 thanks to the choice of q). We write The Hölder's inequality with exponent 2 q and 2 2−q together with the estimates (34) and (33) gives since 2q 2 − q = 2 * (1 − ).

Proofs of the main results
Proof of Theorems 1 and 3. By Lemma 4 the sequence u n is bounded in L m * * (1− ) (Ω) . Moreover, thanks to Lemmas 3 and 5 u n is bounded in W 1,q 0 (Ω) , with q = 2 or q = mN(1− ) N−m(1+ ) depending on whether the hypotheses of Theorem 1 or 3, respectively, are verified. Then up to subsequences, u n converges weakly to some function u which belongs to the same spaces. Moreover, u n converges to u a.e. in Ω.
We fix ∈ C ∞ 0 (Ω) and we take v = in (12). We obtain We define Thanks to the convergence of u n to u a.e. in Ω , the sequence Y n converges to We note that then, by using Hölder's inequality, for any measurable subset A ⊆ Ω we have Finally, the sequence f n converges to f in L m (Ω) , therefore passing to the limit as n → +∞ in (12), u satysfies the integral identity (10). Then, if the assumptions of Theorem 3 hold we conclude that u is a distributional solution of (7) and the thesis follows.
Instead, under the assumptions of Theorem 1, by standard density arguments, we deduce that the integral identity (10) holds also for any function ∈ W 1,2 0 (Ω) , which means that u is a weak solution of (7).

Proof of Theorem 4 Thanks to Lemma 6 the sequence u n is bounded in
. Then there exists a subsequence, not relabelled, which converges to some function u weakly in W 1,q 0 (Ω) , strongly in L p (Ω) , for any 1 ≤ p < q * and a.e. in Ω . We note that, using Hölder's and Sobolev's inequalities, for any measurable subset A ⊆ Ω we have with Y n defined by (37), and the boundedness of the sequence Du n in L q (Ω) implies the equi-integrability of Y n . Up to now, we can argue as in the previous proof and we get the thesis. ◻

Proof of Theorem 5
Here we follow the outlines of [4,5].
Since {u n } is bounded in W 1,1 0 (Ω) up to a subsequence still denoted by {u n } , it converges to some function u strongly in L r (Ω) , for any 1 ≤ r < N N−1 and a.e. in Ω.  Thanks to the almost everywhere convergence of {u n } to u we deduce that v = (1 + |u|) − 1)sgn(u). Now, we will prove that up to subsequences, u n is weakly convergent to u in W 1,1 0 (Ω) . Given k > 0 , let us take |u n | 2 −1+ − k 2 −1+ + sgn(u n ) as test function in (12) (note that 2 − 1 + > 0 since 1 N−1 < < 1 ) and we deduce Note that in the set {|u n | > k} it results and using the Hölder inequality in the right-hand side of (39) we obtain Nonlinear degenerate elliptic equations with a convection… Using the estimate (17) we deduce that holds uniformly with respect to n and, at last, we conclude that the sequence {Du n } is equiintegrable. By the Dunford Pettis Theorem, for any i = 1, … , N there exists is the distributional derivative of u n we have and by virtue of the strong convergence on {u n } to u in L r (Ω) , with 1 ≤ r < N N−1 we can pass to the limit as n → +∞ in the previous equality, obtaining that is, y i = u x i , ∀i = 1, … , N and thus u ∈ W 1,1 0 (Ω). Since < 1 the function g(s) = (1 + |s|) − 1)sgn(s) is Lipschitz continuous and by the chain rule it results almost everywhere in Ω . At last, using the weak convergence of {Dv n } to Dv in L 2 (Ω) we conclude that Now, we can pass to the limit in the approximate problems (12). Let ∈ C ∞ 0 (Ω).We note that or any measurable subset A ⊆ Ω and the boundedness of the sequence Du n in L 1 (Ω) implies the equi-integrability of Y n . Then, passing to the limit as n → +∞ in the approximating problems (12) we obtain that u is a distributional solution of the problem (7). ◻

Proof of Theorem 6
Let k ≥ 0 , we take as test function in (12). Thanks to the assumption (2) and Young's inequality we obtain We observe that the following inequality holds for all s, t positive real numbers and 0 < < N−2 N−1 . Now, using Hölder's inequality, from (46) for any k ≥ 0 we deduce As usual, by assumption |E| ∈ L N (Ω) and Lemma 1 we can choose k ≥ 1 such that therefore By using Hölder's inequality with exponent (1− )2 * 2 on the last term of the right hand side, we get By the choice of the above inequality implies that (50) Nonlinear degenerate elliptic equations with a convection… Now, we can argue as in [4] (see also [5] ) and we deduce that u n ⇀ u weakly in W 1,1 0 (Ω) and Let ∈ C ∞ 0 (Ω) . We note that and the integral in the right-hand side converges to since (54) holds and because 1 − < N N−1 and |M(x)D | is bounded. Due to (53), inequality (38) is also valid for m = 1 therefore we deduce Finally, the sequence f n converges to f in L 1 (Ω) , therefore passing to the limit as n → ∞ in (12), u satysfies the integral identity (10), which means that u is a distributional solution of (7) and thesis of the Theorem 6 holds. ◻

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