Uniqueness for elliptic problems with Hölder--type dependence on the solution

We prove uniqueness of weak (or entropy) solutions for nonmonotone elliptic equations of the type 
\begin{eqnarray} 
-div (a(x,u)\nabla u)=f 
\end{eqnarray} 
in a bounded set $\Omega\subset R^N$ with Dirichlet boundary conditions. The novelty of our results consists in the possibility to deal with cases when $a(x,u)$ is only Holder continuous with respect to $u$.


Introduction.
Let Ω be a bounded subset of R N , N ≥ 1, and consider the nonlinear elliptic Dirichlet problem −div(a(x, u)∇u) = f in Ω , u = 0 on ∂Ω.
In many situations, one is led to consider the case that a(x, u) has either unbounded or singular growth with respect to u. For example, in the paper [17], A. G. Kartsatos and I. V. Skrypnik used the differential operator as a simple example of elliptic operator with strong coefficient growth, where a(x) is a measurable function such that 0 < α ≤ a(x) ≤ β for some α, β ∈ R + . Classical questions as existence or uniqueness of solutions have to be handled with care in order to deal with such examples. Even the formulation itself of the problem, and the notion of solution considered, deserves some attention, since weak solutions may not have sense. To this purpose, the notions of renormalized solution or entropy solution, introduced in [8], [2] respectively, have proved to be suitable, in particular to deal with the case of coefficients with unbounded growth with respect to u. For example, a general existence result of entropy solutions of (1) when f ∈ L 1 (Ω) is proved in [18], which in particular applies to the operator (2). In particular cases, using the growth of the operator, it is also possible to show that such solutions are more regular, for example if f ∈ H −1 (Ω) and the operator is coercive then u ∈ H 1 0 (Ω) and even more, see e.g. [5], [22].
The problem of uniqueness of solutions of (1) has also been the object of several works. First of all, let us recall that the operator A(u) = −div(a(x, u)∇u) is never monotone, unless a(x, u) does not depend on u, see [7]. Therefore the question of uniqueness is not trivial. In the case that a(x, s) is coercive and bounded, uniqueness results for H 1 0 (Ω) solutions have been proved in [1], [24] under the assumption that a(x, u) is (globally) Lipschitz continuous with respect to u. The method used in these latter papers was developed furtherly in [10]; some generalization to the case that a(x, s) is locally Lipschitz continuous with respect to s can be found in [5], [20], still in the context of finite energy solutions (i.e. a(x, u)|∇u| 2 ∈ L 1 (Ω)). Note that in order to consider possibly general growth conditions of the function a(x, s), a generalized concept of solution is needed, to this purpose the use of approximated solutions (so-called SOLA), entropy solutions or renormalized solutions is suitable. In particular, in the Appendix, we present the method of [5] applied to the Dirichlet problem associated to the operator (2).
Finally, in the two papers [3], [21], uniqueness results for entropy solutions (or renormalized solutions) were proved in case that a(x, u) is locally Lipschitz continuous with respect to u under fairly general growth conditions on the modulus of Lipschitz continuity, and assuming only that f ∈ L 1 (Ω). For example, Theorem 1.4 in [21] proves the uniqueness of entropy solutions of (1) when the left hand side is given by the operator (2), and f ∈ L 1 (Ω). Let us mention that some generalization of the above results to p-Laplace type operators are recently given in [16].
In this paper, we consider the case where the dependence of a(x, u) with respect to u is not locally Lipschitz, being possibly singular at some point like in the case of Hölder type continuity. As we mentioned above, most, if not all, of the previous uniqueness results for the model problem (1) are confined to the case that a is locally Lipschitz with respect to u. The case that a is Hölder continuous has been considered previously only in two cases, either for evolution problems or stationary problems with additional zero order terms (see e.g. [1], [13]) or assuming extra regularity of a(x, s) with respect to x (see e.g. [11], and the recent results in [15]). Here, although we do not give a general result for Hölder type nonlinearities, we introduce a new idea which makes it possible at least to deal with several examples of Hölder continuous functions with exponent α ∈ ( 1 2 , 1). The model example of our results can be seen in the following form. We set, for t ∈ R, T 1 (t) = min(1, max(t, −1)). Theorem 1.1. Assume that a(x, s) is a Carathéodory function such that for every s , σ ∈ R and a.e. x ∈ Ω, where α, β > 0, H > 0 and Let f ∈ L 1 (Ω) ∩ H −1 (Ω). Then problem (1) has a unique weak solution u ∈ H 1 0 (Ω).
Theorem 1.1 is clearly modeled on the simplest example of Hölder nonlinearity, given by

UNIQUENESS FOR ELLIPTIC PROBLEMS 1571
Our simple idea is to treat such type of Hölder continuous functions as examples where the modulus of Lipschitz continuity becomes singular at some point. Such idea is vaguely inspired by the assumption used in case of p-Laplace type operators, see e.g. [12]. Of course, Theorem 1.1 admits a generalized version to many extents. First of all, the bound from above on a(x, s) was assumed in Theorem 1.1 just to allow us to use weak solutions. Such bound can be removed up to using a generalized concept of solution, e.g. that of entropy solution. Secondly, the continuity assumption on a(x, s) will also take a more general form allowing for combinations of Lipschitz and/or Hölder functions with possibly different Hölder exponents at different points. The general version of our result will be the following.
We refer the reader to Section 3 for the definition of entropy solution and some discussion concerning assumption (7). Let us note that, following the ideas in [3], [21], further generalizations of these results are possible in at least two directions. On one hand, by relaxing the assumptions on the modulus of Lipschitz continuity at infinity (i.e. relaxing the conditions onω at infinity), on the other hand by considering simply f ∈ L 1 (Ω) (see also the discussion at the end of Section 3). Further possible extensions may also concern the presence of convection terms, at least for nonlinearities which are locally Lipschitz continuous. However, the above extensions require more technicalities which are far beyond the scope of this note, details will appear elsewhere.
On the other hand, we mention that the range θ ∈ ( 1 2 , 1] in Theorem 1.1, as well as the assumption f ∈ L 1 (Ω), are somehow optimal, at least as far as our methods are concerned. We refer to Remark 1 for a short discussion about this point; in particular, Lemma 2.1 would not hold, for the same range of Hölder nonlinearities, only assuming f ∈ H −1 (Ω). 2. The model example of Hölder-like dependence. In this section we study the model example which is the motivation of our results. Let us consider the Dirichlet problem (1), with f ∈ L 1 (Ω) ∩ H −1 (Ω). In particular, if N > 2, any f ∈ L 2N N +2 (Ω) is admitted, due to Sobolev embedding. We assume here (3). In 1 the sup defined here is an essential supremum as far as x ∈ Ω is concerned. particular, the fact that the coefficient a(x, s) is bounded allows us to consider weak solutions, i.e. : where ·, · denotes the duality between H −1 (Ω) and H 1 0 (Ω). A key point in the proof will be played by the following lemma, which explains condition (5). .
Letting go to zero, thanks to Fatou's Lemma we deduce (9).
Remark 1. The condition θ ∈ ( 1 2 , 1] is optimal for Lemma 2.1 to hold. It is enough to consider the case of the Laplace operator and f ≥ 0 to observe that, in the best situation, we have u = O (d(x)) as x → ∂Ω, where d(x) is the distance function to ∂Ω. This is just consequence of the Hopf boundary lemma, stating in addition that |∇u| ≥ γ at ∂Ω for some γ > 0. Therefore, we have, for some δ > 0: and last integral is not finite for every θ ≤ 1 2 . We also stress that the L 1 (Ω) character of f is also essential in the above estimate. As easily shown by one dimensional examples, assuming only f ∈ H −1 (Ω) would not be enough to obtain a similar estimate (at least with the same range of θ).
We can now prove Theorem 1.1, whose proof follows the ideas of [10] in connection with Lemma 2.1.
Proof of Theorem 1.1. Let u, z be two weak solutions of (1). We have then, for any test function ϕ ∈ H 1 0 (Ω): We take ϕ = T ε [z − u], ε > 0, and we have Using the coercivity of a(x, s) and Young's inequality we get which implies, using (4), Using Poincaré inequality we get For every fixed δ > ε we have Remark that The decreasing continuity of the measure implies that from Lemma 2.1 and the fact that u belongs to H 1 0 (Ω) it follows that Passing to the limit as ε → 0 in (11), we deduce meas {x ∈ Ω : |u(x) − z(x)| > δ} = 0 for every δ > 0, hence u = z.

3.
Generalizations. In this section we generalize the example of the previous section, which is of course very special to many regards, in particular a(x, s) was supposed to be singular at only one point. The boundedness assumption on a(x, s) was also not essential but for considering standard weak solutions. Here we assume that a(x, s) only satisfies and that In order to deal with possibly unbounded functions a(x, s), a generalized concept of solution is needed. We recall (see [2]) that a function u (almost everywhere finite in Ω) is said to be an entropy solution of (1) if T k (u) ∈ H 1 0 (Ω) for every k > 0 and where ·, · denotes the duality between H −1 (Ω) and H 1 0 (Ω) and, here and later, by ∇u we denote the generalized gradient as defined in [2] for entropy solutions.
Let us now generalize assumption (4). First of all, we set 2 Note that, thanks to assumption (13), ω ε (s) is a locally bounded function. We assume that ω ε (s) satisfies for some ε 0 > 0.
We start with a standard lemma. The proof is essentially contained in [21], however we give the argument here for completeness. Lemma 3.1. Let u be an entropy solution of (1). Let Ψ(s) be a locally Lipschitz function, nondecreasing, and let Ψ (s) = ψ(s) a.e. in R. If Ψ is bounded, we have 2 note that the sup here is an essential sup as far as x ∈ Ω is concerned; more precisely, Proof. First of all note that, since Ψ is locally Lipschitz and T h (u) ∈ H 1 0 (Ω), by Stampacchia's theorem (see e.g. [19] for a proof) we have Ψ(T h (u)) ∈ H 1 0 (Ω) and ∇Ψ(T h (u)) = ψ(T h (u))∇T h (u) a.e. in Ω. We choose k = Ψ ∞ and ϕ = T h (u) − Ψ(T h (u)), in the definition of entropy solution for u. We have

and (16) is proved.
Remark 2. Let us recall that assuming f ∈ H −1 (Ω) implies that entropy solutions have finite energy, in particular This is a well known fact. Indeed, from the definition of entropy solution we have Using (12) we have Letting k → ∞ yields (17). Proof.
Passing to the limit as ε → 0 in (27), we conclude that Letting k → ∞, sinceω 1 ∈ L 1 (R) we obtain meas {x ∈ Ω : |z − u| > δ} = 0 ∀δ > 0 hence z = u. Let us conclude by observing that the above method can also be used to deal with data f ∈ L 1 (Ω) which may not belong to H −1 (Ω). Indeed, the main ingredient of the arguments used is thatω(u)|∇u| 2 ∈ L 1 (Ω) for an entropy solution u, wherē ω is the function given in (15). It is well known that such regularity holds whenever ω ∈ L 1 (R), only assuming f ∈ L 1 (Ω); in this case a direct extension of the above proof is possible. Another easy case is given if, for instance,ω(s) = a globally Hölder nonlinearity; in that situation, if θ ∈ ( 1 2 , 1] and if f ∈ L m (Ω) with m ≥ 2N θ (N −2+4θ) , thenω(u)|∇u| 2 ∈ L 1 (Ω) as a consequence of Lemma 2.1 and of the regularity results in [9] which ensure u ∈ L N m N −2m (Ω) (if N > 2). However, such examples are far from being optimal, they simply allow for a straightforward extension of the above proof. A general result is possible only coupling the above idea for Hölder-type nonlinearities with the methods used in [3], [21] for locally Lipschitz nonlinearities with large growth at infinity. In particular, an adaptation of such methods can provide directly a general result for L 1 -data. 4. Appendix. In the unpublished paper [5] it is proved the uniqueness of the solutions of the Dirichlet problem assuming, for simplicity, that N > 2. Here we show how the approach of [5] can be adapted to the proof of existence and uniqueness of the solutions of the Dirichlet problem with the differential operator (2) which was treated by I. V. Skrypnik in [17]. Let us recall that the existence can be found also in [18]; for the uniqueness in the framework of entropy solutions, see [21].
Proof. We consider the approximate problems where f n converges to f in L 2N N +2 (Ω). It is quite standard to prove, for every n ∈ N, thanks to Schauder's theorem, the existence of a bounded solution u n . Moreover the use of e 2un − 1 as test function yields, dropping positive terms,

L. BOCCARDO AND A. PORRETTA
Thus the sequence {|e 2un − 1| 2 * } is bounded in L 1 (Ω), which in turn implies that the sequence {e 2un |∇u n |} is bounded in L 2 (Ω). Moreover, using u n as test function and dropping the exponential term, we have and by (29) we deduce that u n is bounded in H 1 0 (Ω). Thus, by the above estimates, there exist u ∈ H 1 0 (Ω) and a subsequence, still denoted by {u n }, such that u n converges weakly to u in H 1 0 (Ω); e 2un ∇u n converges weakly to Y in (L 2 (Ω)) N .
The limit as n → ∞ yields lim sup which implies that (see also [4]) and so u n converges weakly to u in H 1 0 (Ω), e 2un ∇u n converges weakly to e 2u ∇u in (L 2 (Ω)) N .
Then we can pass to the limit in the weak formulation of (31) and we obtain, for every v ∈ H 1 0 (Ω), In particular, we have n→∞ Ω e 2un |∇u n | 2 dx .
Thanks to the strong convergence established, it is possible to pass to the limit when we use T k [u n − ϕ] as test function in (31), so that we conclude that u satisfies (30). Now we prove that the techniques of the above existence proof can be used to show that the solution u obtained (with the above approximation method) is unique. This kind of uniqueness (solutions obtained as limit of approximation) is strongly related to the entropy solutions also used in this paper (see [14], [2], [23]). This proof follows the techniques of [5]. Let {g n } be any sequence of smooth functions converging to f in L 2N N +2 (Ω) and let w n be the weak solutions of the Dirichlet problem w n ∈ H 1 0 (Ω) : −div([a(x) + e 2wn ]∇w n ) = g n (x) . The above proof says that w n ∈ H 1 0 (Ω) ∩ L ∞ (Ω) converges in H 1 0 (Ω) to w, solution of Using T ε [u n − w n ] as test function in the equations of u n and w n , and subtracting, we have