Regularizing effect of the interplay between coefficients in some nonlinear Dirichlet problems with distributional data
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Regularizing effect of the interplay between coefficients in some nonlinear Dirichlet problems with distributional data David Arcoya1 · Lucio Boccardo2 · Luigi Orsina2 Received: 10 October 2019 / Accepted: 23 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove that the solution u of Dirichlet problem (1.1) has exponential summability under the only assumption that there exists R > 0 such that |F(x)|2 ≤ R a(x) ; furthermore, we prove the boundedness of u under the slightly stronger assumption that there exists R > 0 such that |F(x)|p ≤ R a(x) , p > 2. Keywords Nonlinear elliptic equations · Regularizing effect · Interplay between coefficients Mathematics Subject Classification 35J60 · 35B51 · 35B65
1 Introduction and statement of the results In this paper, we study the existence of regular (with respect to the summability or boundedness) weak solutions of the problem
u ∈ W01,2 (Ω) ∶ −div(M(x, ∇u)) + a(x) u = −div(F).
(1.1)
where Ω is a bounded set in ℝN and −div(M(x, ∇u)) is a classical nonlinear differential operator, defined by a Carathéodory function M(x, 𝜉) satisfying, for some 0 < 𝛼 ≤ 𝛽 , and for almost every x in Ω,
* Luigi Orsina [email protected] David Arcoya [email protected] Lucio Boccardo [email protected] 1
Departamento de Análisis Matemático, Universidad de Granada, Granada, Spain
2
Dipartimento di Matematica, “Sapienza” Università di Roma, Rome, Italy
13
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D. Arcoya et al.
{
M(x, 𝜉) 𝜉 ≥ 𝛼|𝜉|2 , |M(x, 𝜉)| ≤ 𝛽|𝜉|, ∀𝜉 ∈ ℝN , [M(x, 𝜉) − M(x, 𝜂)](𝜉 − 𝜂) > 0, ∀𝜉, 𝜂 ∈ ℝN , 𝜉 ≠ 𝜂.
(1.2)
Our key assumption is that the function a(x) and the vector-valued function F(x) are such that
0 ≤ a(x) ∈ L1 (Ω),
(1.3)
∃ R > 0 such that |F(x)|2 ≤ R a(x) .
(1.4)
The existence of bounded solutions for (1.1) can be proved, without assumption (1.4), if |F| belongs to Lm (Ω) , for some m > N . This is a consequence of the positivity of a (assumption (1.3)) and of Stampacchia-type estimates (see [12]): see “Appendix.” Note that, in our case, since a belongs to L1 (Ω) it follows from (1.4) that |F| belongs to L2 (Ω) , so that (once again by the fact that a is positive), existence of a solution u in W01,2 (Ω) can be proved using classical arguments (see the first part of Theorem 2.1 in Sect. 2 and “Appendix”). We shall 2 prove in the second part of the cited theorem that equ − 1 belongs to W01,2 (Ω) for every 𝛼 q < R. This result must be compared with the corresponding one in [1] where the same problem is studied replacing the term −div(F) with function source f (x) ∈ L1 (Ω) . In this case, existence of a bounded solution in W01,2 (Ω) , and not only an exponentially summable one, is obtained showing the regularizing effect of the term a(x) u when condition (1.4) holds true. Recent works studying this effect can be found in [2] (see also [5–7, 9–11]). In case of problem (1.1), the boundedness of solutions under assumption (1.4) (as it happens for Lebesgue data in [1]) remains as an o
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