Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data
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Calculus of Variations
Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data Marie-Françoise Bidaut-Véron1 · Quoc-Hung Nguyen2 · Laurent Véron1 Received: 30 June 2018 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the equation −div(A(x, ∇u)) = |u|q1 −1 u|∇u|q2 + μ where A(x, ∇u) ∼ |∇u| p−2 ∇u in some suitable sense, μ is a measure and q1 , q2 are nonnegative real numbers and satisfy q1 + q2 > p − 1. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity. Mathematics Subject Classification 31C15 · 35J62 · 35J92 · 35R06 · 45G15
Contents 1 Introduction and main results 2 Estimates on potential . . . . 3 Renormalized solutions . . . 4 Proof of the main results . . . References . . . . . . . . . . . .
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1 Introduction and main results This article is devoted to the study of existence of solutions of some second order quasilinear equations with measure data with a source-reaction term involving the function and its
Communicated by M. Struwe.
B
Quoc-Hung Nguyen [email protected] Marie-Françoise Bidaut-Véron [email protected] Laurent Véron [email protected]
1
Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais, Tours, France
2
ShanghaiTech University, 393 Middle Huaxia Road, Pudong, Shanghai 201210, China 0123456789().: V,-vol
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gradient. First we consider the problem with a Radon measure μ in R N in the whole space − div(A(x, ∇u)) = |u|q1 −1 u |∇u|q2 + μ in R N .
(1.1)
In this setting, (x, ξ ) → A(x, ξ ) from R N × R N to R N is a Carathéodory vector field satisfying for almost all x ∈ R N the growth and ellipticity conditions (i)
|A(x, ξ )| + |ξ ||∇ξ A(x, ξ )|| ≤ 1 |ξ | p−1 for all ξ ∈ R N , p−2
A(x, ξ ) − A(x, η), ξ − η ≥ 2 (|ξ |2 + |η|2 ) 2 |ξ − η|2 for all ξ, η ∈ R N , |A(x, ξ ) − A(y, ξ )| ≤ 1 |x − y|α0 |ξ | p−1 for all ξ ∈ R N , A(x, λξ ) = |λ| p−2 λA(x, ξ ) for all (λ, ξ ) ∈ R × R N , (1.2) −2 where 1 ≥ 2 > 0 are constants and 3N 2N −1 < p < N , and where q1 , q2 > 0 satisfy q1 + q2 > p − 1, and α0 ∈ (0, 1). The special case A(x, ξ ) = |ξ | p−2 ξ gives rise to the standard p-Laplacian p u = div (|∇u| p−2 ∇u). Note that these conditions imply that A(x, 0) = 0 for a.e. x ∈ R N , and (ii) (iii) (iv)
∇ξ A(x, ξ )λ, η ≥ 2
p−2 2
2 |ξ | p−2 |
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