A priori estimates for elliptic equations with reaction terms involving the function and its gradient
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Mathematische Annalen
A priori estimates for elliptic equations with reaction terms involving the function and its gradient Marie-Françoise Bidaut-Véron1 · Marta Garcia-Huidobro2 · Laurent Véron1 Received: 10 November 2018 / Revised: 10 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study local and global properties of positive solutions of −u = u p + M |∇u|q in a domain of R N , in the range min{ p, q} > 1 and M ∈ R. We prove a priori estimates and existence or non-existence of ground states for the same equation. Mathematics Subject Classification 35J62 · 35B08 · 35B45
Contents 1 Introduction . . . . . . . . . . . . 2 The direct Bernstein method . . . . 2.1 Proof of Theorems A, A and C 2.1.1 Proof of Theorem A . . 2.1.2 Proof of Theorem A . . 2.1.3 Proof of Theorem C . . 2.2 Proof of Theorems B and B . 2.2.1 Proof of Theorem B . . 2.2.2 Proof of Theorem B . . 3 The refined Bernstein method . . . 3.1 Proof of Theorem D . . . . . 4 The integral method . . . . . . . . 4.1 Preliminary inequalities . . . .
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Communicated by Y. Giga.
B
Laurent Véron [email protected] Marie-Françoise Bidaut-Véron [email protected] Marta Garcia-Huidobro [email protected]
1
Laboratoire de Mathématiques et Physique Théorique, Université de Tours, 37200 Tours, France
2
Departamento de Matematicas, Pontifica Universidad Catolica de Chile, Casilla 307, Correo 2, Santiago de Chile, Chile
123
M.-F. Bidaut-Véron et al. 4.2 Proof of Theorem E . . . . . . . . . . . . . 5 Radial ground states . . . . . . . . . . . . . . . 5.1 Energy functions . . . . . . . . . . . . . . 5.1.1 Exponential perturbations . . . . . . 5.1.2 Pohozaev–Pucci–Serrin type functions 5.2 Some known results in the case M < 0 . . . 5.3 The case M > 0 . . . . . . . . . . . . . . . +2 5.3.1 The case M > 0, 1 < p ≤ N N −2 . . . +2 5.3.2 The case M > 0, p > N N −2 6 Separable solutions . . . . . . . . . . . 6.1 Constant solutions . . . . . . . . . 6.2 Bifurcations . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .
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