The existence of solutions to a class of quasilinear elliptic problems on noncompact Riemannian manifolds, with finite volume, is investigated. Boudary value problems, with homogeneous Neumann conditions, in possibly irregular Euclidean domains are included as a special instance. A nontrivial solution is shown to exist under an unconventional growth condition on the right-hand side, which depends on the geometry of the underlying manifold. The identification of the critical growth is a crucial step in our analysis, and entails the use of the isocapacitary function of the manifold. A condition involving its isoperimetric function is also provided.
Quasilinear elliptic equations on noncompact Riemannian manifolds / Barletta, Giuseppina; Cianchi, Andrea; Maz'Ya, Vladimir. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - STAMPA. - 273:(2017), pp. 3426-3462. [10.1016/j.jfa.2017.08.018]
Quasilinear elliptic equations on noncompact Riemannian manifolds
CIANCHI, ANDREA;
2017
Abstract
The existence of solutions to a class of quasilinear elliptic problems on noncompact Riemannian manifolds, with finite volume, is investigated. Boudary value problems, with homogeneous Neumann conditions, in possibly irregular Euclidean domains are included as a special instance. A nontrivial solution is shown to exist under an unconventional growth condition on the right-hand side, which depends on the geometry of the underlying manifold. The identification of the critical growth is a crucial step in our analysis, and entails the use of the isocapacitary function of the manifold. A condition involving its isoperimetric function is also provided.I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.