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We discuss the existence of entire (i.e. defined on the whole space) subsolutions of fully nonlinear degenerate elliptic equations, giving necessary and sufficient conditions on the coefficients of the lower order terms which extend the classical Keller--Osserman conditions for semilinear elliptic equations. Our analysis shows that, when the conditions of existence of entire subsolutions fail, a priori upper bounds for local subsolutions can be obtained., CONTENTS 1. Introduction 469 2. Radial Solutions and Comparison Principle 472 3. Existence of Entire Subsolutions and a Priori Upper Bounds 474 References 477 1. Introduction In this note, we [...]

Francesco M Quaranta-Leoni,1,2 Maria Grazia Fiorino,1 Flavia Quaranta-Leoni,3 Matteo Di Marino1 1Orbital and Adnexal Service, Villa Tiberia Hospital – GVM Care & Research, Rome, 00137, Italy; 2Oftalmoplastica Roma, Rome, 00197, Italy; 3Catholic University of the Sacred Heart, Rome, 00168, ItalyCorrespondence: Francesco M Quaranta-Leoni Email fquarantaleoni@gmail.comAbstract: Anophthalmic socket syndrome determines functional deficits and facial deformities, and may lead to poor psychological outcomes. This review aims to comprehensively evaluate the features of the syndrome, based on literature review and authors’ clinical and surgical experience. An electronic database (PubMed,MEDLINE and Google Scholar) search of all articles written in English and non-English language with abstract translated to English on anophthalmic socket syndrome was performed. Data reviewed included demographics, presentations, investigations, management, complications and outcomes. Different types of orbital implants were evaluated; the management of implant exposure was examined; different orbital volume enhancement procedures such as secondary implantation, subperiosteal implants and the use of fillers in anophthalmic patients were described; the problems related to socket contraction were outlined; the treatment options for chronic anophthalmic socket pain and phantom eye syndrome were assessed; the most recent advances in the management of congenital anophthalmia were described. Current clinical evidence does not support a specific orbital implant; late exposure of porous implants may be due to pegging, which currently is seldom used; filler absorption in the orbit appears to be faster than in the dermis, and repeated treatments could be a potential source of inflammation; socket contraction results in significant functional and psychological disability, and management is challenging. Patients affected by anophthalmic socket pain and phantom eye syndrome need specific counseling. It is auspicable to use a standardized protocol to treat children affected by clinical congenital anophthalmia; dermis fat graft is a suitable option in these patients as it helps continued socket expansion. Dermis fat graft can also address the volume deficit in case of explantation of exposed implants and in contracted sockets in both children and adults. Appropriate clinical care is essential, as adequate prosthesis wearing improves the quality of life of anophthalmic patients.Keywords: orbital implants, socket surgery, contracted socket, phantom eye syndrome, congenital anophthalmia

We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller-Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions., UDC 517.956 CONTENTS 1. Introduction 2. Pucci and Partial Laplace Operators 3. Sign Propagation and the Generalized Principal Eigenvalue 4. Entire Subsolutions References 1. Introduction Nonlinear degenerate elliptic equations arise [...]