Blow-up solutions of some nonlinear elliptic problems

posed in a baunded domain fi of R~ with smooth boundary 8 SI wiih Dirichiel dala u100 = O, and a continuous, positive, increasing and convex funetion f an [O,oc) such thai f(s)/s — 00 as a —. oc>. Under Ihese conditions Ihere is a maximal or extremal value of the parameter A > O such thaI ihe problem has a solution. We invesligate Ihe exisience and properties of Ihe corresponding extrema) solutions when Ihe>’ are unhaunded (i.e., singular or blow-up solutions). We characterize ihe singular Hí extremal solulions and ihe extrenial value by a eriterioncansisiing of twa condiiions: (i) they musí be ener~’ saluiions, mal in L~; (Ii) tite>’ musí sntisfy a Hardy inequalil>’ which transíates Ihe fact thai ihe firsí eigenvalue of the linearized aperator is nannegalive. In arder to apply ibis characlerizatiat to ihe typical examples arising in Ihe lilerature we need an improved version of ihe cíassical Hardy inequalil>’ wiih besí constaní. We aiablish such a resulí as a simultaneous generalization of Hardy’s and Poincaré’s inequaUiies for ah dirnensions n > 2. A striking prapert>’ of sorne examples of unbaunded exiremal solutiona is the fact thai tite hinearization of ihe prablem araund them happens to be formalí>’ invertible and nevertheless tite npplication of ihe Inverse and Imphicii Funetion iheorems falís to produce Ihe usual exislence ar continuation resulís. We consider ihis question and explain tite pitenomenon as a lack of appropriate funetional setting.