ABSTRACT
Recall that there exists singular solutions to PDEs of the form (1.3). Take, for example,
us(x) =−2 ln |x | , (4.1) solving (1.3) in the unit ball B ⊂ RN , N ≥ 3, for the nonlinearity f (u) = λeu, with λ = λs := 2(N − 2). Also recall that, when N ≥ 10, us is the monotone limit of a curve of (regular) stable solutions uλ, λ < λs. So, us is stable. In addition, even if our interest rested solely in studying the nicest possible solutions-say the smooth, radial, and stable ones-no a priori bound holds true in full generality, as examplified by the aforementioned curve uλ.
