Nonexistence of positive supersolutions to some nonlinear elliptic problems

Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms

“…There exist also two unstable trajectories T 1,j unst , j=3,4, converging to P 1,M when t → −∞, associated to a positive characteristic value λ and the common slope at P 1,M is 2 p−1 −λ . We assume that T 1,3 st is locally below L and T 1,4 st locally above L, thus, in a neighbourhood of P 1,M , these trajectories belong also to the regions C and A. In particular the trajectory T Next we look for the existence of limit cycles.…”

“…The aim of this article is to study local and global properties of positive radial solutions of the equation 1) in R N or R N \ {0} where p > 1 and M is a real parameter. This is a particular case of the following class of equations − ∆u = |u| p−1 u + M |∇u| q , (1.2) where q > 1 which has been the subject or many works in the radial case when M < 0, where a basic observation is that the two terms |u| p−1 u and M |∇u| q are in competition.…”

A priori estimates for elliptic equations with reaction terms involving the function and its gradient

“…Note that this does not imply the non-existence of ground state. In [1] Alarcón, García-Melián and Quass study the equation − ∆u = |∇u| q + f (u), (1.19) in an exterior domain of R N emphasizing the fact that positive solutions are super harmonic functions. They prove that if 1 < q ≤ N N −1 and if f is positive on (0, ∞) and satisfies lim sup s→0 s −p f (s) > 0, (1.20) for some p > N N −2 , then (1.19) admits no positive supersolution.…”

“…When q is critical with respect to p the situation is more delicate since the value of M plays a fundamental role. Our first statement is a particular case of a more general result in [1], but with a simpler proof which allows us to introduce techniques that we use later on.…”

Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

“…− ∆u = f (u) in R N (2) with f (t) = t p , p > 1 was considered. Later, some other works have dealt with the same problem, either considering alternative proofs to that in [25] (see [16]) or obtaining similar results for more general nonlinearities (cf.…”