Positive Solutions to Semilinear Elliptic Equations With Critical Lower Order Terms

Abstract

We study the existence and nonexistence of positive (super) solutions to a semilinear elliptic equation −∆u − Ax |x| 2 • ∇u − B |x| 2 u = c |x| σ u p in cone-like domains of R N. On the plane R 2 we determine the set of (p, σ) such that the equation has no positive (super) solutions, depending on the parameters A, B ∈ R and the geometry of the domain.

Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity

“…whence (43) follows at once. Suppose now that the exponents α and β satisfy the assumptions of the theorem.…”

“…where L stands here for some second-order elliptic operator, specified in the study, while Ω is either the entire R N , or a cone, or an exterior domain; we refer to the works [7,8,10,16,17,20,24,25,27,29,31,35,39,41,42,58,60,61,64,68] in which L coincides with the standard Laplacian, to [11,12,14,19,21,44,47,63] where L is its nonlinear counterpart, the p-Laplacian, and to [13,[34][35][36]43,45,48,50,54,55] where more general linear or nonlinear elliptic operators are considered. Nevertheless, the above list is by no means exhaustive and the reader who wishes to get a panoramic view of this fascinating field should also consult the extensive treatise [49], as well as the very recent surveys [28] and [37].…”

“…[10,16,17,20,23,24,26,31,39,58]), as well as on the derivation of pointwise estimates like Phragmén-Lindelöf-type bounds or Harnacktype estimates (cf. [34][35][36][43][44][45]), or (iii) pointwise estimates via nonlinear potential theory (cf. [54,55]).…”

Semilinear elliptic equations for the fractional Laplacian with Hardy potential

“…However substantial difficulties have to be overcome due to the nonlocal structure of the fractional Laplacian. Nonexistence result of nonlinear elliptic problems using comparison principles have been obtained in [1], [2] [39], [36], [32], [37], [31] and the references therein. For the existence result, in the supercritical case N +2s−2α N −2s−2α > p ≥ N +2s N −2s , we have an explicit solution constructed via ϑ α .…”

Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

“…Further references in this direction are [5], [9], [12], [13]. We also quote [2], [3] [27], [29], [23], [21], [24], [20].…”