Second-order semilinear elliptic inequalities in exterior domains

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

“…The techniques in those works usually involve careful integral estimates and/or sophisticated analysis of related nonlinear ODE's. A different approach to nonlinear Liouville type theorems goes to back to an earlier paper by Kondratiev and Landis [26] and was recently developed in the context of semilinear equations (p = 2) in [27][28][29][30]. The approach is based on the pointwise Phragmén-Lindelöf type bounds on positive super-harmonic functions and related Hardy-type inequalities.…”

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

“…Remark 13. One may interpret condition (36) in Theorem 11 as saying that a(.) is sufficiently "large" with respect to b(.).…”

“…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]).…”

Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains

“…[3,4,7,8,21,23,24] and the references therein). In particular, in [13] it was shown that the critical exponent p * = N N −2 is stable with respect to the change of the Laplacian by a second-order uniformly elliptic divergence type operator with measurable coefficients, perturbed by a potential, for a sufficiently wide class of potentials (see also [14] for equations of type (1.3) in exterior domains in the presence of first order terms). In [15,16] equation (1.3) and the corresponding equation with the divergence type elliptic operators was studied on cone-like domains, and it was shown that the value of the critical exponent is dependent on the geometry of the domain even in the case of the Laplacian, and on the behaviour of the coefficients of the elliptic operator at infinity.…”