Liouville-type results for semilinear elliptic equations in unbounded domains

Abstract

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: −a_ij(x)∂_ij u(x) − q_i(x)∂_i u(x) = f (x,u(x)), x ∈ R^N. The aim is to prove uniqueness of positive- bounded solutions—Liouville-type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the linearized operator, which guarantees the existence result as well as the Liouville property. Then, following another approach, we establish other results relying on the sign of the principal eigenvalue of the linearized operator about u = 0, of some limit operator at infinity which we define here. This framework will be seen to be the most general one.We also derive the large time behavior for the associated evolution equation

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

“…In particular, the results of [7] yield that the equation associated with b ≡ 0, f as in (3.3) and an arbitrary a satisfying (H) is monostable. First, in this framework, critical travelling waves are time-increasing and unique up to translation in time, as in the ignition-type setting.…”

“…We refer to [7] for more general conditions on the coefficients guaranteeing the monostability of the equation. In particular, the results of [7] yield that the equation associated with b ≡ 0, f as in (3.3) and an arbitrary a satisfying (H) is monostable.…”

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

“…Note that since C 1 c (Ω) is dense in W 1,p 0 (Ω) with respect to W 1,p norm, the infimum in (1.4) can be taken over C 1 c (Ω). When Ω is an arbitrary (possibly unbounded) domain, following Berestycki et al [7,8,9,11,28], we define This type of eigenvalue was first introduced in a celebrated work of BerestyckiNirenberg-Varadhan [8] for second order operators in bounded (not necessarily smooth) domains, and then was developed to second order operators in unbounded domains [7,9,11]. An important feature of the notion of generalized principal eigenvalue is that if Ω is a smooth and bounded domain, λ(K V , Ω) coincides with the principal eigenvalue λ 1,V (Ω), while if Ω is unbounded λ(K V , Ω) is well defined and can be expressed by a variational formula.…”

“…An important feature of the notion of generalized principal eigenvalue is that if Ω is a smooth and bounded domain, λ(K V , Ω) coincides with the principal eigenvalue λ 1,V (Ω), while if Ω is unbounded λ(K V , Ω) is well defined and can be expressed by a variational formula. For related definitions of generalized principal eigenvalues, the reader is referred to [7,8,9,11] for linear operators, [4,28] for fully nonlinear operators and [13] for singular fully nonlinear operators. To our knowledge, no investigation of generalized principal eigenvalue for quasilinear operators has been previously obtained.…”

“…Therefore, we only deal with decaying solutions of (1.1). Thus, when 1 < p < 2 the class of functions under consideration is [7,18,27].…”