Liouville type theorems for Hardy–Hénon equations with concave nonlinearities

Abstract

In this paper, we are concerned with the Hardy-Hénon equations −Δ = | | and Δ 2 = | | with ∈ ℝ and ∈ (0, 1]. Inspired by Serrin and Zou [25], we prove Liouville theorems for nonnegative solutions to the above Hardy-Hénon equations (Theorem 1.1 and Theorem 1.3), that is, the unique nonnegative solution is ≡ 0. K E Y W O R D S bi-harmonic, concave nonlinearity, Hardy-Hénon equations, Liouville theorems, nonnegative solutions, super-harmonic property M S C ( 2 0 1 0 ) Primary: 35B53, Secondary: 35J61, 35J91 1084

Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains

“…The nonlinear terms in (1.10) is called critical if p = p s (a) := n+α+2a n−α (:= +∞ if n = α), subcritical if 0 < p < p s (a) and supercritical if p s (a) < p < +∞. Liouville type theorems for equations (1.10) (i.e., nonexistence of nontrivial nonnegative solutions) in the whole space R n , the half space R n + and bounded domains Ω have been extensively studied (see [1,2,3,4,5,7,10,13,15,16,17,18,19,20,21,23,28,29,33,36,37,38,39] and the references therein). For other related properties on PDEs (1.10) and Liouville type theorems on systems of PDEs of type (1.10) with respect to various types of solutions (e.g., stable, radial, singular, nonnegative, sign-changing, • • • ), please refer to [1,3,6,12,14,16,18,22,27,28,29,32,35,39] and the references therein.…”

“…Liouville type theorems for equations (1.10) (i.e., nonexistence of nontrivial nonnegative solutions) in the whole space R n , the half space R n + and bounded domains Ω have been extensively studied (see [1,2,3,4,5,7,10,13,15,16,17,18,19,20,21,23,28,29,33,36,37,38,39] and the references therein). For other related properties on PDEs (1.10) and Liouville type theorems on systems of PDEs of type (1.10) with respect to various types of solutions (e.g., stable, radial, singular, nonnegative, sign-changing, • • • ), please refer to [1,3,6,12,14,16,18,22,27,28,29,32,35,39] and the references therein. These Liouville theorems, in conjunction with the blowing up and re-scaling arguments, are crucial in establishing a priori estimates and hence existence of positive solutions to non-variational boundary value problems for a class of elliptic equations on bounded domains or on Riemannian manifolds with boundaries (see [4,15,17,19,24,…”

Liouville type theorem for critical order Lane-Emden-Hardy equations in $\mathbb{R}^n$

“…We say equations (1.2) have critical order if α = n and non-critical order if 0 < α < n. Being essentially different from the non-critical order equations, the fundamental solution c n ln 1 |x−y| of (−∆) n 2 changes its signs in critical order case α = n, thus the integral representation in terms of the fundamental solution can't be deduced directly from the super poly-harmonic properties. Liouville type theorems for equations (1.2) (i.e., nonexistence of nontrivial nonnegative solutions) have been quite extensively studied (see [1,2,4,6,7,16,17,21,23,25,26,28,32] and the references therein). It is crucial in establishing a priori estimates and existence of positive solutions for non-variational Dirichlet problems of a class of elliptic equations (see [22,27]).…”

“…, there are also lots of literatures on Liouville type theorems for general fractional order or higher order Hardy-Hénon equations (1.2), for instance, Bidaut-Véron and Giacomini [1], Chen and Fang [3], Dai and Qin [16], Gidas and Spruck [21], Mitidieri and Pohozaev [25], Phan [26], Phan and Souplet [28] and many others. For Liouville type theorems on systems of PDEs of type (1.2) with respect to various types of solutions (e.g., stable, radial, nonnegative, sign-changing, • • • ), please refer to [1,17,18,24,26,27,29,31] and the references therein.…”

Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres

“…We say equations (1.4) have critical order if α = n and non-critical order if 0 < α < n. The nonlinear terms in (1.4) is called critical if p = p c (a) := n+α+2a n−α (:= ∞ if n = α) and subcritical if 0 < p < p c (a). Liouville type theorems for equations (1.4) (i.e., nonexistence of nontrivial nonnegative solutions) in the whole space R n and in the half space R n + have been extensively studied (see [1,4,5,6,8,12,13,16,18,22,23,24,26,30,36,37,41,44,46,47,52,53] and the references therein). For Liouville type theorems and related properties on systems of PDEs of type (1.4) with respect to various types of solutions (e.g., stable, radial, nonnegative, sign-changing, • • • ), please refer to [1,24,28,35,40,44,45,50,51] and the references therein.…”