Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in $\mathbb{R}^{n}$

Abstract

In this paper, we consider the critical order Hardy-Hénon equationswhere n ≥ 4 is even, −∞ < a < n, and 1 < p < +∞. We first prove a Liouville theorem (Theorem 1.1), that is, the unique nonnegative solution to this equation is u ≡ 0. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem 1.4 and 1.5). Our results seem to be the first Liouville theorem, a priori estimates, and existence on the critical order equations in higher dimensions (n ≥ 3). Extensions to super-critical order Hardy-Hénon equations and inequalities will also be included (Theorem 1.7 and 1.9).

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.…”

“…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 [4,15,17,19,24,33,35,37]).…”

Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity

“…) as |x| → +∞. For more literatures on the quantitative and qualitative properties of solutions to fractional order or higher order conformally invariant PDE and IE problems, please refer to [3,6,7,11,16,20,21,24,37,38,39,42,43,44,45,46,54,64,65] and the references therein.…”

“…with the finite total curvature R 2 u 4 (x)dx < +∞, i.e., system (1.1) with p = 3 2 . For more literatures on the classification of solutions and Liouville type theorems for various PDE and IE problems via the methods of moving planes or spheres and the method of scaling spheres, please refer to [8,13,14,16,19,20,21,22,23,24,25,26,28,33,35,36,37,38,39,40,41,45,50,53,54,61,63,64,65,70,71,73,75,76,77,78] 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.…”