Corrigendum to “Some Liouville theorems for Hénon type elliptic equations” [J. Funct. Anal. 262 (4) (2012) 1705–1727]

SynopsisLiouville type theorems are obtained for bounded entire solutions of equations of the form Δ2u − q(x)Δu + p(x)u = 0 by means of subharmonic functionals and Green type inequalities.

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Multiple positive solutions for a class of concave–convex elliptic problems in ℝN involving sign-changing weight, II

Communications in Contemporary Mathematics ◽

10.1142/s021919971450045x ◽

2015 ◽

Vol 17 (05) ◽

pp. 1450045 ◽

Cited By ~ 10

Author(s):

Yisheng Huang ◽

Tsung-Fang Wu ◽

Yuanze Wu

Keyword(s):

Positive Solutions ◽

Elliptic Equations ◽

Elliptic Problems ◽

Unbounded Domains ◽

Multiple Positive Solutions ◽

Three Solutions ◽

Semilinear Elliptic ◽

Two Parameters ◽

Concentration Behavior ◽

Funct Anal

In this paper, we study the following concave–convex elliptic problems: [Formula: see text] where N ≥ 3, 1 < q < 2 < p < 2* = 2N/(N - 2), λ > 0 and μ < 0 are two parameters. By using several variational methods and a perturbation argument, we obtain three positive solutions to this problem under the predefined conditions of fλ(x) and gμ(x), which simultaneously extends the result of [T. Hsu, Multiple positive solutions for a class of concave–convex semilinear elliptic equations in unbounded domains with sign-changing weights, Bound. Value Probl. 2010 (2010), Article ID 856932, 18pp.; T. Wu, Multiple positive solutions for a class of concave–convex elliptic problems in ℝN involving sign-changing weight, J. Funct. Anal. 258 (2010) 99–131]. We also study the concentration behavior of these three solutions both for λ → 0 and μ → -∞.

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Sharp Liouville Theorems

Advanced Nonlinear Studies ◽

10.1515/ans-2020-2111 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Salvador Villegas

Keyword(s):

Elliptic Equations ◽

Sufficient Conditions ◽

Unbounded Domains ◽

Necessary And Sufficient Conditions ◽

Qualitative Properties ◽

Liouville Theorems ◽

Necessary And Sufficient

AbstractConsider the equation {\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in {\mathbb{R}^{N}}, where {\varphi>0}. Berestycki, Caffarelli and Nirenberg proved in [H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 69–94] that if there exists {C>0} such that \int_{B_{R}}(\varphi\sigma)^{2}\leq CR^{2} for every {R\geq 1}, then σ is necessarily constant. In this paper, we provide necessary and sufficient conditions on {0<\Psi\in C([1,\infty))} for which this result remains true if we replace {CR^{2}} by {\Psi(R)} in any dimension N. In the case of the convexity of Ψ for large {R>1} and {\Psi^{\prime}>0}, this condition is equivalent to \int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty.

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