Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities

Wan, Haitao; Li, Xiaohong; Li, Bo; Shi, Yongxiu

doi:10.1016/j.nonrwa.2018.07.021

Cited by 7 publications

(5 citation statements)

References 40 publications

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“…(1.1) is unique for some appropriate assumptions as regards f and b. Inspired by the ideas of the authors in [1,17,18], Wan [47,48] and Wan et al [49] used a perturbation method and a truncation technique to study the asymptotic behavior and uniqueness of entire large solutions of Eq. (1.1).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…(1.1). In particular, the authors in [49] found some necessary and sufficient conditions for these rapidly and regularly varying functions and proved some sharp uniqueness results for entire large solutions when f belongs to a more general class of functions. Part II (μ = 0).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…Later, Giarrusso [21] and [22] further extended the above results when f belongs to a more general class of functions. When Motivated by the work of [2] and [47][48][49], in this paper, we determine the exact asymptotic behavior of entire large solutions to Eq. (1.1) at infinity in R N , and we show that the convection term μb(x)|∇u(x)| q does not affect the asymptotic behavior under certain conditions.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

Wan

2018

Bound Value Probl

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In this paper, we analyze the blow-up rates and uniqueness of entire large solutions to the equation u = a(x)f (u) + μb(x)|∇u| q , x ∈ R N (N ≥ 3), where μ > 0, q > 0 and a, b ∈ C α loc (R N) (α ∈ (0, 1)). The weight a is nonnegative, b is able to change sign in R N , and f ∈ C 1 [0, ∞) is positive and nondecreasing on (0, ∞) and rapidly or regularly varying at infinity. Additionally, we investigate the uniqueness of entire large solutions.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

Wan

2018

Bound Value Probl

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“…where b, h, φ, ψ are continuous functions. Besides, under the Keller-Osserman condition and its extension, many excellent results have been obtained, such as the existence and uniqueness of the nonnegative viscosity solutions [9], the existence and uniqueness of blow up solutions [10], the existence and uniqueness of entire blow up solutions [11], the existence of entire classical weak solutions of the differential inequality [12], the existence and uniqueness of solutions with strong isolated singularity [13], and the existence of solutions for the boundary blow up problem in one dimensional case [14]. we define the k-Hessian operator (k � 1, 2, .…”

Section: Introductionmentioning

confidence: 99%

New Results on the Radial Solutions to a Class of Nonlinear k‐Hessian System

Wang

Zhang

2022

Journal of Mathematics

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This paper investigates the positive radial solutions of a nonlinear k -Hessian system. Λ S k 1 / k λ D 2 z 1 S k 1 / k λ D 2 z 1 = b x φ z 1 , z 2 , x ∈ ℝ N Λ S k 1 / k λ D 2 z 2 S k 1 / k λ D 2 z 2 = h x ψ z 1 , z 2 , x ∈ ℝ N , where Λ is a nonlinear operator and b , h , φ , ψ are continuous functions. With the help of Keller–Osserman type conditions and monotone iterative technique, the positive radial solutions of the above problem are given in cases of finite, infinite, and semifinite. Our results complement the work in by Wang, Yang, Zhang, and Baleanu (Radial solutions of a nonlinear k -Hessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simul. 91(2020), 105396).

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“…For other related insight on Eq. (1.1), please refer to [2], [8]- [7], [15]- [16]. In this paper, by structuring a comparison function, we establish the new asymptotic behavior of large solutions to problem (1)-(2) (including the case of p = 2) when f ∈ N RV p−1 .…”

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confidence: 99%

Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case

Shi

Wan

2021

era

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This paper is considered with the quasilinear elliptic equation ∆pu = b(x)f (u), u(x) > 0, x ∈ Ω, where Ω is an exterior domain with compact smooth boundary, b ∈ C(Ω) is non-negative in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) is positive and increasing on (0, ∞) which satisfies a generalized Keller-Osserman condition and is regularly varying at infinity with critical index p − 1. By structuring a new comparison function, we establish the new asymptotic behavior of large solutions to the above equation in the exterior domain. We find that the lower term of f has an important influence on the asymptotic behavior of large solutions. And then we further establish the uniqueness of such solutions.

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Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities

Cited by 7 publications

References 40 publications

Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

New Results on the Radial Solutions to a Class of Nonlinear k‐Hessian System

Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case

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