Nonexistence Results for Hessian Inequality

Ou, Qianzhong

doi:10.4310/maa.2010.v17.n2.a5

Cited by 14 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When p is not larger than the Serrin exponent, (1.1) has no negative kadmissible solution (cf. [15], [22] and [23]). Thus, we always assume in this paper that p is larger than the Serrin exponent p > p se .…”

Section: Regular Solutionsmentioning

confidence: 99%

On critical exponents of ak-Hessian equation in the whole space

Wang¹,

Lei²

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equationwith radial structure, where n ≥ 3, 1 < k < n/2 and p > 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research k-Hessian equations without radial structure.

show abstract

Section: Regular Solutionsmentioning

confidence: 99%

On critical exponents of ak-Hessian equation in the whole space

Wang¹,

Lei²

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

show abstract

“…When q is not larger than the Serrin (type critical) exponent: q ≤ nk n−2k , (4.1) has no negative solution (cf. [26][27][28]). Furthermore, if R(x) is double bounded, namely there exists C > 1 such that C −1 ≤ R(x) ≤ C, we can see that the analogous result still holds.…”

Section: K-hessian Equationsmentioning

confidence: 97%

Critical conditions and finite energy solutions of several nonlinear elliptic PDEs inRn

Lei

2015

Journal of Differential Equations

View full text Add to dashboard Cite

This paper is concerned with the critical conditions of nonlinear elliptic equations and the corresponding integral equations involving Riesz potentials and Bessel potentials. We show that the equations and some energy functionals are invariant under the scaling transformation if and only if the critical conditions hold. In addition, the Pohozaev identity shows that those critical conditions are the necessary and sufficient conditions for existence of the finite energy positive solutions. Finally, we discuss respectively the existence of the negative solutions of the k-Hessian equations in the subcritical case, critical case and supercritical case. Here the Serrin type critical exponent and the Sobolev type critical exponent play key roles.

show abstract

“…(ii) Let δ = 0 first and then 0 < δ < 1 for α = k * . In case (i), the following integral estimate had been deduced by the author (see (3.10) in [6]):…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…The nonexistence results were deduced in [7,8] from sharp pointwise estimates of solutions in terms of Wolff potentials. Later in [6], the author reproved some of Phuc-Verbitsky's results by a very different method -by using the argument of integration by parts via careful choices of the test functions.…”

Section: Introductionmentioning

confidence: 99%