# Singular solutions to $k$-Hessian equations with fast-growing nonlinearities

@inproceedings{JoaoMarcosdo2021SingularST, title={Singular solutions to \$k\$-Hessian equations with fast-growing nonlinearities}, author={'O JoaoMarcosdo and Evelina Shamarova and Esteban da Silva}, year={2021} }

We study a class of elliptic problems, involving a k-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular…

## References

SHOWING 1-10 OF 25 REFERENCES

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

- Mathematics
- 2018

A bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2017

We consider the following semilinear elliptic equation: where B 1 is the unit ball in ℝ d , d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λ…

Stable solutions to semilinear elliptic equations are smooth up to dimension $9$

- MathematicsActa Mathematica
- 2020

In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$.
This result, that was only…

Singular Solutions of Elliptic Equations with Iterated Exponentials

- MathematicsThe Journal of Geometric Analysis
- 2019

We construct positive singular solutions for the problem $$-\Delta u=\lambda \exp (e^u)$$ - Δ u = λ exp ( e u ) in $$B_1\subset {\mathbb {R}}^n$$ B 1 ⊂ R n ( $$n\ge 3$$ n ≥ 3 ), $$u=0$$ u = 0 on…

Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems

- Mathematics
- 1975

Abstract : Continuation and variational methods are developed to construct positive solutions for nonlinear elliptic eigenvalue problems. The class of equations studied contain in particular models…

Geometry of phase space and solutions of semilinear elliptic equations in a ball

- Mathematics
- 2007

We consider the problem {-Δu=u p +λu in in B, u>0 in B, u=0 on ∂B, where B denotes the unit ball in R N , N > 3, A > 0 and p > 1. Merle and Peletier showed that for p > N+2/N-2 there is a unique…

Some Positone Problems Suggested by Nonlinear Heat Generation

- Mathematics
- 1967

There is much current interest in boundary value problems containing positive linear differential operators and monotone functions of the dependent variable, see for example, M.A. Krasnosel'ski [1]…

Stable Solutions of Elliptic Partial Differential Equations

- Mathematics
- 2011

Defining Stability Stability and the variations of energy Linearized stability Elementary properties of stable solutions Dynamical stability Stability outside a compact set Resolving an ambiguity The…