Abstract
In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.
Highlights
Let D2u be a square matrix of second-order partial derivatives for the scalar-valued function u, which is called the Hessian matrix in mathematics
When k = N, the k-Hessian operator reduces to the Monge–Ampère operator det D2u, and if k = 1, Sk(λ(D2u)) turns into a Laplace operator, which implies that the k-Hessian operator constructs a discrete collection of partial differential operators including the Monge–Ampère operator det D2u and the Laplace operator ∆u as special cases
To the best of our knowledge, no result has been reported on the existence and nonexistence of blow-up solutions for the k-Hessian equation [1], and this is the first paper using the iterative method to study the k-Hessian equation involving a nonlinear operator
Summary
Let D2u be a square matrix of second-order partial derivatives for the scalar-valued function u, which is called the Hessian matrix in mathematics. We consider the existence and nonexistence of blow-up solutions for the following k-Hessian equation with a nonlinear operator:. Many existing work for the k-Hessian equation is devoted to constructing mathematical theory rather than modeling or exploring new applications. To the best of our knowledge, no result has been reported on the existence and nonexistence of blow-up solutions for the k-Hessian equation [1], and this is the first paper using the iterative method to study the k-Hessian equation involving a nonlinear operator.
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