Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations

Abstract

In this paper, we establish the existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations in a ball. Under some suitable local growth conditions for nonlinearity, several new results are obtained by using the fixed-point theorem.

A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

“…where p(s) > 0, s ∈ (0, +∞). Then According to Theorem 1, the k-Hessian equation (14) has no positive entire blow-up solutions. According to Theorem 2, the k-Hessian equation 14has infinitely many positive entire blow-up radial solutions.…”

“…If p(r) = 4/(r3 (1 + r 4 ) 2 ), the k-Hessian equation(14) has no positive entire blow-up solutions. (ii) If p(r) = e r (4r −3 + r −2 ), the k-Hessian equation(14) has infinitely many positive entire blow-up radial solution.In fact, here k = 3, N = 7, f (x, u) = p(|x|)u 1/4 , andB(x) = x 2 . Thus we have L −1 (x) = x 1/3 and f (s, cu) = p(s)(cu) 1/4 p(s)c 1/3 u 1/2 = c 1/3 f (s, u)for all (s, u) ∈ (0, +∞) × [0, +∞), c ∈ (0, 1], which implies that (A) holds.…”

Uniqueness of Iterative Positive Solution to Nonlinear Fractional Differential Equations with Negatively Perturbed Term

“…Fractional calculus differential equations are an important branch of differential equations. In recent years, it has attracted the interest of many researchers and has become a hot-button issue [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Compared with the integer order, it has a wider range of applications as it can be used to describespecific problems more precisely, such as the problem in complex analysis, polymer rheology, physical chemistry, electrical networks, and many other branches of science, For specific applications, see [15,16,[20][21][22][23][24][25][26][27][28].…”

Positive solutions to n-dimensional $\alpha _{1}+\alpha _{2}$ order fractional differential system with p-Laplace operator

“…It is well known that p-Laplace operator has deep background in analyzing mechanics, chemical physics, dynamic systems, etc. In the last ten years, fractional boundary value problems with p-Laplace operator have been widely studied, and there have been some excellent results on the existence, nonexistence, uniqueness, multiplicity of the solutions and positive solutions, we refer the readers to [7][8][9][10][11][12][13][14] and the references therein.…”