Existence of entire positivek-convex radial solutions to Hessian equations and systems with weights

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

“…Here we wish to mention the works of Bao-Ji-Li [2], Jacobsen [11], Bao-Li [12], Lazer and McKenna [14, (the case k = N )], Salani [16] and Zhang-Zhou [19] which will be useful in our proofs. It is interesting to note that in our results the dimension of the space R N affect the properties of the solution of the equation and system which in the case of the classical Laplace operator and the Monge-Ampére operator this condition doesn't appear in any works.…”

“…Motivated by the recent work of Zhang-Zhou [19] we are interested in proving the following theorems:…”

On the solutions to weakly coupled system of $\boldsymbol{k_i}$-Hessian equations

“…In this paper, we shall establish the existence and multiplicity of nontrivial radial k-admissible solutions of the weakly coupled degenerated system (1.1). It is worth to notice that the system (1.1) contains a variety of different k-Hessian equations which is significantly different from that in [5,7,23,29] such that the problem we considered can contain Laplace equations and Monge-Ampère equations at the same time. This kind of system can represent the coupling of different types of elliptic equations, which makes our problem more comprehensive and more applicable.…”

A remark on the existence of solutions to a (k1,k2)-Hessian system with convection term

“…In recent years equations of the type (1) have been the subject of rather deep investigations since appears from many branches of mathematics and applied mathematics. For more surveys on these questions we advise the paper of Alves and Holanda [1], Bao-Ji and Li [3], Bandle and Giarrusso [2], Cirstea and Rȃdulescu [4], Clément-Manásevich and Mitidieri [5], De Figueiredo and Jianfu [6], Galaktionov and Vázquez [7], Ghergu and Rȃdulescu [8], Salani [19], Ji and Bao [12], Jian [11], Peterson and Wood [16], Quittner [17], Li and Yang [13], Li-Zhang and Zhang [14], Viaclovsky [20,21] and not the last Zhang and Zhou [22].…”

“…To simplify the presentation we wish to mention that our purpose is to complete and improve all the results in [22] for the more general system (1). By analogy with the work of Zhang and Zhou [22] we introduce the following notations…”