Nonexistence of radial entire solutions of elliptic systems.
The author considers the quasi-variational ordinary differential system \[ [\nabla G(u,u')]'- \nabla_u G(u,u')+ Q(t, u,u')= f(t,u) \] on \(J= (T,\infty)\), \(T\geq 0\), where \(\nabla= \nabla_u\) denotes the gradient operator with respect to the second variable of the function \(G\). Also, he applies the above result to the study of nonexistence of radial solutions of elliptic systems of the general form \[ \text{div}(g(u) A(| D_u|) D_u)- \nabla_u g(u){\mathcal A}(| D_u|)= f(| x|,u),\quad x\in\mathbb{R}^n, \] which is an extension of the results of Naito-Usami, where \(D_u\) denotes the Jacobian matrix and \({\mathcal A}(s)= \int^s_0 \sigma A(\sigma)\,d\sigma\). The results are very nice.
quasi-variational ordinary differential system
scientific article; zbMATH DE number 1902853