Existence and symmetry for elliptic equations in \(\mathbb{R}^{n}\) with arbitrary growth in the gradient (Q518554)

The paper deals with the existence and symmetry properties of solutions decaying to zero at infinity for the semilinear elliptic equation \(\Delta u+g(x,u,Du)=0\) in \({\mathbb R}^n\) (\(n\geq 3\)), where \(g:{\mathbb R}^n\times{\mathbb R}\times{\mathbb R}^n\rightarrow {\mathbb R}\) includes a large class of nonlinear functions, such as functions with polynomial- and exponential-type growths in their second and third variables. Under natural hypotheses, the authors establish sufficient conditions for the existence of solutions. Positivity and symmetry of solutions are also considered in the present paper, as well as the asymptotic behavior of solutions and their gradients.