This extensive paper is concerned with stability properties of radially symmetric solitary waves solutions for nonlinear evolution equations
Suppose is 2 or 3, , , , as , and . The author considers a smooth function in (1) of the form , . Then, if and decrease to 0 at infinity, we have
Under the assumption above, the equation (3) possesses a positive, radially symmetric solution . The function is called stable if for all , there is such that, if and is a solution of (1), with , then for all , where is the set of such that .
The main result of this paper states that if the curve is with values in , there exist , such that , , and the null space of the linearized operator is spanned by , then is stable if and only if , where and with , and . Moreover, if we define the stability of for equation (2) in the same way as we did above for the equation (1), then the same result as for (1) holds for the equation (2).