On elliptic equations related to self-similar solutions for nonlinear heat equations
The author studies the nonlinear elliptic equation \[ (*)\quad \Delta w- ()x\cdot \nabla w-\alpha w+| w|^{p-1} w=0\quad in\quad {\mathbb{R}}^ n, \] where \(p>1\) is fixed and \(\alpha\geq 0\) is a parameter. This equation is obtained from the nonlinear heat equation \(u_ t-\Delta u-| u|^{p-1} u=0\) by introducing a suitable similarity space variable and separating the time variable. Only global positive solutions are considered which are radial in the sense that they depend on \(r=| x|\) alone. A constant positive solution only exists for \(\alpha =1/(p-1)\). It turns out that this value is a ``bifurcation value'' of the parameter \(\alpha\), the precise meaning of which is given by the following main content of the paper: (a) If \(\alpha >1/(p-1)\) and \(n<2(p+1)/(p-1)\), then there exists a positive radially decreasing global solution w(r) of (*). (b) If \(\alpha\leq 1/(p-1)\) and \(n\leq 2(p+1)/(p-1)\), no such solution exists. Moreover, in case (a) the asymptotic decay of w(r) is not exponential but of finite order with respect to r. The elegant proofs make use of the shooting method, the variational method, and a certain integral identity.
scientific article; zbMATH DE number 4017461