On the minimal energy solution in a quasilinear elliptic equation

Abstract

In this paper we seek a positive, radially symmetric and energy minimizing solution of an $m$-Laplacian equation, $-div$ $(|\nabla u|^{m-2}|\nabla u) = h(u)$. In the variational sense, the solutions are the critical points of the associated functional called the energy, $J(v) = \frac{1}{m} \int_{\mathbf{ R}^N} \vert \nabla v \vert^{m}-\int_{\mathbf{R}^N}H(v) dx,$ where $H(v)=\int_{0}^{v}h(t) dt$. A positive, radially symmetric critical point of $J$ can be obtained by solving the constrained minimization problem; minimize$\{\int_{\mathbf{ R}^N}$ $ |\nabla u|^m dx \mid \int_{\mathbf{ R}^N}H(u)dx=1\}.$ Moreover, the solution minimizes $J(v).$