An ergodic control problem arising from the principal eigenfunction of an elliptic operator
The paper considers the equation (*) \(-(1/2)\Delta v+(1/2)|\nabla v|^ 2+\alpha v-V(x)=0\) in the whole space \(\mathbb{R}^ n\). It is assumed that \(\alpha>0\) and that the function \(V\) is smooth nonnegative and \(V(x)\to\infty\) as \(| x|\to\infty\). By means of special transformation \(v=-\log u\) and the method of super- and subsolutions there are established the existence of nonnegative solutions of (*) and the close relationship between (*) and the eigenvalue problem of the Schrödinger operator \(-(1/2)\Delta+V\). Particulary, it is shown that the limit of solutions of (*) as \(\alpha\to 0\) can be expressed in terms of the principal eigenvalue and eigenfunction of the Schrödinger operator.
scientific article; zbMATH DE number 9372