Hermite elliptic equations
The following eigen-value problem of the Hermite nonlinear elliptic equation \[ -\Delta u+x^2u=\lambda f(x,u),\quad u\in H^1(\mathbb R^n),\tag{*} \] where \(f(x,u)\) is nondecreasing, Lipschitz and \(|f(x,t)|\leq c|t|^q\), \(1<q<N-1\), \(N=2n/(n-2)\) is considered. The problem is treated by an inverse power method adopted by the author to unbounded domains. It is proved (Theorem 7) that there exists a weak solution \(u\in H^1_{x^2}\) (\(\|u\|^2=\int|\nabla u|^2+x^2u^2 dx\)) of the problem (*). In the linear case \(f(x,u)=f(x),\lambda=1 \) an explicit series solution to the problem (*) is given.
scientific article; zbMATH DE number 1810183