Existence of prescribed \(L^2\)-norm solutions for a class of Schrödinger-Poisson equation
Summary: By using the standard scaling arguments, we show that the infimum of the following minimization problem: \[ I_{\rho^2}= \text{inf}\Biggl\{(1/2) \int_{\mathbb{R}^3} |\nabla u|^2 dx+ (1/4) \iint_{\mathbb{R}^3} (|u(x)|^2|u(y)|^2/|x- y|)\,dx\,dy-(1/p) \int_{\mathbb{R}^3} |u|^p dx: u\in B_\rho\Biggr\} \] can be achieved for \(p\in (2,3)\) and \(\rho>0\) small, where \(B_\rho:=\{u\in H^1(\mathbb{R}^3):\| u\|_2= \rho\}\). Moreover, the properties of \(I_{\rho^2}/\rho^2\) and the associated Lagrange multiplier \(\lambda_\rho\) are also given if \(p\in (2,8/3]\).
nonlinear Schrödinger-Poisson type equation
scientific article; zbMATH DE number 6306797
scientific article