On the number of solutions of nonlinear Schrödinger equations and on unique continuation
Consider the following equations: \[ (- \Delta+ q(x)) u= \lambda u\pm f(x, u), \] where \(q\in L^\infty(\mathbb{R}^n)\) is real-valued and where \(f: \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}\) satisfies ``quasi-polynomials'' conditions. The author studies critical values of \(\varepsilon\) for the stability of radial solutions. Since \(q\in L^\infty\), the operator \(- \Delta+ q(x)\) has a unique selfadjoint extension \(H\) and the equation above is written as \[ (H- \lambda) u= \pm f(x, u). \] The author derives that there are infinitely many non-zero weak solutions, if \(\lambda\) is in the ``spectral gap'' and \(I(f):= \{x\in \mathbb{R}^n\mid f(x, s)\neq 0 \forall s\}\) has positive Lebesgue measure, and obtains bounds of these weak solutions.
scientific article; zbMATH DE number 729101