Quantum mechanics

Ground state

Pure mathematics

Mathematical physics

Statistics

Energy (signal processing)

Dimension (graph theory)

Scalar field

Mathematical optimization

Supercritical fluid

Thermodynamics

Physics

Nonlinear system

Scalar (mathematics)

Mathematics

Geometry

Lagrange multiplier

Combinatorics

Compact space

Calculus of Variations and Partial Differential Equations

In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $\mu\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.

A mass supercritical problem revisited