Uniqueness of excited states to \(-\Delta u + u - u^3 = 0\) in three dimensions (Q6596278)

The radial solutions of the PDE \N\[\N\Delta u+f(u)=0\N\]\Nin three dimensions correspond to the solutions of the singular ODE \N\[\N\ddot{y}(t)+\frac{2}{t}\dot{y} (t)+f(y(t))=0,\quad y(0)=b,\quad \dot{y}(0)=0, \tag{1}\N\]\Nvia the identification \(t=r\) (where \(r\) is the radial variable). Equation (1) can be viewed as a damped oscillator with a time-dependent friction coefficient.\N\NThere exists a smooth solution of (1), denoted by \(y_b\), for all \(b\in \mathbb{R}\) and all \(t\ge0\) and it is unique. A \textit{bound state} is a nonzero solution with \(y_b(t)\to 0\) (which gives rise to a nontrivial solution \(u\in H^1(\mathbb{R}^3)\) of the PDE), a \textit{ground state} is a positive bound state and a \textit{\(n\)-th excited state} is a bound state with exactly \(n\) zero crossings.\N\NThe interesting problem addressed in the paper is the uniqueness of the excited states of (1) for a cubic nonlinearity. The main result is the following:\N\NTheorem. The first twenty excited states of (1) are unique for \(f(y)=y^3-y\).\N\NThe proof is computer-assisted and is based on a combination of the analytical properties of (1) with rigorous calculations using interval arithmetic. The authors limited themselves up to the \(20\)-th excited state due to computation time but readers are invited to verify uniqueness of higher excited states using the same arguments.\N\NOverall, the paper is well written, the strategy of the proof is clearly stated and the computer code is publicly available. Furthermore, the authors expect that their method will also work for more general nonlinearities and other dimensions.