Nonlinear problems related to the Thomas-Fermi equation.

*(English)*Zbl 1150.35406This beautiful paper contains results obtained over the period 1975–77 (most of them announced at various meetings), which however have remained unpublished. It is inspired by a paper of E.H. Lieb and B. Simon on the Thomas-Fermi equation. In fact, the authors are concerned with a grand generalization of the Thomas-Fermi equation, namely the following nonlinear multivalued elliptic equation:
\[
\begin{aligned} -\Delta u+\gamma(u-\lambda)\ni-\Delta V{\text{ in }}\mathbb{R}^N,\cr u(x)\to 0{\text{ as }}| x| \to\infty.\end{aligned}(P)
\]
The parameter \(\lambda\) is unknown as well, so we need one more condition, which, according to the original case, should be the following constraint:
\[
\int_{\mathbb{R}^N}\Delta(u-V)dx=I,(C)
\]
where \(I\) is a nonnegative given real number. Here \(\gamma=(\partial j)^{-1}\), where \(j:\mathbb{R}\to [0,+\infty]\) is a given convex function such that \(j(0)=0\) and \(j(r)=+\infty\) for \(r< 0\), and \(V:\mathbb{R}^N\to\mathbb{R}\) is a given measurable function.

Inspired by the paper of Lieb and Simon, the authors associate a suitable energy functional to problem (P) and then consider the corresponding minimization problem. In this way, in some situations, problem (P) has a variational formulation, namely the corresponding Euler equation. The relationship between the variational formulation and the Euler equations is completely clarified here. When the variational formulation is meaningless the authors approach problem (P) directly.

Basically, the main result of the paper amounts to saying that there exists a critical value \(\lambda_1\) of \(\lambda\) in \([0,+\infty]\) which separates the values of \(\lambda\) for which we have at least one solution for (P) from those for which we have no solution. Specifically, if \(\lambda>\lambda_1\) (and \(\lambda> +\infty\)), then there is a unique solution of problem (P); if \(\lambda< \lambda_1\), then there is no solution of (P). The non-existence case is an effect of the nonlinear character of the problem. The authors obtain supplementary information about the critical value \(\lambda_1\) in the particular case when \(-\Delta V\in L^1(\mathbb{R}^N)\). Two problems raised by Lieb and Simon in their paper are solved in the more general framework here. One of them is a min-max principle for the parameter \(\lambda\) (which can be interpreted as a Lagrange multiplier associated with the constraint (C) in the minimization problem) and the uniqueness of the extremals.

Inspired by the paper of Lieb and Simon, the authors associate a suitable energy functional to problem (P) and then consider the corresponding minimization problem. In this way, in some situations, problem (P) has a variational formulation, namely the corresponding Euler equation. The relationship between the variational formulation and the Euler equations is completely clarified here. When the variational formulation is meaningless the authors approach problem (P) directly.

Basically, the main result of the paper amounts to saying that there exists a critical value \(\lambda_1\) of \(\lambda\) in \([0,+\infty]\) which separates the values of \(\lambda\) for which we have at least one solution for (P) from those for which we have no solution. Specifically, if \(\lambda>\lambda_1\) (and \(\lambda> +\infty\)), then there is a unique solution of problem (P); if \(\lambda< \lambda_1\), then there is no solution of (P). The non-existence case is an effect of the nonlinear character of the problem. The authors obtain supplementary information about the critical value \(\lambda_1\) in the particular case when \(-\Delta V\in L^1(\mathbb{R}^N)\). Two problems raised by Lieb and Simon in their paper are solved in the more general framework here. One of them is a min-max principle for the parameter \(\lambda\) (which can be interpreted as a Lagrange multiplier associated with the constraint (C) in the minimization problem) and the uniqueness of the extremals.

Reviewer: Catalin Popa (Iaşi)

##### MSC:

35Q55 | NLS equations (nonlinear Schrödinger equations) |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

82B10 | Quantum equilibrium statistical mechanics (general) |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

37M99 | Approximation methods and numerical treatment of dynamical systems |