Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on \({\mathbb{R}}^ 3\)
The authors discuss the existence and uniqueness of positive classical solutions u(x) of the problem \[ (1)\quad -\Delta u-Z | x|^{-1} u+a(u)=0\quad in\quad {\mathbb{R}}^ 3,\quad u(x)\to 0\quad as\quad x\to \infty. \] In (1), Z is a positive real constant and the function a(.) satisfies some hypothesis. The following important results are obtained. First, if \(u\in E\) is a solution of (1), then (i) u is continuous in \({\mathbb{R}}^ 3\), more precisely \(u\in C^{0,\alpha}\) all \(\alpha <1\); (ii) u is bounded in \({\mathbb{R}}^ 3\) and u(x)\(\to 0\) as \(| x| \to \infty\); (iii) \(u\in H^ 2\); (iv) either \(u\equiv 0\) or \(u>0\) everywhere \(u\in C^ 2\) except at \(x=0\); (v) if u has radial symmetry, it satisfies the following cusp condition at the origin: (2) \(2(d/dr)u(0)+Zu(0)=0.\) Besides, they only consider the radial equation \[ (3)\quad -u''-(2/r)u'- (Z/r)u+a(u)=0,\quad u'=(du/dr),\quad etc. \] It is also proved that there is at most one positive solution of (3), (2) which goes to zero at infinity, under some hypothesis. Finally, they further apply the existence and uniqueness results to the Thomas-Fermi-Dirac-von Weizsäcker equation without electrostatic repulsion.
Thomas-Fermi-Dirac-von Weizsäcker equation
scientific article; zbMATH DE number 3960036