Structure of positive radial solutions to scalar field equations with harmonic potential
Here, the nonlinear elliptic equation \[ -\bigtriangleup u+(\lambda + |x|^2)u - |u|^{p-1}u=0,\;x\in \mathbb{R}^n, \tag{1} \] is considered, with \(\lambda\in \mathbb{R}\) and \(p>1\). The authors prove that there exists at least one solution to (1) satisfying \[ u\in \Sigma, \;u(x)>0 \;\text{ for} x\in \mathbb{R}^n, \tag{2} \] with \[ \Sigma := \{ u\in L_2(\mathbb{R}^n): \int_{\mathbb{R}^n} (|u|^2+|\bigtriangledown u|^2 +|xu|^2)dx <\infty \}. \] They are interested in radial solutions (\(u=u(r)\) with \(r=|x|\)) to (1), (2) and therefore, they study the initial value problem \[ u''+ {{n-1}\over r}u'-(\lambda + r^2)u +|u|^{p-1}u=0\quad u(0)=\alpha,\;u'(0)=0, \tag{3} \] with \(\alpha>0, r>0\). Here, \(n\) need not be restricted to integers and can be a real number. The authors prove the existence of a unique global solution \(u(r)\in C^2([0,\infty))\) to (3), which is denoted by \(u(r;\alpha)\). Then they define a crossing solution as a solution having a zero in \((0,\infty)\) and an entirely positive solution as a solution which is positive on \([0,\infty)\). They provide the following main result: Under the assumption \(n>2, \lambda >-n\) and \(1<p<{{n+2}\over{n-2}}\) for \(n\geq \)3, \({{4-n}\over{n-2}}<p<{{n+2}\over{n-2}}\) for \(2<n<3\), there exists a unique positive number \(\alpha_0\) such that, for every \(\alpha \in (\alpha_0,\infty)\), \(u(r;\alpha)\) is a crossing solution and, for every \(\alpha \in (0,\alpha_0]\), \(u(r;\alpha)\) is an entirely positive solution. The asymptotic behaviour of the solutions is given, as well. Consequently, for \(n\geq 3, \lambda>-n\) and \(1<p<(n+2)/(n-2)\), the authors get that (1), (2) has a unique solution.
scientific article; zbMATH DE number 1763740