On the existence and uniqueness of a solution to a stationary problem with free boundary (Q2768795)

Using the variational method, the authors prove existence, uniqueness and space localization of solutions to the problem \(v''=x^{p}v^{q}\), \(x\in \mathbb{R}_{+}\), with \(0<q<1\), \(p\geq 1\); \(v(x)\geq 0\), \(x\in \mathbb{R}_{+}\); \(v(0)=v_0>0\); and \(v,v'\to 0\) as \(x\to\xi\) or \(x\to\infty\). The authors study properties of the solution to the considered problem and, in particular, they prove that any limit point \(u\) of a minimizing sequence to the functional \(\Phi(u)=\int_{0}^{\infty}[{u'}^2/2+x^{p}|u|^{q+1}/(q+1)] dx\) is a generalized solution to the considered problem.