Bubble tower solutions for supercritical elliptic problem in \(\mathbb{R}^N\) (Q2807093)

The authors consider positive solutions to the problem \(-\Delta u+u=u^p+\lambda u^q\) in \(\mathbb{R}^N\), \(N\geq 3\), such that \(u\to 0\) as \(|x|\to \infty\), where \(p=(N+2)/(N-2)+\varepsilon\); \(1<q<(N+2)/(N-2)\) if \(N\geq 4\), whereas \(3<q<5\) if \(N=3\); \(\lambda>0\) is a positive constant while \(\varepsilon>0\) is a small positive parameter. Under the above assumptions, they prove that given any integer \(k\) there exists a radial solution \(u_\varepsilon\) which blows up at the origin as the sum of \(k\) bubbles, provided that \(\varepsilon>0\) is sufficiently small. More precisely, letting \(w\) denote the unique positive solution of \(-\Delta w=w^{(N+2)/(N-2)}\), then near the origin \(u_\varepsilon\) behaves like the sum of \(k\) suitably scaled copies of \(w\).NEWLINENEWLINETheir approach is perturbative, based on a well known finite dimensional variational Lyapunov-Schmidt reduction (the aforementioned scalings being the finite dimensional unknowns). Interestingly enough, the analysis is carried out in the equivalent formulation of the problem resulting by applying the Emden-Fowler transformation.