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Bubble tower solutions for supercritical elliptic problem in \(\mathbb{R}^N\). (English) Zbl 1358.35039

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 \(w\).
Their 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.

MSC:

35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
35J08 Green’s functions for elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35B09 Positive solutions to PDEs
35B44 Blow-up in context of PDEs
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