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Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains. (English) Zbl 1295.35223

Let \(\Omega_t\subset {\mathbb R}^N\) be a tubular domain that expands as \(t\rightarrow +\infty\) and assume that \(p\) is a real number such that \(1<p<\infty\) if \(N=2\) and \(1<p<(N+2)/(N-2)\) if \(N\geq 3\). This paper deals with the existence of multi-bump solutions for the Lane-Emden equation \[ -\Delta u=u^p\quad\text{in}\;\Omega_t, \] subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega_t\).
The main result establishes the existence of positive multi-bump solutions without the non-degeneracy condition for the limit problem. The proof combines various refined variational arguments.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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