## 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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### References:

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