Multi-bump solutions for a Kirchhoff-type problem
In this paper, we study the existence of solutions for the Kirchhoff problem M(∫ℝ3|∇u|2dx+∫ℝ3(λa(x)+1)u2dx)(-Δu+(λa(x)+1)u)=f(u)$M\Biggl (\int _{\mathbb {R}^{3}}|\nabla u|^{2}\, dx + \int _{\mathbb {R}^{3}} (\lambda a(x)+1)u^{2}\, dx\Biggl ) (- \Delta u + (\lambda a(x)+1)u) = f(u)$ in ℝ3, u ∈ H1(ℝ3)...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2016-02-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2015-0101 |
| Summary: | In this paper, we study the existence of solutions for the Kirchhoff problem
M(∫ℝ3|∇u|2dx+∫ℝ3(λa(x)+1)u2dx)(-Δu+(λa(x)+1)u)=f(u)$M\Biggl (\int _{\mathbb {R}^{3}}|\nabla u|^{2}\, dx + \int _{\mathbb {R}^{3}} (\lambda a(x)+1)u^{2}\, dx\Biggl ) (- \Delta u + (\lambda a(x)+1)u) = f(u)$ in ℝ3, u ∈ H1(ℝ3)
Assuming that the nonnegative function a(x) has a potential well with int (a -1({0})) consisting of k disjoint components Ω1,Ω2,...,Ωk and the nonlinearity f(t) has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by using variational methods. |
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| ISSN: | 2191-9496 2191-950X |