### Abstract

In this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator L_{K} with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem {L_{K}u + λu + f(x, u) = 0 in Ω u = 0 in ℝ^{n}\Ω, where λ is a real parameter and the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. This equation has a variational nature, and so its solutions can be found as critical points of the energy functional J_{λ} associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when λ < λ_{1} and λ ≥ λ_{1}, where λ_{1} denotes the first eigenvalue of the operator-L_{K}. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian {(-δ)^{s}u - λu = f(x, u) in Ω u = 0 in ℝ^{n}\Ω. Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.

Original language | English |
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Pages (from-to) | 2105-2137 |

Number of pages | 33 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 33 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 May 2013 |

Externally published | Yes |

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## Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*33*(5), 2105-2137. https://doi.org/10.3934/dcds.2013.33.2105