### Abstract

We consider the following variational inclusions system of the form- △ u + u ∈ ∂_{1} F (u, v) in R^{N},- △ v + v ∈ ∂_{2} F (u, v) in R^{N}, with u, v ∈ H^{1} (R^{N}), where F : R^{2} → R is a locally Lipschitz function and ∂_{i} F (u, v) (i ∈ {1, 2}) are the partial generalized gradients in the sense of Clarke. Under various growth conditions on the nonlinearity F we study the existence of nonzero weak solutions of the above system (in the sense of hemivariational inequalities), which are critical points of an appropriate locally Lipschitz function defined on H^{1} (R^{N}) × H^{1} (R^{N}). The main tool used in the paper is the principle of symmetric criticality for locally Lipschitz functions.

Original language | English |
---|---|

Pages (from-to) | 1578-1594 |

Number of pages | 17 |

Journal | Nonlinear Analysis |

Volume | 65 |

Issue number | 8 |

DOIs | |

Publication status | Published - Oct 15 2006 |

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### Keywords

- Cerami condition
- Hemivariational inequalities
- Locally Lipschitz functions
- Palais-Smale condition
- Principle of symmetric criticality
- Variational inclusions system

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Mathematics(all)

### Cite this

**Existence of nonzero weak solutions for a class of elliptic variational inclusions systems in R ^{N}.** / Kristály, A.

Research output: Contribution to journal › Article

^{N}',

*Nonlinear Analysis*, vol. 65, no. 8, pp. 1578-1594. https://doi.org/10.1016/j.na.2005.10.033

}

TY - JOUR

T1 - Existence of nonzero weak solutions for a class of elliptic variational inclusions systems in RN

AU - Kristály, A.

PY - 2006/10/15

Y1 - 2006/10/15

N2 - We consider the following variational inclusions system of the form- △ u + u ∈ ∂1 F (u, v) in RN,- △ v + v ∈ ∂2 F (u, v) in RN, with u, v ∈ H1 (RN), where F : R2 → R is a locally Lipschitz function and ∂i F (u, v) (i ∈ {1, 2}) are the partial generalized gradients in the sense of Clarke. Under various growth conditions on the nonlinearity F we study the existence of nonzero weak solutions of the above system (in the sense of hemivariational inequalities), which are critical points of an appropriate locally Lipschitz function defined on H1 (RN) × H1 (RN). The main tool used in the paper is the principle of symmetric criticality for locally Lipschitz functions.

AB - We consider the following variational inclusions system of the form- △ u + u ∈ ∂1 F (u, v) in RN,- △ v + v ∈ ∂2 F (u, v) in RN, with u, v ∈ H1 (RN), where F : R2 → R is a locally Lipschitz function and ∂i F (u, v) (i ∈ {1, 2}) are the partial generalized gradients in the sense of Clarke. Under various growth conditions on the nonlinearity F we study the existence of nonzero weak solutions of the above system (in the sense of hemivariational inequalities), which are critical points of an appropriate locally Lipschitz function defined on H1 (RN) × H1 (RN). The main tool used in the paper is the principle of symmetric criticality for locally Lipschitz functions.

KW - Cerami condition

KW - Hemivariational inequalities

KW - Locally Lipschitz functions

KW - Palais-Smale condition

KW - Principle of symmetric criticality

KW - Variational inclusions system

UR - http://www.scopus.com/inward/record.url?scp=33745862726&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745862726&partnerID=8YFLogxK

U2 - 10.1016/j.na.2005.10.033

DO - 10.1016/j.na.2005.10.033

M3 - Article

VL - 65

SP - 1578

EP - 1594

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 8

ER -