### Abstract

We study PDI's of the type - Δ Φu ∈ ∂ C_{f}(x, u) subject to Dirichlet boundary condition in a bounded domain Ω ⊂ R^{N} Lipschitzboundary ∂ Ω. Here, Φ: R→ [0,∞) is the N-function defined by Φ(t) := ∫_{0} ^{t} a(|s|)s ds, with a: (0,∞) → (0,∞,1) a prescribed function, not necessarily differentiable, and ΔΦu := div(a(|∇u|)∇u) is the Φ -Laplacian. In addition, f: Ω × R→R is a locally Lipschitz function with respect to the second variable and ∂C denotes the Clarke subdifferential. Using a direct variational method and a nonsmooth version of the Mountain Pass Theorem the existence of nontrivial weak solutions is established. A multiplicity alternative is also proved without imposing an Ambrosetti-Rabinowitz type condition. More precisely, we show that our problem possesses either at least two nontrivial weak solutions or a rich family of negative eigenvalues. Several examples which highlight the applicability of our theoretical results are also provided.

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

Pages (from-to) | 523-554 |

Number of pages | 32 |

Journal | Advances in Differential Equations |

Volume | 23 |

Issue number | 7-8 |

Publication status | Published - júl. 1 2018 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Advances in Differential Equations*,

*23*(7-8), 523-554.

**Weak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaces.** / Costea, Nicuşor; Moroşanu, Gheorghe; Varga, C.

Research output: Article

*Advances in Differential Equations*, vol. 23, no. 7-8, pp. 523-554.

}

TY - JOUR

T1 - Weak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaces

AU - Costea, Nicuşor

AU - Moroşanu, Gheorghe

AU - Varga, C.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - We study PDI's of the type - Δ Φu ∈ ∂ Cf(x, u) subject to Dirichlet boundary condition in a bounded domain Ω ⊂ RN Lipschitzboundary ∂ Ω. Here, Φ: R→ [0,∞) is the N-function defined by Φ(t) := ∫0 t a(|s|)s ds, with a: (0,∞) → (0,∞,1) a prescribed function, not necessarily differentiable, and ΔΦu := div(a(|∇u|)∇u) is the Φ -Laplacian. In addition, f: Ω × R→R is a locally Lipschitz function with respect to the second variable and ∂C denotes the Clarke subdifferential. Using a direct variational method and a nonsmooth version of the Mountain Pass Theorem the existence of nontrivial weak solutions is established. A multiplicity alternative is also proved without imposing an Ambrosetti-Rabinowitz type condition. More precisely, we show that our problem possesses either at least two nontrivial weak solutions or a rich family of negative eigenvalues. Several examples which highlight the applicability of our theoretical results are also provided.

AB - We study PDI's of the type - Δ Φu ∈ ∂ Cf(x, u) subject to Dirichlet boundary condition in a bounded domain Ω ⊂ RN Lipschitzboundary ∂ Ω. Here, Φ: R→ [0,∞) is the N-function defined by Φ(t) := ∫0 t a(|s|)s ds, with a: (0,∞) → (0,∞,1) a prescribed function, not necessarily differentiable, and ΔΦu := div(a(|∇u|)∇u) is the Φ -Laplacian. In addition, f: Ω × R→R is a locally Lipschitz function with respect to the second variable and ∂C denotes the Clarke subdifferential. Using a direct variational method and a nonsmooth version of the Mountain Pass Theorem the existence of nontrivial weak solutions is established. A multiplicity alternative is also proved without imposing an Ambrosetti-Rabinowitz type condition. More precisely, we show that our problem possesses either at least two nontrivial weak solutions or a rich family of negative eigenvalues. Several examples which highlight the applicability of our theoretical results are also provided.

UR - http://www.scopus.com/inward/record.url?scp=85046767886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046767886&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85046767886

VL - 23

SP - 523

EP - 554

JO - Advances in Differential Equations

JF - Advances in Differential Equations

SN - 1079-9389

IS - 7-8

ER -