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

Nicuşor Costea, Gheorghe Moroşanu, C. Varga

Research output: Article

Abstract

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.

Original languageEnglish
Pages (from-to)523-554
Number of pages32
JournalAdvances in Differential Equations
Volume23
Issue number7-8
Publication statusPublished - júl. 1 2018

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Orlicz-Sobolev Spaces
Sobolev spaces
Differential Inclusions
Dirichlet
Weak Solution
Solvability
Clarke Subdifferential
Locally Lipschitz Function
Partial
Mountain Pass Theorem
Direct Method
Variational Methods
Dirichlet Boundary Conditions
Differentiable
Bounded Domain
Multiplicity
Denote
Eigenvalue
Alternatives
Boundary conditions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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

In: Advances in Differential Equations, Vol. 23, No. 7-8, 01.07.2018, p. 523-554.

Research output: Article

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