Differential and Integral Equations

Multiple solutions for operator equations involving duality mappings on Orlicz-Sobolev spaces

George Dinca and Pavel Matei

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Let $X$ be a real reflexive, smooth and separable Banach space having the Kadeč-Klee property and compactly imbedded in the real Banach space $Y$ and let $G:Y\rightarrow \mathbb{R} $ be a differentiable functional. By using the "fountain theorem" (Bartsch [3]), we will study the multiplicity of solutions for the operator equation \[ J_{\varphi}u=G^{\prime}(u)\text{,} \] where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Article information

Differential Integral Equations, Volume 21, Number 9-10 (2008), 891-916.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R20: Partial operator-differential equations (i.e., PDE on finite- dimensional spaces for abstract space valued functions) [See also 34Gxx, 47A50, 47D03, 47D06, 47D09, 47H20, 47Jxx]
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47J30: Variational methods [See also 58Exx]


Dinca, George; Matei, Pavel. Multiple solutions for operator equations involving duality mappings on Orlicz-Sobolev spaces. Differential Integral Equations 21 (2008), no. 9-10, 891--916. https://projecteuclid.org/euclid.die/1356038591

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