A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids

By M. Mihăilescu, Vicentiu D. Rădulescu. Proceedings of the Royal Society A, 2006

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We study the boundary value problem in , u=0 on , where is a smooth bounded domain in and is a -Laplace type operator, with . We prove that if λ is large enough then there exist at least two non-negative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/2ceff680bfe0cf6ce4005be590d748b89c127f6f

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Citations in the corpus

Recent developments in problems with nonstandard growth and nonuniform ellipticity (2021, background): “This is the case of the study of eigenvalues [66,174], see also a related papers in this volume [4,8,14,53,181].”
On the removability of isolated singular points for elliptic equations involving variable exponent (2015, background): “Since Kováčik and Rákosník first studied the Lp(x) spaces andWk,p(x) spaces in [12], many results have been obtained concerning these kinds of variable exponent spaces, see examples in [4, 7, 8, 10, 11, 16, 17, 21].”
Solutions of nonlinear problems involving p(x)-Laplacian operator (2015, background): “similar with the assumptions accepted in [14], but not the same; for example, in [14] the authors assume that (a(x, ξ) − a(x, ψ)) ⋅ (ξ − ψ) ≥ 0 for all x ∈ Ω and ξ, ψ ∈ RN ,”
On a nonhomogeneous quasilinear eigenvalue problem in sobolev spaces with variable exponent (2006, methodology): “Recent qualitative properties of solutions to quasilinear problems in Sobolev spaces with variable exponent have been obtained by Alves and Souto [2], Chabrowski and Fu [6], and Mihăilescu and Rădulescu [19].”
On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting (2016, neutral): “E-mail addresses: mmihailes@yahoo.com (M. Mihăilescu), vicentiu.radulescu@imar.ro (V. Rădulescu), dusan.repovs@guest.arnes.si (D. Repovš).”
Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent (2008, background): “4 of [22] can be used in order to show that Jλ is weakly lower semicontinuous on E.”
Remarks on eigenvalue problems involving the p(x)-Laplacian (2009, background): “In [8,13,28,30] some eigenvalue problems of other form, different from (1.”
Overview of differential equations with non-standard growth (2010, neutral): “In [150], continuing on the path of [151, 154], Mihăilescu studied a homogeneous Dirichlet problem given by ∆p1(·)u+ ∆p2(·)u = s ( −λ |u|m(x)−2u+ |u|q(x)−2u ) , (6.”
Multiple solutions for a class of p(x)-Laplacian equations in involving the critical exponent (2010, background): “In recent years, differential equations and variational problems with p(x)growth conditions have been studied extensively (see for example [1, 2, 3, 6, 13, 15]).”