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

2006 ◽  
Vol 462 (2073) ◽  
pp. 2625-2641 ◽  
Author(s):  
Mihai Mihăilescu ◽  
Vicenţiu Rădulescu
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Electrorheological Fluids ◽  
Mountain Pass Lemma ◽  
Multiplicity Result ◽  
Smooth Bounded Domain ◽  
Degenerate Problem ◽  
Laplace Type ◽  
Nonlinear Degenerate

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.

scholarly journals Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

2006 ◽  
Vol 4 (3) ◽  
pp. 225-242 ◽  
Author(s):  
Teodora-Liliana Dinu
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Eigenvalue Problems ◽  
Mountain Pass Lemma ◽  
Smooth Bounded Domain ◽  
Nonlinear Eigenvalue Problems ◽  
First Case ◽  
Symmetric Version

We study the boundary value problem-div⁡((|∇u|p1(x)-2+|∇u|p2(x)-2)∇u)=f(x,u)inΩ,u=0on∂Ω, whereΩis a smooth bounded domain inℝN. We focus on the cases whenf±(x,  u)=±(-λ|u|m(x)-2u+|u|q(x)-2u), wherem(x)≔max⁡⁡{p1(x),p2(x)}<q(x)<N⋅m(x)N-m(x)for anyx∈Ω̅. In the first case we show the existence of infinitely many weak solutions for anyλ>0. In the second case we prove that ifλis large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with aℤ2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.

Multiple solutions for a fourth-order elliptic equation of Kirchhoff type with variable exponent

2019 ◽  
Vol 13 (05) ◽  
pp. 2050096 ◽  
Author(s):  
Nguyen Thanh Chung
Keyword(s):  
Continuous Function ◽  
Fourth Order ◽  
Variable Exponent ◽  
Mountain Pass Theorem ◽  
Multiplicity Result ◽  
Hölder Continuous ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Elliptic Equations

In this paper, we consider a class of fourth-order elliptic equations of Kirchhoff type with variable exponent [Formula: see text] where [Formula: see text], [Formula: see text], is a smooth bounded domain, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is the operator of fourth-order called the [Formula: see text]-biharmonic operator, [Formula: see text] is the [Formula: see text]-Laplacian, [Formula: see text] is a log-Hölder continuous function and [Formula: see text] is a continuous function satisfying some certain conditions. A multiplicity result for the problem is obtained by using the mountain pass theorem and Ekeland’s variational principle provided [Formula: see text] is small enough.

GEOMETRY OF SOBOLEV SPACES WITH VARIABLE EXPONENT AND A GENERALIZATION OF THE p-LAPLACIAN

2009 ◽  
Vol 07 (04) ◽  
pp. 373-390 ◽  
Author(s):  
GEORGE DINCA ◽  
PAVEL MATEI
Keyword(s):  
Sobolev Space ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Uniformly Convex ◽  
Smooth Bounded Domain ◽  
Generalized Lebesgue Space

Let Ω ⊂ ℝN, N ≥ 2, be a smooth bounded domain. It is shown that: (a) if [Formula: see text] and ess inf x ∈ y p(x) > 1, then the generalized Lebesgue space (Lp (·)(Ω), ‖·‖p(·)) is smooth; (b) if [Formula: see text] and p(x) > 1, [Formula: see text], then the generalized Sobolev space [Formula: see text] is smooth. In both situations, the formulae for the Gâteaux gradient of the norm corresponding to each of the above spaces are given; (c) if [Formula: see text] and p(x) ≥ 2, [Formula: see text], then [Formula: see text] is uniformly convex and smooth.

scholarly journals Nonlocal approximations to anisotropic Sobolev norms

2021 ◽  
pp. 1-20
Author(s):  
Ivan Cinelli ◽  
Gianluca Ferrari ◽  
Marco Squassina
Keyword(s):  
Image Processing ◽  
Nonlinear Elasticity ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Electrorheological Fluids ◽  
Variable Exponent Sobolev Spaces ◽  
Sobolev Norms

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent p ( x ) reaches the value 1.

scholarly journals Nonlocal characterizations of variable exponent Sobolev spaces

2021 ◽  
pp. 1-22
Author(s):  
Gianluca Ferrari ◽  
Marco Squassina
Keyword(s):  
Elasticity Theory ◽  
Nonlinear Elasticity ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Singular Limit ◽  
Electrorheological Fluids ◽  
Nonlinear Elasticity Theory ◽  
Limit Formula ◽  
Variable Exponent Sobolev Spaces ◽  
Anisotropic Case

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.

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