Journal ArticleDOI
Nonlinear elliptic equations with variable exponent: Old and new
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In this paper, the qualitative analysis of solutions to nonlinear elliptic problems of the type { − div A (x, ∇ u ) = λ | u | q (x ) − 2 u in Ω u = 0 on ∂ Ω, where Ω is a bounded or an exterior domain of R N and q is a continuous positive function.Abstract:
In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type { − div A ( x , ∇ u ) = λ | u | q ( x ) − 2 u in Ω u = 0 on ∂ Ω , where Ω is a bounded or an exterior domain of R N and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane–Emden equation and we focus on new phenomena which are due to the presence of variable exponents.read more
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Recent developments in problems with nonstandard growth and nonuniform ellipticity
TL;DR: In this article, the authors provide an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators.
Journal ArticleDOI
On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent
TL;DR: In this article, a variational analysis of a class of nonlocal fractional problems with variable exponents was performed in the context of non-homogeneous Laplace operators, and the abstract results established in this paper are applied in the variational analyses of a related non-local operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator.
Journal ArticleDOI
Double phase anisotropic variational problems and combined effects of reaction and absorption terms
TL;DR: In this paper, the existence of multiple solutions for the quasilinear equation − div A (x, ∇ u ) + V (x ) | u | α ( x ) − 2 u = f ( x, u ) in R N, which involves a general variable exponent elliptic operator in divergence form.
Journal ArticleDOI
Regularity for minimizers for functionals of double phase with variable exponents
TL;DR: In this paper, the exponents are functions of $x$ and partly generalize their regularity results for minimizers of such functionals, and the results are extended to the case that exponents of exponents can be functions of functions of the same type.
Journal ArticleDOI
Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves
TL;DR: In this paper, a class of double phase energy functionals arising in the theory of transonic flows were studied and their main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient.
References
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Lebesgue and Sobolev Spaces with Variable Exponents
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Orlicz Spaces and Modular Spaces
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Electrorheological Fluids: Modeling and Mathematical Theory
TL;DR: In this paper, a mathematical framework for modeling electrorheological fluids with shear-dependent viscosities is presented for steady flows and unsteady flows, respectively, and stable flows.