Solutions of an anisotropic nonlocal problem involving variable exponent

‎‎In this paper‎, ‎we prove the existence of infinitely many solutions of a system of boundary value problems involving flux boundary conditions in anisotropic variable exponent Sobolev spaces‎, ‎by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces‎.

In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain D⊂RN (N≥3) with homogeneous Dirichlet boundary conditions. A key feature of the system is the presence of a Hardy-type singular term with a variable exponent, where δ(x) represents the distance from x to the boundary ∂D. By employing a critical point theorem in the framework of variable exponent Sobolev spaces, we establish the existence of a weak solution whose norm vanishes at zero.

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.

We introduce a new class of operators that extend both generalized Laplace operators and generalized mean curvature operators. We start the discussion on general anisotropic systems with variable exponents that involve our operators, then we focus on a specific example of such system, we show that it admits a unique weak solution and we complete our work with some comments on other related systems. The newly introduced operators are appropriate for the study conducted in the anisotropic spaces with variable exponents, but at the end of the paper we also provide their versions corresponding to the studies conducted in the anisotropic Sobolev spaces with constant exponents, or in the isotropic variable exponent Sobolev spaces, since, to the best of our knowledge, they represent a novelty even for the classical Sobolev spaces.

In this paper, we shall be concerned with the study of the following quasilinear anisotropic elliptic Dirichlet problems of the type \begin{document}$\begin{equation} -\mbox{div }\;a(x,u,\nabla u) = f -\mbox{div } F \qquad \mbox{in}\quad \Omega,\quad\quad\quad\quad\quad(1) \end{equation}$ \end{document} where $ f\in L^{1}(\Omega) $ and $ F \in \prod_{i = 1}^{N} L^{p'_{i}(\cdot)}(\Omega), $ and $ a_{i}(x,u,\xi) $ are Carathéodory functions from $ \Omega \times I\!\!R \times I\!\!R^{N} $ into $ I\!\!R $, which satisfy assumptions of growth, coercivity and strict monotonicity. We prove the existence of entropy solutions for the quasilinear elliptic equation associated to the unilateral problem by relying on the penalization method, in the anisotropic variable exponent Sobolev spaces. Our approach is also based on the techniques of monotone operators in Banach spaces, the existence of weak solutions, and some approximations methods. The problems of the type $ (1) $ are very interesting from the purely mathematical point of view. On the other hand, such equations $ (1) $ appear in different contexts, in particular, the mathematical description of motions of the non-newtonien fluids; we quote for instance the electro-rheological fluids; the deformation of membrane constrained by an obstacle, the image processing and other various physical applications.

In this paper, we prove the existence and uniqueness of entropy solution to nonlinear parabolic problem with homogeneous Dirichlet boundary conditions with $$L^1$$ -data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. The general assumptions and the nonlinear semigroup theory are considered to prove the existence and uniqueness of mild solution satisfying the $$L^1$$ -comparison principle. Moreover, under the same general assumptions and some a-priori estimations of the sequence of mild solutions, we obtain the existence and uniqueness of weak solution. Finally, we prove the existence and uniquess of the renormalized solution which is equivalent to the existence and uniqueness of entropy solution.

In this paper, we establish existence of infinitely many weak solutions for a class of quasilinear stationary Kirchhoff‐type equations, which involves a general variable exponent elliptic operator with critical growth. Precisely, we study the following nonlocal problem: where Ω is a bounded smooth domain of , with homogeneous Dirichlet boundary conditions on ∂Ω, the nonlinearity is a continuous function, is a function of the class is a continuous function, whose properties will be introduced later, λ is a positive parameter and . We assume that , where is the critical Sobolev exponent. We will prove that the problem has infinitely many solutions and also we obtain the asymptotic behavior of the solution as λ → 0+. Furthermore, we emphasize that a difference with previous researches is that the conditions on a(·) are general overall enough to incorporate some interesting differential operators. Our work covers a feature of the Kirchhoff's problems, that is, the fact that the Kirchhoff's function M in zero is different from zero, it also covers a wide class of nonlocal problems for p(x) > 1, for all . The main tool to find critical points of the Euler–Lagrange functional associated with this problem is through a suitable truncation argument, concentration‐compactness principle for variable exponent found in Bonder and Silva (2010), and the genus theory introduced by Krasnoselskii. The result of this paper extends or complements, or else completes recent papers and is new in several directions for the stationary Kirchhoff equations involving the p(x)‐Laplacian type operators.

On a PDE involving the [formula omitted]-Laplace operator

In this paper, we prove the existence and the uniqueness of renormalized solution to a nonlinear multivalued parabolic problem β(u)t - div a(x,∇ u) ∋ f , with homogeneous Dirichlet boundary conditions and L1-data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. Some a-priori estimates are used to obtain our results.

In this paper, we establish a new abstract functional space where K is a uncertain weighted function and p is a variable exponent. Based on the properties of this space, we consider the existence and regularity of weak solutions for non‐local fractional differential inclusion with homogeneous Dirichlet boundary conditions. Under a suplinear growth condition we obtain the existence of weak solutions, the compactness and Hölder regularity of the solution set using set‐valued analysis and the surjectivity principle of pseudomonotonicity. Furthermore, the existence of extremal solutions and a relaxation result is discussed.

In the beginning of the 90’s Kovacik and Rakosnik [17] introduced variable exponent Lebesgue and Sobolev spaces. In fact, generalized Lebesgue and Sobolev spaces are special cases of so-called Orlicz-Musielak spaces, and in this form their investigation goes back a bit further, to Orlicz [20], Hudzik [15], and Musielak [18], see also Sharapudinov [23]. During the last couple of years Lebesgue and Sobolev spaces with variable exponent have been studied at an increasing pace by Diening [4, 5], Edmunds and Rakosnik [6, 7], Fan, Shen and Zhao [9, 10], Cruz-Uribe, Fiorenze and Neugebauer [3], Kokilasvili and Samko [16], and Nekvinda [19], among others. One area where these spaces have found applications is the study of electrorheological fluids, as described in the book of Růžicka [22]. A mathematical application is the study of variational integrals with non-standard growth, see the papers by Acerbi and Mingione [1, 2]. Sobolev capacity for fixed exponent spaces has found a great number of uses (e.g. the monographs by Evans and Gariepy [8] and Heinonen, Kilpelainen, and Martio [14]). It was introduced into the study of variable exponent spaces in [12] and has been applied to the investigation of zero boundary values of Sobolev functions in [13]. In [12] we required the assumption 1 < ess inf p ≤ ess sup p < ∞ of the variable exponent p to guarantee that our set-function is indeed a Choquet capacity. This is unsatisfactory, since there is no reason to expect this condition to be of relevance in this context. In this paper we show that the lower inequality needs to hold only locally. In particular we show in Corollary 4.2 that if the exponent p is continuous, then zero capacity sets enjoy the usual subadditivity property.

Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian

The Dirichlet problem is considered in arbitrary domains for a class of second-order anisotropic elliptic equations with variable nonlinearity exponents and right-hand sides in . It is proved that an entropy solution exists in anisotropic Sobolev spaces with variable exponent. It is proved that the entropy solution obtained is a renormalized solution of the problem under consideration. Bibliography: 37 titles.

We establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic equations involving the anisotropic -Laplace operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev spaces and our main tool is the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz.

In this chapter, we present the variable exponent Lebesgue spaces defined by Orlicz1) in 1931. Although the variable exponent Lebesgue spaces are introduced in 1931, it began to be actively studied in 1990s. We especially used the resources by Diening, Harjulehto, Hasto, Ruzicka 2011; Fan, Zhao 2001; Kovacik, Rakosnik 1991; Cruz-Uribe, Fiorenza 2013; Radulescu and Repovs 2015 for this section. The notion variable exponent Lebesgue and Sobolev spaces is directly related to the classical Lebesgue and Sobolev spaces where the constant p is replaced with the function p(.) which may depend on a variable. Further properties of these spaces are introduced and analyzed in that chapter.