Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type

This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent Δ p ( x ) 2 u − M ( ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x ) Δ p ( x ) u + | u | p ( x ) − 2 u = λ f ( x , u ) + μ g ( x , u ) in Ω , u = Δ u = 0 on ∂ Ω , $$\begin{array}{} \left\{\begin{array}{} \Delta^2_{p(x)}u-M\bigg(\displaystyle\int\limits_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,\text{d} x \bigg)\Delta_{p(x)} u + |u|^{p(x)-2}u = \lambda f(x,u)+\mu g(x,u) \quad \text{ in }\Omega,\\ u=\Delta u = 0 \quad \text{ on } \partial\Omega, \end{array}\right. \end{array}$$ where p − := inf x ∈ Ω ¯ p ( x ) > max 1 , N 2 , λ > 0 $\begin{array}{} \displaystyle p^{-}:=\inf_{x \in \overline{\Omega}} p(x) \gt \max\left\{1, \frac{N}{2}\right\}, \lambda \gt 0 \end{array}$ and μ ≥ 0 are real numbers, Ω ⊂ ℝ N (N ≥ 1) is a smooth bounded domain, Δ p ( x ) 2 u = Δ ( | Δ u | p ( x ) − 2 Δ u ) $\begin{array}{} \displaystyle \Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u) \end{array}$ is the operator of fourth order called the p(x)-biharmonic operator, Δ p(x) u = div(|∇u| p(x)–2∇u) is the p(x)-Laplacian, p : Ω → ℝ is a log-Hölder continuous function, M : [0, +∞) → ℝ is a continuous function and f, g : Ω × ℝ → ℝ are two L 1-Carathéodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions.

A new analytic solution of complex Langevin differential equations

PurposeIn this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.Design/methodology/approachThe methodology is based on the geometric function theory.FindingsThe authors present a new analytic function for a class of complex LDEs.Originality/valueThe authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.

Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

In this paper, we study the solution set of the following Dirichlet boundary equation: − div a 1 x , u , D u + a 0 x , u = f x , u , D u in Musielak-Orlicz-Sobolev spaces, where a 1 : Ω × ℝ × ℝ N ⟶ ℝ N , a 0 : Ω × ℝ ⟶ ℝ , and f : Ω × ℝ × ℝ N ⟶ ℝ are all Carathéodory functions. Both a 1 and f depend on the solution u and its gradient D u . By using a linear functional analysis method, we provide sufficient conditions which ensure that the solution set of the equation is nonempty, and it possesses a greatest element and a smallest element with respect to the ordering “≤,” which are called barrier solutions.

Global Hölder continuity of solutions to quasilinear equations with Morrey data

We deal with general quasilinear divergence-form coercive operators whose prototype is the [Formula: see text]-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical [Formula: see text]-result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.

Existence of renormalized solutions for some quasilinear elliptic Neumann problems

This paper is devoted to study some nonlinear elliptic Neumann equations of the type { A u + g ( x , u , ∇ u ) + | u | q ( ⋅ ) - 2 u = f ( x , u , ∇ u ) in Ω , ∑ i = 1 N a i ( x , u , ∇ u ) ⋅ n i = 0 on ∂ Ω , \left\{ {\matrix{ {Au + g(x,u,\nabla u) + |u{|^{q( \cdot ) - 2}}u = f(x,u,\nabla u)} \hfill & {{\rm{in}}} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{a_i}(x,u,\nabla u) \cdot {n_i} = 0} } \hfill & {{\rm{on}}} \hfill & {\partial \Omega ,} \hfill \cr } } \right. in the anisotropic variable exponent Sobolev spaces, where A is a Leray-Lions operator and g(x, u, ∇u), f (x, u, ∇u) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.

On application of differential subordination for Carathéodory functions

New sufficient conditions involving the properties of analytic functions to belong to the class of Carathéodory functions are investigated. Certain univalence and starlikeness conditions are deduced as special cases of main results.

The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

In this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula

We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum ${\textsf{E}}$ and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as ${\textsf{E}}$-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them “MCMV.” Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.