Nonlinear Elliptic Research Articles

Method of Weierstrass Elliptic Functions for Finding Exact Solutions of Nonlinear Third Order Elliptic Systems on a Plane

Method of Weierstrass Elliptic Functions for Finding Exact Solutions of Nonlinear Third Order Elliptic Systems on a Plane

Quaternary Boundary Optimal Control Problem for Quaternary Nonlinear Elliptic System with Constraints

Background: Boundary optimal control problems governed by nonlinear elliptic systems are complex, involving equality and inequality constraints. Objective: This paper examines a quaternary boundary optimal control vector problem (QBOCVP) regulated by a quaternary nonlinear elliptic system (QNES) and subject to equality and inequality constraints (EINC). Methods: A weak formulation of the QBOCVP is developed, along with a mathematical representation of the quaternary adjoint equations (QAEs) associated with the QNES. Results: An existence theorem for a QBOCV addressing the constrained problem is established and rigorously proven under appropriate assumptions. The QAEs corresponding to the QNES are mathematically formulated. The Fréchet derivative (FD) of the cost function (CF) and the EINC is also derived. Furthermore, the necessary condition theorem (NCTH) and the sufficient condition theorem (SCTH) for optimality are presented and proved. Conclusions: This work provides a rigorous analysis of the QBOCVP with EINC controlled by QNES. It establishes the existence theorem and optimality conditions, providing a theoretical framework for addressing such constrained problems.

Global Solution for a Fourth-Order Nonlinear Elliptic Equation

Objective: In this work, the existence of a global solution for a fourth-order nonlinear elliptic equation with boundary conditions is studied. Method: Variational methods are applied to the proposed model, naturally using the Banach Fixed Point Contraction Theorem. Results and Discussion: The conclusion of the result obtained is that for the nonlinear elliptic equation of the fourth order they have equivalent norms in the conveniently defined spaces. On the other hand, the existence of a solution exists whenever Banach's fixed-point theorem can be applied. Implications of the research: Apply in the first university years the application of global solution for a fourth-order nonlinear elliptic equation in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of global solution for a fourth-order nonlinear elliptic equation.

On the Solvability of an Essentially Nonlinear Elliptic Differential Equation with Nonlocal Boundary Conditions

On the Solvability of an Essentially Nonlinear Elliptic Differential Equation with Nonlocal Boundary Conditions

Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations

Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations

Existence and Convergence of Solutions for Nonlinear Elliptic Systems on Graphs

Existence and Convergence of Solutions for Nonlinear Elliptic Systems on Graphs

New Regularity Estimates for the Extremal Solution of Nonlinear Elliptic Problems Involving the Fractional Laplacian with General Nonlinearity

New Regularity Estimates for the Extremal Solution of Nonlinear Elliptic Problems Involving the Fractional Laplacian with General Nonlinearity

Solving Nonlinear Elliptic Inverse Source, Coefficient and Conductivity Problems by the Methods with Bases Satisfying the Boundary Conditions Automatically

Solving Nonlinear Elliptic Inverse Source, Coefficient and Conductivity Problems by the Methods with Bases Satisfying the Boundary Conditions Automatically

Anisotropic Nonlinear Elliptic Systems with Variable Exponents, Degenerate Coercivity and Data

Anisotropic Nonlinear Elliptic Systems with Variable Exponents, Degenerate Coercivity and Data

Regularity Conditions for Nonlinear Elliptic Systems with Quadratic Nonlinearities in the Gradient

We consider nonlinear elliptic systems of equations with nondiagonal principal matrices and quadratic nonlinearities in the gradient. Under a one-sided condition on the strongly nonlinear terms, we study the local regularity of possibly unbounded weak solutions.

Asymptotic Behavior of Positive Solutions to a Nonlinear Elliptic Coupled System on an Exterior Domain

Asymptotic Behavior of Positive Solutions to a Nonlinear Elliptic Coupled System on an Exterior Domain

Entropy Solutions of the Dirichlet Problem for Some Nonlinear Elliptic Degenerate Second-Order Equations

Entropy Solutions of the Dirichlet Problem for Some Nonlinear Elliptic Degenerate Second-Order Equations

A Strongly Nonlinear Elliptic Problem with Generalized Growth in Musielak Spaces

In this article, we prove an existence theorem of renormalized solutions for nonlinear elliptic problem of the type $$\begin{aligned} -\mathop {\mathrm{div}}\>{\mathcal {A}}(x,u,\nabla u)-\mathop {\mathrm{div}}\varPhi (x,u)+{\mathcal {H}}(x,u,\nabla u)= f \quad \hbox {in }{\varOmega }, \end{aligned}$$ where the first lower-order term $$\varPhi $$ satisfies only a generalized natural growth condition without any supplementary assumptions. The approach does not require any particular type of growth condition on $$\varPhi $$ .

О суммируемости решений задачи Дирихле для нелинейных эллиптических уравнений с правой частью из классов, близких к $L^1$

О суммируемости решений задачи Дирихле для нелинейных эллиптических уравнений с правой частью из классов, близких к $L^1$

Summability of Solutions of the Dirichlet Problem for Nonlinear Elliptic Equations with Right-Hand Side in Classes Close to L1

Summability of Solutions of the Dirichlet Problem for Nonlinear Elliptic Equations with Right-Hand Side in Classes Close to L1

Nonlinear Elliptic Systems with Coupled Gradient Terms

In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type: $$ \left \{ \textstyle\begin{array}{r@{\quad }c@{\quad }l@{\quad }l@{\quad }l} -\Delta u & = & |\nabla v|^{q}+\lambda f & \text{in } &\Omega , -\Delta v& = &|\nabla u|^{p}+\mu g &\text{in } &\Omega , u=v&=& 0 & \text{on } &\partial \Omega , u,v& \geq & 0 & \text{in } &\Omega , \end{array}\displaystyle \right . $$ where $\Omega $ is a bounded domain of $\mathbb{R}^{N}$ and $p, q\ge 1$ . $f,g$ are nonnegative measurable functions with additional hypotheses and $\lambda , \mu \ge 0$ . This extends previous similar results obtained in the case where the right-hand sides are potential and gradient terms, see (Abdellaoui et al. in Appl. Anal. 98(7):1289–1306, [2019], Attar and Bentifour in Electron. J. Differ. Equ. 2017:1, [2017]).

Capacity Solution to a Nonlinear Elliptic Coupled System in Orlicz–Sobolev Spaces

We analyze the existence of a capacity solution to the following nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, $$\begin{aligned} \left\{ \begin{array}{ll} -Au= \rho (u)|\nabla \varphi |^2 &{}\quad \mathrm{in}\ \Omega , \\ \mathop {\mathrm{div}}\nolimits (\rho (u)\nabla \varphi ) =0&{}\quad \mathrm{in}\ \Omega , \\ \varphi =\varphi _0 &{}\quad \hbox {on } \partial \Omega , \\ u=0&{}\quad \mathrm{on}\, {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset {\mathbb {R}}^d$$, $$d\ge 2$$ and $$\displaystyle Au=-\mathop {\mathrm{div}}\nolimits a(x,u,\nabla u)$$ is a Leray–Lions operator defined on $$W_0^{1}L_M(\Omega )$$, M is a N-function which does not have to satisfy a $$\Delta _2$$ condition. Therefore, we work with generalized Orlicz–Sobolev spaces which are not necessarily reflexive. The function $$\varphi _0$$ is given. The proof combines truncation methods, monotonicity techniques and regularizing methods in Orlicz spaces. We introduce a sequence of approximate problems which converges (up to a subsequence) in a certain sense to a capacity solution in the context of non-reflexive Orlicz–Sobolev spaces.

The Continuous Classical Boundary Optimal Control of a Couple Nonlinear Elliptic Partial Differential Equations with State Constraints

This paper is concerned with, the proof of the existence and the uniqueness theorem for the solution of the state vector of a couple of nonlinear elliptic partial differential equations using the Minty-Browder theorem, where the continuous classical boundary control vector is given. Also the existence theorem of a continuous classical boundary optimal control vector governing by the couple of nonlinear elliptic partial differential equation with equality and inequality constraints is proved. The existence of the uniqueness solution of the couple of adjoint equations associated with the considered couple of the state equations with equality and inequality constraints is studied. The necessary conditions theorem and the sufficient conditions theorem for optimality of the couple of nonlinear elliptic equations with equality and inequality constraints are proved using the Kuhn-Tucker-Lagrange multipliers theorems

Bifurcating Solutions of a Nonlinear Elliptic Neumann Problem on Large Spherical Caps

Bifurcating Solutions of a Nonlinear Elliptic Neumann Problem on Large Spherical Caps

Entropy Solution to Nonlinear Elliptic Problem with Non-local Boundary Conditions and L¹-data

Entropy Solution to Nonlinear Elliptic Problem with Non-local Boundary Conditions and L¹-data