Class Of Elliptic Problems Research Articles

Elliptic equations with critical and supercritical growth at the boundary

In this study, we consider two classes of elliptic problems with nonlinear boundary conditions of concave–convex type. In the first problem, we obtain two nonzero and nonnegative solutions when the nonlinear term exhibits critical growth. In the second problem, we obtain infinitely many solutions (with no prescribed sign) by assuming that the nonlinearity is even and subcritical near the origin but with no growth condition at infinity.

Nehari manifold for a Schrödinger equation with magnetic potential involving sign-changing weight function

We consider the following class of elliptic problems − Δ A u + u = a λ ( x ) | u | q − 2 u + b μ ( x ) | u | p − 2 u , x ∈ R N , where 1 < q < 2 < p < 2 ∗ = 2 N N − 2 N ≥ 3 , a λ ( x ) is a sign-changing weight function, b μ ( x ) is continuous, 0 $ ]]> λ > 0 and 0 $ ]]> μ > 0 are real parameters, u ∈ H A 1 ( R N ) and A : R N → R N is a magnetic potential. Exploring the relationship between the Nehari manifold and fibering maps, we will discuss the existence, multiplicity and regularity of solutions.

On singular elliptic boundary value problems via a harmonic analysis approach

We consider a class of elliptic problems in the half-space R + n with nonhomogeneous boundary conditions containing nonlinearities and critical singular potentials. We obtain existence and regularity results by means of a harmonic analysis approach based on a framework of weighted spaces in Fourier variables. This framework seems to be new in the context of elliptic boundary value problems and allows us to consider Hardy's potential λ 1 / | x | 2 in R + n and Kato's potential λ 2 / | x ′ | on the boundary ∂ R + n , as well as their versions with multiple poles, without using the so-called Kato and Hardy inequalities. Singular boundary forcing terms can also be addressed. Moreover, our results cover supercritical nonlinearities, such as ± u p in R + n and ± u q on ∂ R + n with integers p > 2 ⁎ − 1 and q > 2 ⁎ − 1 .

AN INEXACT NEWTON METHOD WITH INNER PRECONDITIONED CG FOR NON-UNIFORMLY MONOTONE ELLIPTIC PROBLEMS

The present paper introduces an inexact Newton method, coupled with a preconditioned conjugate gradient method in inner iterations, for elliptic operators with non-uniformly monotone upper and lower bounds. Convergence is proved in Banach space level. The results cover real-life classes of elliptic problems. Numerical experiments reinforce the convergence results.

A Bayesian Numerical Homogenization Method for Elliptic Multiscale Inverse Problems

A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. Assuming a known micro structure, our aim is to recover a macroscopic scalar parameterization of the microscale tensor. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings.

Numerical Homogenization and Model Order Reduction for Multiscale Inverse Problems

A new numerical method based on numerical homogenization and model order reduction is introduced for the solution of multiscale inverse problems. We consider a class of elliptic problems with highl...

Some nonlinear elliptic problems with [formula omitted] coefficients

In this paper we focus on a class of elliptic problems with coefficients in L1(Ω). We also analyze the interplay between the principal term and the lower order ones and the regularizing effect that can be obtained from this interaction.

On a fractional degenerate Kirchhoff-type problem

In this paper, we study a highly nonlocal parametric problem involving a fractional-type operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Our approach is of variational nature; by working in suitable fractional Sobolev spaces which encode Dirichlet homogeneous boundary conditions, and exploiting an abstract critical point theorem for smooth functionals, we derive the existence of at least three weak solutions to our problem for suitable values of the parameters. Finally, we provide a concrete estimate of the range of these parameters in the autonomous case, by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain. The methods adopted here can be exploited to study different classes of elliptic problems in presence of a degenerate nonlocal term.

Multiplicity of Multi-Bump Type Nodal Solutions for A Class of Elliptic Problems with Exponential Critical Growth in ℝ2

AbstractIn this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problemswhereλ ∈(0, ∞),fis a continuous function with exponential critical growth andV: ℝ2→ℝ is a continuous function verifying some hypotheses.

Multiplicity of solutions for elliptic problems of p-Kirchhoff type with critical exponent

In this paper we consider a class of elliptic problems of p-Kirchhoff type with critical exponent in bounded domains and new results as regards the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method.

Nonlinear elliptic problem related to the Hardy inequality with singular term at the boundary

Let Ω ⊂ ℝNbe a bounded regular domain of ℝNand 1 &lt; p &lt; ∞. The paper is divided into two main parts. In the first part, we prove the following improved Hardy inequality for convex domains. Namely, for all [Formula: see text], we have [Formula: see text] where d(x) = dist (x, ∂Ω), [Formula: see text] and C is a positive constant depending only on p, N and Ω. The optimality of the exponent of the logarithmic term is also proved. In the second part, we consider the following class of elliptic problem [Formula: see text] where 0 &lt; q ≤ 2* - 1. We investigate the question of existence and nonexistence of positive solutions depending on the range of the exponent q.

Two Algorithms for a Class of Elliptic Problems in Shape Optimization

We propose two algorithms for elliptic boundary value problems in shape optimization. With the finite element method, the optimization problem is replaced by a discrete variational problem. We give rules and use them to decide which elements are to be reserved. Those rules are determined by the optimization; as a result, we get the optimal design in shape. Numerical examples are provided to show the effectiveness of our algorithms.

Homogenization and correctors of a class of elliptic problems in perforated domains

Homogenization and correctors of a class of elliptic problems in perforated domains

On a class of elliptic problems with indefinite nonlinearities

In this paper we apply variational and sub-supersolution methods to study the existence and multiplicity of nonnegative solutions for a class of indefinite semilinear elliptic problems that depend on a parameter. The results on the existence of solutions do not impose any growth condition at infinity on the term which depends on the parameter. To derive such results, first we find a positive supersolution by solving an auxiliary problem. Then we use a truncation argument and a global minimization method. The main hypothesis for the existence of two nonzero solutions is that the indefinite term is the product of a weight function, having a thick zero set, and a nonlinear function which satisfies the Ambrosetti–Rabinowitz superlinear condition. Results for some corresponding indefinite problems are also established.

Existence and multiplicity of solutions for a class of p &amp; q elliptic problems with critical exponent

AbstractThis paper is concerned with the existence and multiplicity of nontrivial solutions to the following class of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$p \&amp; q$\end{document} elliptic problems with critical exponent where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega \subset \mathbb {R}^{N}$\end{document} is a bounded smooth domain, N ≥ 3, λ is a positive parameter and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$a:\mathbb {R}\rightarrow \mathbb {R}$\end{document} is a function of C1 class. Here 1 &lt; γ &lt; N, γ &lt; s &lt; γ* or 1 &lt; s &lt; γ, where γ will be stated later.

Existence of solution for two classes of elliptic problems in [formula omitted] with zero mass

In this paper we show the existence of a positive solution for the following class of elliptic equations−Δu=K(x)f(u)in RN where f is a continuous function with quasicritical growth and K is a nonnegative function verifying some conditions of two kinds. We analyze the problem with K asymptotically periodic, K periodic and K belonging to some Lr.

STRICTLY CONVERGENT ALGORITHM FOR AN ELLIPTIC EQUATION WITH NONLOCAL AND NONLINEAR BOUNDARY CONDITIONS

The paper describes a formally strictly convergent algorithm for solving a class of elliptic problems with nonlinear and nonlocal boundary conditions, which arise in modeling of the steady-state conductive-radiative heat transfer processes. The proposed algorithm has two levels of iterations, where inner iterations by means of the damped Newton method solve an appropriate elliptic problem with nonlinear, but local boundary conditions, and outer iterations deal with nonlocal terms in boundary conditions.

A variational binary level-set method for elliptic shape optimization problems

We present a variational binary level-set method to solve a class of elliptic problems in shape optimization. By the ‘ersatz material’ approach, which amounts to fill the holes by a weak phase, the original shape optimization model is approximated by a two-phase optimization problem. Under the binary level-set framework, we need to optimize a smooth functional under a binary constraint. We propose an augmented Lagrangian method to solve the constrained optimization problem. Numerical results are presented and compared with those obtained by level-set methods, which demonstrate the robustness and efficiency of our method.

Uniqueness of singular solution of semilinear elliptic equation

In this paper, we study asymptotic behavior of solution near 0 for a class of elliptic problem. The uniqueness of singular solution is established.

A global compact result for a semilinear elliptic problem with Hardy potential and critical non-linearities on ℝ N

In this paper, we are concerned with the following class of elliptic problems: $$ \left\{ \begin{gathered} - \Delta u - \mu \frac{u} {{|x|^2 }} + a(x)u = |u|^{2* - 2} u + k(x)|u|^{q - 2} u, \hfill \\ u \in H^1 (\mathbb{R}^N ), \hfill \\ \end{gathered} \right. $$ where 2* = 2N/(N - 2) is the critical Sobolev exponent, 2 < q < 2*, \( 0 \leqslant \mu < \bar \mu \triangleq \frac{{(N - 2)^2 }} {4} \), a(x), k(x) ∈ C(ℝN). Through a compactness analysis of the functional corresponding to the problems (*), we obtain the existence of positive solutions for this problem under certain assumptions on a(x) and k(x).