Solutions Of Singular Elliptic Equations Research Articles

Existence of solutions for singular elliptic equations with mixed boundary conditions

In this article, we investigate generalized solutions for elliptic equations involving double-phase ( p ( x ) , q ( x ) ) -Laplacian operators with Hardy potential in variable exponent spaces. ( p ( x ) , q ( x ) ) -Laplacian operators include p-Laplacian, q-Laplacian, p ( x ) -Laplacian and q ( x ) -Laplacian operators. Additionally, ( p ( x ) , q ( x ) ) -Laplacian elliptic equations with singular coefficients under mixed boundary conditions are seldom mentioned in previous work. Some new theorems of the existence on the generalized solutions are reestablished for such equations via variational methods when the nonlinearity satisfies suitable hypotheses in variable exponent Lebesgue spaces.

Asymptotic behavior and existence of solutions for singular elliptic equations

We study the asymptotic behavior, as $$\gamma $$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is $$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$ where $$\Omega $$ is an open, bounded subset of $${\mathbb {R}}^{N}$$ and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f: either strictly positive on every compactly contained subset of $$\Omega $$ or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with $$\begin{aligned} -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text { in }\Omega . \end{aligned}$$

Singular Solutions of Elliptic Equations with Iterated Exponentials

We construct positive singular solutions for the problem $$-\Delta u=\lambda \exp (e^u)$$ in $$B_1\subset {\mathbb {R}}^n$$ ($$n\ge 3$$), $$u=0$$ on $$\partial B_1$$, having a prescribed behaviour around the origin. Our study extends the one in Miyamoto (J Differ Equ 264:2684–2707, 2018) for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials.

Singular solutions of elliptic equations on a perturbed cone

In this work we obtain positive singular solutions of{−Δu(y)=u(y)p in y∈Ωt,u=0 on y∈∂Ωt, where Ωt is a sufficiently small C2,α perturbation of the cone Ω:={x∈RN:x=rθ,r>0,θ∈S} where S⊂SN−1 has a smooth nonempty boundary and where p>1 satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.

Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball

In this article we obtain positive singular solutions of(1)−Δu=|∇u|p in Ω,u=0 on ∂Ω, where Ω is a small C2 perturbation of the unit ball in RN. For NN−1<p<2 we prove that if Ω is a sufficiently small C2 perturbation of the unit ball there exists a singular positive weak solution u of (1). In the case of p>2 we prove a similar result but now the positive weak solution u is contained in C0,p−2p−1(Ω‾) and yet is not in C0,p−2p−1+ε(Ω‾) for any ε>0.

Point Rupture Solutions of Singular Elliptic Equations in N-D

Point Rupture Solutions of Singular Elliptic Equations in N-D

On the multiplicity of non radial solutions for singular elliptic equations

The main goal of this work is to analyze the existence and the multiplicity of non radial solutions for a Dirichlet problem associated to an elliptic singular partial differential equation. Our approach is based on a variational method.

Optimal Quadrilateral Finite Elements on Polygonal Domains

We propose three quadrilateral mesh refinement algorithms to improve the convergence of the finite element method approximating the singular solutions of elliptic equations, which are due to the non-smoothness of the domain. These algorithms result in graded meshes consisting of convex and shape-regular quadrilaterals. With analysis in weighted spaces, we provide the selection criteria for the grading parameter, such that the optimal convergence rate can be recovered for the associated finite element approximation. Various numerical tests verify the theory. In addition to the bi-k elements, we also investigate the serendipity elements on the graded quadrilateral meshes in the numerical experiments.

Positive solutions for singular elliptic equations with mixed Dirichlet-Neumann boundary conditions

MSC (2010) 35D30, 35J20, 35J75 In this paper, we investigate the existence of positive solutions for singular elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Sobolev-Hardy critical exponents and Hardy terms by using the concentration compactness principle, the strong maximum principle and the Mountain Pass lemma. We also prove, under complementary conditions, that there is no nontrivial solution if the domain is star-shaped with respect to the origin. C

Nonexistence of solutions for singular elliptic equations with a quadratic gradient term

This paper concerns the nonexistence of solutions for singular elliptic equations with a quadratic gradient term. The main results complement and partly extend some works by Arcoya et al. (2009) [1]. As a by-product of the main results, we fill in a gap in one of their works.

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.

On singular elliptic equations involving a concave term and critical Caffarelli‐Kohn‐Nirenberg exponent

AbstractIn this paper, we establish the existence of multiple positive solutions for singular elliptic equations involving a concave term and critical Caffarelli‐Kohn‐Nirenberg exponent. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev–Hardy exponents

Variational methods are used to prove the multiplicity of positive solutions for the following singular elliptic equation:where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂ Ω, λ &gt; 0 , $1\le q&lt;2$, $0\le\mu&lt;\bar{\mu}=(N-2)^2/4$, 0 ≤ s &lt; 2, 2*(s)=2(N−s)/(N−2) and f and g are continuous functions on $\bar{\varOmega}$, that change sign on Ω.

Singular solutions of elliptic equations with nonstandard growth

AbstractWe prove a removability result for nonlinear elliptic equations withp (x)‐type nonstandard growth and estimate the growth of solutions near a nonremovable isolated singularity. To accomplish this, we employ a Harnack estimate for possibly unbounded solutions and the fact that solutions with nonremovable isolated singularities are p (x)‐superharmonic functions (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Multiple Positive Solutions for Singular Elliptic Equations with Concave-Convex Nonlinearities and Sign-Changing Weights

We study existence and multiplicity of positive solutions for the following Dirichlet equations: in , on , where is a bounded domain with smooth boundary , , , , , and are continuous functions on which are somewhere positive but which may change sign on .

On stabilization of solutions of singular elliptic equations

We study linear and quasi-linear elliptic equations containing the Bessel operator with respect to a selected variable (so-called special variable). The well-posedness of the nonclassical Dirichlet problem (with the additional condition of evenness with respect to the special variable) in the half-space is proved, an integral representation of the solution is constructed, and a necessary and sufficient condition of stabilization is established. The stabilization is understood as follows: the solution has a finite limit as the independent variable tends to infinity along the direction orthogonal to the boundary hyperplane.

On Uniqueness in the General Inverse Transmisson Problem

In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.

Singular solutions of elliptic equations and the determination of conductivity by boundary measurements

We improve some results on uniqueness and stability in the determination of the coefficient a in the equation (i) div( a grad u) = 0 in Ω, when all possible pairs of Dirichlet and Neumann data on ∂Ω are known. We also treat cases of anisotropic equations. Our method relies on the construction of solutions to (i) having an isolated singularity with prescribed asymptotic behaviour.

Entire solutions of singular elliptic equations

On etudie les solutions entieres positives de l'equation elliptique singuliere −Δu=f(x)u − λ, λ>0, ou f∈C loc α(R n ), 0 0 pour |x|>0

A class of strongly degenerate elliptic operators

Using a weighted Poincaré inequality, we study (ω1,…,ωn)-elliptic operators. This method is applied to solve singular elliptic equations with boundary conditions in W1,2. We also obtain a result about the regularity of solutions of singular elliptic equations. An application to (ω1,…,ωn)-parabolic equations is given.