Singular Elliptic Boundary Value Problem Research Articles

On a degenerate singular elliptic problem

AbstractIn this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by where Ω is a bounded smooth domain in with , w belongs to the Muckenhoupt class for some , f is a nonnegative function belonging to some Lebesgue space and .

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 .

On Connection between Solutions in Different Weighted Spaces to One Singular Elliptic Boundary Value Problem

On Connection between Solutions in Different Weighted Spaces to One Singular Elliptic Boundary Value Problem

The role of the Hilbert metric in a class of singular elliptic boundary value problems in convex domains

In a recent paper [7], we were led to consider a distance over a bounded open convex domain. It turns out to be the so-called Thompson metric, which is equivalent to the Hilbert metric. It plays a key role in the analysis of existence and uniqueness of solutions to a class of elliptic boundary-value problems that are singular at the boundary.

A note on regularity of solutions for degenerate singular elliptic boundary value problems

We prove the $$C^{1,\beta }$$ -boundary regularity and a comparison principle for weak solutions of the problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u-\lambda \psi _{p}(u)=f(x)&{}\quad \text {in }\Omega , \\ u=0&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb {R}^{N},N>1\ $$ with smooth boundary $$\partial \Omega ,\ \ \Delta _{p}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u),\psi _{p}(u)=|u|^{p-2}u,p>1,\ $$ and f is allowed to be unbounded.

Existence of Weak Solutions to a Class of Singular Elliptic Equations

This paper is concerned with the existence of solutions to the following singular elliptic boundary value problem involving \({p}\)-Laplace operator $$-{\rm div}(|\nabla u|^{p-2} \nabla u)=\frac{h}{u^\gamma} \text{in } \Omega, \quad u > 0\text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega.$$ Here, \({\Omega\subset \mathbb{R}^N(N\geq3)}\) is a bounded domain with smooth boundary, and \({h}\) is a positive \({L^1}\) function on \({\Omega}\). A “compatibility condition” on the couple \({(h(x),\gamma)}\) is given for the problem to have at least one solution. More precisely, it is shown that the problem admits at least one solution if and only if there exists a \({u_0\in W_0^{1,p}(\Omega)}\) such that \({\int_\Omega hu_0^{1-\gamma} \mathrm{d}x < \infty}\). This generalizes a previous result obtained by Sun and Zhang (Calc Var Partial Differ Equ 49:909–922, 2014) who considered the case \({p=2}\).

A three solution theorem for singular nonlinear elliptic boundary value problems

We establish a three solution theorem for singular elliptic boundary value problems of the form −Δu=f(u)uβ in Ω, u=0 on ∂Ω where Ω is a bounded domain in RN, N≥1 with a smooth boundary ∂Ω. Here f:[0,∞)→[0,∞) is a C1 function in [0,∞) with f(0)>0 and β∈(0,1). In particular if there exist two pairs of sub-supersolutions (ψ1,ϕ1) and (ψ2,ϕ2) where ψ1≤ψ2≤ϕ1, ψ1≤ϕ2≤ϕ1 with ψ2≰ϕ2, and ψ2, ϕ2 are strict sub and supersolutions, then we establish existence of three solutions u1, u2 and u3 for the above boundary value problem such that u1∈[ψ1,ϕ2], u2∈[ψ2,ϕ1] and u3∈[ψ1,ϕ1]∖([ψ1,ϕ2]∪[ψ2,ϕ1]). Our results extend the work in [1] (and also in [17]) where such multiplicity results were discussed in the non-singular case β=0. Further our results strengthen the multiplicity results in [12] and [13] for such singular problems (β≠0), namely, establish an additional third solution. We also establish a sub-super solution theorem for infinite semipositone problems (i.e. the case when f(0)<0).

The second order estimate for the solution to a singular elliptic boundary value problem

We study the second order estimate for the unique solution near the boundary to the singular Dirichlet problem -?u = b(x)g(u); u &gt; 0; x ? ?, u|?? = 0, where ? is a bounded domain with smooth boundary in RN, g ? C1((0,?),(0?)), g is decreasing on (0,?) with lim s?0+ g(s) = 1 and g is normalized regularly varying at zero with index ? (? &gt; 1), b ? C?(??) (0 &lt; ? &lt; 1), is positive in ?, may be vanishing on the boundary. Our analysis is based on Karamata regular variation theory.

The second expansion of the solution for a singular elliptic boundary value problem

In this paper we analyze the second expansion of the unique solution near the boundary to the singular Dirichlet problem − Δ u = b ( x ) g ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = 0 , where Ω is a bounded domain with smooth boundary in R N , g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) , g is decreasing on ( 0 , ∞ ) with lim s → 0 + g ( s ) = ∞ and g is normalised regularly varying at zero with index − γ ( γ > 1 ), b ∈ C α ( Ω ¯ ) , is positive in Ω, may be vanishing on the boundary.

Existence of positive solutions of singular elliptic boundary value problems in a ball

This paper deals with existence theorems of positive solutions for singular elliptic boundary value problems in a ball. The results of this paper are generalizations of those proved by authors cited in the references.

Solvability of a singular boundary value problem in a domain of the Lobachevskii plane

Under study is some singular elliptic boundary value problem in a domain of the Lobachevskii plane with an isolated boundary point. We introduce the new function spaces that coincide with the spaces of Sobolev-Nikol’skii-Besov type outside the singular point, and the concept of σ-trace at the singular point. The main result is a proof that the singular boundary value problem is uniquely solvable.

Singular elliptic and parabolic problems and a class of free boundary problems

AbstractIn this short contribution we report on some recently made progress in the analytical treatment of a class of Free Boundary Problems (FBPs) characterized by the initial onset of a phase. The analysis is motivated by a model problem arising in the description of case II diffusion in polymers. The main local well‐posedness and regularity theorem is obtained by means of fundamental results concerning the underlying singular elliptic and parabolic boundary value problems. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Boundary behavior of solutions to some singular elliptic boundary value problems

Let Ω be a bounded domain with smooth boundary in R N ( N ≥ 1 ) . For the more general weight b , some nonlinearities f and singularities g , by two kinds of nonlinear transformations, a new perturbation method, which was introduced by García Melián in [J. García Melián, Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal. 67 (2007) 818–826], and comparison principles, we show that the boundary behavior of solutions to a boundary blow-up elliptic problem Δ w = b ( x ) f ( w ) , w > 0 , x ∈ Ω , w | ∂ Ω = ∞ and a singular Dirichlet problem − Δ u = b ( x ) g ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = 0 has the same form under the nonlinear transformations, which can be determined in terms of the inverses of the transformations.

Critical points and positive solutions of singular elliptic boundary value problems

Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems.

Critical singular problems on infinite cones

We prove existence results for non autonomous perturbations of critical singular elliptic boundary value problems. The non singular case was treated by Tarantello (Ann. Inst. H. Poincaré, Analyse Non-linéaire 9 (1992) 281) for bounded domains; here the singular weight allows for unbounded domains as cones and give rise to a different non compactness picture (as was first remarked by Caldiroli and Musina (Calc. Variations PDE 8 (1999) 365)).

Positive solution of a singular non-linear elliptic boundary value problem

The singular elliptic boundary value problem: Δu+g(x)u α+h(x)u −β=0 in Ω, u>0 in Ω, u=0 on ∂ Ω is considered in this paper, where α∈(0,1), β>0, Ω is a bounded region in R N with a smooth boundary. Under a set of suitable assumptions including that h( x)⩾0 in Ω and supp g −⊂⊂ supp h , it is proved that there exists a classic solution u, and when g( x)⩾0 (or g( x)⩽0) in whole Ω, then the solution u is unique.

Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem

The following singular elliptic boundary value problem is studied: Δu+λu −γ+u p=0 in Ω, u>0 in Ω, u=0 on ∂Ω, where Ω⊂R N (N⩾3) is a bounded domain with smooth boundary ∂Ω, 0<γ<1<p⩽ N+2 N−2 , and λ>0 is a real parameter. The existence, multiplicity and asymptotic behavior (as p→1) of solutions of this equation are discussed by combining variational and sub-supersolution methods.

A 2-D free boundary problem with onset of a phase and singular elliptic boundary value problems

Numerous models of industrial processes, such as diffusion in glassy polymers or solidification phenomena, lead to general one phase free boundary value problems with phase onset.The classical well-posedness of a fast diffusion approximation to the concerned free boundary value problems is proved. The analysis is performed via a singular change of variables leading to a singular system in a fixed domain. An existence and regularity theory for classical solutions is developed for the relevant underlying class of singular elliptic boundary value problems and is then used to prove the well-posedness for the models considered in which these are coupled to Hamilton-Jacobi or to parabolic evolution equations.

On a singular quasilinear elliptic boundary value problem in a ball

On a singular quasilinear elliptic boundary value problem in a ball

An existence theorem for a singular quasilinear elliptic boundary value problem with boundary blow-up

An existence theorem for a singular quasilinear elliptic boundary value problem with boundary blow-up