Elliptic Problems Research Articles

Existence of Solutions for p(x)-Laplacian Elliptic BVPs on a Variable Sobolev Space Via Fixed Point Theorems

In this paper, we prove some existence theorems for elliptic boundary value problems within the p(x)-Laplacian on a variable Sobolev space. For this purpose, the main problem is transformed into a fixed point problem and then fixed point arguments such as Schaefer’s and Schauder’s theorems are used. Our approach involves fewer stringent assumptions on the nonlinearity function than the prior findings. An interesting example is presented to examine the validity of the theoretical findings.

Singular elliptic measure data problems with irregular obstacles

We investigate elliptic irregular obstacle problems with p-growth involving measure data. Emphasis is on the strongly singular case 1<p≤2−1/n, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates.

Efficient attribute-based strong designated verifier signature scheme based on elliptic curve cryptography.

In an attribute-based strong designated verifier signature, a signer who satisfies the access structure signs the message and assigns it to a verifier who satisfies the access structure to verify it, which enables fine-grained access control for signers and verifiers. Such signatures are used in scenarios where the identity of the signer needs to be protected, or where the public verifiability of the signature is avoided and only the designated recipient can verify the validity of the signature. To address the problem that the overall overhead of the traditional attribute-based strong designated verifier signature scheme is relatively large, an efficient attribute-based strong designated verifier signature scheme based on elliptic curve cryptography is proposed, as well as a security analysis of the new scheme given in the standard model under the difficulty of the elliptic curve discrete logarithm problem (ECDLP). On the one hand, the proposed scheme is based on elliptic curve cryptography and uses scalar multiplication on elliptic curves, which is computationally lighter, instead of bilinear pairing, which has a higher computational overhead in traditional attribute-based signature schemes. This reduces the computational overhead of signing and verification in the system, improves the efficiency of the system, and makes the scheme more suitable for resource-constrained cloud end-user scenarios. On the other hand, the proposed scheme uses LSSS (Linear Secret Sharing Schemes) access structure with stronger access policy expression, which is more efficient than the "And" gate or access tree access structure, making the computational efficiency of the proposed scheme meet the needs of resource-constrained cloud end-users.

Semilinear elliptic problems via the nonlinear Rayleigh quotient with two nonlocal nonlinearities

It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: { − Δ u + V ( x ) u = μ ( I α 1 ∗ | u | q ) | u | q − 2 u − λ ( I α 2 ∗ | u | p ) | u | p − 2 u in R N , u ∈ H 1 ( R N ) , where p > q , λ , μ > 0 , α 1 ≤ α 2 ; α 1 , α 2 ∈ ( 0 , N ) , N ≥ 3 ; p ∈ ( 2 α 2 , 2 α 2 ∗ ) ; q ∈ ( 2 α 1 , 2 α 1 ∗ ) , 2 α j = ( N + α j ) / N and 2 α j ∗ = ( N + α j ) / ( N − 2 ) , j = 1 , 2 . Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp μ n > 0 and λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each μ > μ n where λ ∈ ( 0 , min ( λ ∗ , λ ∗ ) ) . The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each μ < μ n and λ > 0 .

On nonminimizing solutions of elliptic free boundary problems

On nonminimizing solutions of elliptic free boundary problems

Global existence and blowup of solutions for nonlinear Love equation

In this paper, we are concerned with the nonlinear Love equation, which describes some sort of “propagation process” in physics. Firstly, we establish the local existence and uniqueness of the solution by using the contraction mapping principle and the regularity theory for the elliptic problem. Secondly, we construct a family of potential wells to obtain a threshold for the existence of global solutions and blowup in a finite time of the solution in both cases with subcritical and critical initial energy. Additionally, we provide the decay rate of the global solution and estimate the lifespan of a blow‐up solution. Moreover, at the supercritical initial energy level, we provide a sufficient condition for initial data leading to blow‐up results. This paper, along with the previous work of A. N. Carvalho, J. Cholewa, Local well‐posedness, asymptotic behavior, and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc. 361 (5) (2009) 2567‐2586, provides us with a comprehensive and systematic study of the dynamic behavior of the solution to the nonlinear Love equation.

Existence, sign and asymptotic behaviour for a class of integro-differential elliptic type problems

Existence, sign and asymptotic behaviour for a class of integro-differential elliptic type problems

Marcinkiewicz Estimates for Solutions of Some Elliptic Problems with Singular Data

Marcinkiewicz Estimates for Solutions of Some Elliptic Problems with Singular Data

Regularizing effect of absorbtion term in singular and degenerate elliptic problems

Dans cet article, nous étudions l’existence et la régularité des solutions au problème singulier suivant\\\begin{equation}\left\{\begin{array}{lll}-\displaystyle\mbox{div} \big(a(x,u)\vert\nabla u\vert^{p-2}\nabla u\big) + \vert u\vert^{s-1}u =h(u)f &amp;\mbox{ in } \Omega \\&amp;u\geq 0 &amp;\mbox{ in }\Omega \\&amp;u=0 &amp;\mbox{ on } \delta\Omega\\\end{array}\right.\end{equation} prouvant que le terme d’ordre inférieur $u\vert u\vert^{s-1}$ a des effets régularisants sur les solutions dans le cas d’un opérateur elliptique à coercivité dégénérée.

Nonlinear elliptic problem in exterior domains: exact boundary behavior of blow-up positive solutions

Nonlinear elliptic problem in exterior domains: exact boundary behavior of blow-up positive solutions

Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method’s key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete H1 and L2 norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

A pressure-projection pre-conditioning multi-fractional-step method for Navier–Stokes Flow in Porous Media

A new pressure-projection pre-conditioning multi-fractional-step (PPP-MFS) method for incompressible Navier–Stokes flow in porous media is presented. This fractional step method is applied to decouple the pressure and the velocity; thereby, overcoming the computational costs and difficulty associated with the resolution of the nonlinear term in the Navier–Stokes equation for fine geometric models. Specifically, time evolution is decomposed into a sequence of multi-fractional solution steps. In the first step, an elliptic problem is solved for pressure (p) with a no-slip boundary condition. This gives the Stokes pressure and velocity fields. In the second step, the obtained pressure p is then projected onto the field p∗ and used to solve for velocity field (u) required for the pre-conditioning of the solution to the Navier–Stokes equation. The pressure and velocity fields are obtained from the solution of the Navier–Stokes equation in the third step. Numerical and geometric discretisation of porous samples were carried out using finite-element method. For flow in simple channel models represented by two- and three-dimensions and in systems with high conductivity, the Stokes and Navier–Stokes numerical solutions produced close pressure and velocity field approximations. For flow around a cylinder, computation time is consistently higher in the Navier–Stokes equation by a factor of 2 with a pronounced non-symmetric pressure field at high mesh refinements. This computation time is desirable given that Navier–Stokes computation without preconditioning can be orders of magnitude more expensive.

Analysis of solution to an elliptic free boundary value problem equipped with a ‘bad’ data

We will study a free boundary value problem driven by a source term which is quite irregular. In the process, we will establish a monotonicity result, and the regularity of solutions to the free boundary value problem under consideration.

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

In this article we study the fundamental solutions or “α-harmonic functions” for some nonlinear positive homogeneous nonlocal elliptic problems in conical domains, such asF(u)=0inCω,u=0inRN∖Cω, where ω is a proper C2 domain in SN−1 for N≥2, Cω:={x:x≠0,|x|−1x∈ω} is the cone-like domain related to ω, and F is an extremal fully nonlinear integral operator. We prove the existence of two fundamental solutions that are homogeneous and do not change signs in the cone; one is bounded at the origin and the other at infinity.As an application, we use the fundamental solutions obtained to prove Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler equation in the formF(u)+up=0inCω,u=0inRN∖Cω. We also prove a generalized Hopf type lemma in domains with corners. Most of our results are new even when F is the fractional Laplacian operator.

Desingularization of 2D elliptic free-boundary problem with non-autonomous nonlinearity

In this paper, we consider the existence and limiting behaviour of solutions to a semilinear elliptic equation arising from confined plasma problem in dimension two \[ \begin{cases} -\Delta u=\lambda k(x)f(u) &amp; \text{in}\ D,\\ u= c &amp; \displaystyle\text{on}\ \partial D,\\ \displaystyle - \int_{\partial D} \frac{\partial u}{\partial \nu}\,{\rm d}s=I, \end{cases} \] where $D\subseteq \mathbb {R}^2$ is a smooth bounded domain, $\nu$ is the outward unit normal to the boundary $\partial D$ , $\lambda$ and $I$ are given constants and $c$ is an unknown constant. Under some assumptions on $f$ and $k$ , we prove that there exists a family of solutions concentrating near strict local minimum points of $\Gamma (x)=({1}/{2})h(x,\,x)- ({1}/{8\pi })\ln k(x)$ as $\lambda \to +\infty$ . Here $h(x,\,x)$ is the Robin function of $-\Delta$ in $D$ . The prescribed functions $f$ and $k$ can be very general. The result is proved by regarding $k$ as a $measure$ and using the vorticity method, that is, solving a maximization problem for vorticity and analysing the asymptotic behaviour of maximizers. Existence of solutions concentrating near several points is also obtained.

High Energy Solutions for p-Kirchhoff Elliptic Problems with Hardy–Littlewood–Sobolev Nonlinearity

This article deals with the study of the following Kirchhoff–Choquard problem: M∫RN|∇u|p(-Δp)u+V(x)|u|p-2u=∫RNF(u)(y)|x-y|μdyf(u),inRN,u>0,inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\begin{array}{cc} \\displaystyle M\\left( \\, \\int \\limits _{{\\mathbb {R}}^N}|\ abla u|^p\\right) (-\\Delta _p) u + V(x)|u|^{p-2}u = \\left( \\, \\int \\limits _{{\\mathbb {R}}^N}\\frac{F(u)(y)}{|x-y|^{\\mu }}\\,dy \\right) f(u), \\;\\;\ ext {in} \\; {\\mathbb {R}}^N,\\\\ u > 0, \\;\\; \ ext {in} \\; {\\mathbb {R}}^N, \\end{array} \\end{aligned}$$\\end{document}where M models Kirchhoff-type nonlinear term of the form M(t)=a+btθ-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M(t) = a + bt^{\ heta -1}$$\\end{document}, where a,b>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a, b > 0$$\\end{document} are given constants; 1<p<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<N$$\\end{document}, Δp=div(|∇u|p-2∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _p = \ ext {div}(|\ abla u|^{p-2}\ abla u)$$\\end{document} is the p-Laplacian operator; potential V∈C2(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V \\in C^2({\\mathbb {R}}^N)$$\\end{document}; f is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for θ∈1,2N-μN-p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in \\left[ 1, \\frac{2N-\\mu }{N-p}\\right) $$\\end{document} via the Pohožaev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.

Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems

Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems

Hexagonal finite differences for the two-dimensional variable coefficient Poisson equation

For many years, finite differences in hexagonal grids have been developed to solve elliptic problems such as the Poisson and Helmholtz equations. However, these schemes are limited to constant coefficients, which reduces their usefulness in many applications. The main challenge is accurately approximating the diffusive term. This paper presents examples of both successful and unsuccessful attempts to obtain accurate finite differences based on a hexagonal stencil with equilateral triangles to approximate two-dimensional Poisson equations. Local truncation error analysis reveals that a second-order scheme can be achieved if the derivative of the diffusive coefficient is included. Finally, we provide numerical examples to verify the accuracy of the proposed methods.

On some anisotropic unilateral elliptic problems with measure data

Here, we are concerned with the existence results for some nonlinear  unilateral elliptic problem in generalized  Sobolev space.