Solutions Of Quasilinear Elliptic Equations Research Articles

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

We study regularity for solutions of quasilinear elliptic equations of the form \mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F} in bounded domains in \mathbb{R}^n . The vector field \mathbf{A} is assumed to be continuous in u , and its growth in \nabla u is like that of the p -Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions u to the equation under a small BMO condition in x for \mathbf{A} . As a consequence, we obtain that \nabla u is in the classical Morrey space \mathcal{M}^{q,\lambda} or weighted space L^q_w whenever |\mathbf{F}|^{1/(p-1)} is respectively in \mathcal{M}^{q,\lambda} or L^q_w , where q is any number greater than p and w is any weight in the Muckenhoupt class A_{q/p} . In addition, our two-weight estimate allows the possibility to acquire the regularity for \nabla u in a weighted Morrey space that is different from the functional space that the data |\mathbf{F}|^{1/(p-1)} belongs to.

Multiplicity of Radially Symmetric Small Energy Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the L ∞ -bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization method. Employing this, we provide the existence of multiple small energy radially symmetric solutions whose L ∞ -norms converge to zero. We utilize the modified functional method and the dual fountain theorem as the main tools.

Regularity of solutions of quasi-linear elliptic equations with [formula omitted] coefficients

Let D be an bounded region in Rn. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being Δnu=Vun−1. The coefficients are assumed to be in the space Llogm⁡L(D) for m>n−1. Using a Moser iteration argument coupled with the Moser-Trudinger inequality, a local L∞ bound on the solution u is proven. A Harnack-type inequality is then proven. These results are shown to be sharp with respect to m. Then essential continuity of u is proven, and away from the boundary a bound on the modulus of continuity.

Positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term

We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension N involving the N-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term. In such case, variational methods cannot be applied. Our approach is based on approximation scheme, where we consider a new class of normed spaces of finite dimension. As a particular case, we extended the result achieved by De Araujo and Montenegro [2016] for any N>2.

Harnack Inequality for Quasilinear Elliptic Equations in Generalized Orlicz-Sobolev Spaces

In this paper we prove, by a new method, the Harnack inequality for positive solutions of quasilinear elliptic equations in the generalized Orlicz-Sobolev space setting. Our approach is based on the usage of the Φ-functions associated to generalized Φ-functions and the Moser’s iteration technique. As a consequence, we obtain the Holder continuity of bounded solutions of such equations.

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

AbstractIn this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

含Φ-Laplace算子的拟线性椭圆型方程的无穷多解

含Φ-Laplace算子的拟线性椭圆型方程的无穷多解

A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.

The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

We prove the existence of extremal solutions of the following quasilinear elliptic problem -∑i=1N∂/∂xiai(x,u(x),Du(x))+g(x,u(x),Du(x))=0 under Dirichlet boundary condition in Orlicz-Sobolev spaces W01LM(Ω) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term g:Ω×R×RN→R is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.

Two positive solutions for quasilinear elliptic equations with singularity and critical exponents

In this paper, we consider the quasilinear elliptic equation with singularity and critical exponents \t\t\t{−Δpu−μ|u|p−2u|x|p=Q(x)|u|p∗(t)−2u|x|t+λu−s,in Ω,u>0,in Ω,u=0,on ∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta_{p}u-\\mu \\frac{ \\vert u \\vert ^{p-2}u}{ \\vert x \\vert ^{p}}=Q(x) \\frac{ \\vert u \\vert ^{p^{*}(t)-2}u}{ \\vert x \\vert ^{t}}+\\lambda u^{-s}, &\\text{in }\\Omega , \\\\ u>0, & \\text{in }\\Omega , \\\\ u=0, &\\text{on }\\partial \\Omega , \\end{cases} $$\\end{document} where Delta_{p}= operatorname {div}(|nabla u|^{p-2}nabla u) is a p-Laplace operator with 1< p< N. p^{*}(t):=frac{p(N-t)}{N-p} is a critical Sobolev–Hardy exponent. We deal with the existence of multiple solutions for the above problem by means of variational and perturbation methods.

Infinitely many solutions for quasilinear elliptic equations with lack of symmetry

In this paper we look for weak solutions of the quasilinear elliptic model problem −div(A(x,u)∇u)+12At(x,u)|∇u|2=g(x,u)+h(x)in Ω,u=0on ∂Ω,where Ω⊂RN is a bounded domain, N≥2, the real terms A(x,t), At(x,t)=∂A∂t(x,t) and g(x,t) are Carathéodory functions on Ω×R and h:Ω→R is a given measurable map.We prove that, even if At(x,t)≢0, under suitable assumptions infinitely many solutions exist in spite of the lack of symmetry. A suitable supercritical growth is allowed for the nonlinear term g(x,t).We use a variant of the variational perturbation techniques introduced by Rabinowitz in Rabinowitz (1982) but by means of a weak version of the Cerami–Palais–Smale condition.

Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions

This paper studies regularity estimates in Lebesgue spaces for gradients of weak solutions of a class of general quasilinear equations of p-Laplacian type in bounded domains with inhomogeneous conormal boundary conditions. In the considered class of equations the leading terms are vector-valued functions that are measurable in the x-variable and that depend in a nonlinear way on the solution and on its gradient. This class of equations consists of the well-known class of degenerate p-Laplace equations for $$p >1$$ . Under some sufficient conditions, we establish local interior, local boundary, and global $$W^{1,q}$$ -regularity estimates for weak solutions with $$q>p$$ , assuming that the weak solutions are in the John–Nirenberg BMO space. The paper therefore improves available results because it removes the boundedness or continuity assumptions on solutions. Our results also unify and cover known results for equations in which the principals are only allowed to depend on x-variable and gradient of solution variable. More than that, this paper gives a method to treat non-homogeneous boundary value problems directly without using any form of translations that is sometimes complicated due to the nonlinearities.

Location of Nodal solutions for quasilinear elliptic equations with gradient dependence

Existence and regularity results for quasilinear elliptic equations driven by \begin{document}$(p, q)$\end{document} -Laplacian and with gradient dependence are presented. A location principle for nodal (i.e., sign-changing) solutions is obtained by means of constant-sign solutions whose existence is also derived. Criteria for the existence of extremal solutions are finally established.

Hölder Continuity of Solutions for Quasilinear Elliptic Equations with Gradient Terms and Measures

We consider quasi-linear second order elliptic differential equations with gradient terms and study Holder continuity of solutions of the equation. Also, as an application we investigate removable sets for Holder continuous solutions of a certain equation.

Removable sets for continuous solutions of quasilinear elliptic equations with nonlinear source or absorption terms

This paper establishes removable singularity theorems for nonnegative continuous solutions of quasilinear elliptic equations with nonlinear source or absorption terms when an exceptional set is conditioned in terms of the regularity of Hausdorff measure and a uniform Minkowski property. These weaker conditions enable us to consider some fractal sets as an exceptional set.

Ground solutions for critical quasilinear elliptic equations via Pohožaev manifold method

The paper deals with the following problem:where is a parameter, and , . Under some certain assumptions on V(x), we establish the existence of ground-state solutions for quasi-linear elliptic equations with Sobolev exponent via the method of Pohožaev manifold, the monotonic trick and global compactness lemma. Some recent results are extended.

Existence of three solutions for quasilinear elliptic equations: an Orlicz-Sobolev space setting

In this paper, we establish the existence of three weak solutions for quasilinear elliptic equations in an Orlicz-Sobolev space via an abstract result recently obtained by Ricceri in [13].

Location of solutions for quasi-linear elliptic equations with general gradient dependence

Location of solutions for quasi-linear elliptic equations with general gradient dependence

Infinitely many solutions for quasilinear elliptic equations involving [formula omitted]-Laplacian in [formula omitted

In this paper, under some superquadratic conditions made on the nonlinearity f, we use variational approaches to establish the existence of infinitely many solutions to quasilinear elliptic equations with (p,q)-Laplacian −Lpu−Lqu+a(x)|u|p−2u+b(x)|u|q−2u=f(x,u)inR, where 1<q<p, Lpu=(|u′|p−2u′)′+(|(u2)′|p−2(u2)′)′u,Lqu=(|u′|q−2u′)′+(|(u2)′|q−2(u2)′)′u.

Existence of solutions for quasilinear elliptic equations with Hardy potential

In this paper, we consider the following quasilinear elliptic equation with Hardy potential and Dirichlet boundary condition: −∑i,j=1NDj(aij(x,u)Diu)+12∑i,j=1NDsai,j(x,u)DiuDju−λ|x|−2u=f(x,u)inΩ, where Ω ⊂ ℝN(N ≥ 3) is a smooth bounded domain, Di=∂∂xi,Dsaij(x,s)=∂∂saij(x,s), and 0≤λ&amp;lt;λ∗:=(N−22)2, and λ|x|−2 is called the Hardy potential. By using the perturbation method, we prove the existence of infinitely many solutions for the above problem.