Results for degenerate nonlinear elliptic equations involving a Hardy potential

Consider the problem

     

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.     

On positive solution for a class of degenerate quasilinear elliptic positone/semipositone systems

This paper deals with the existence and nonexistence of positive weak solutions of degenerate quasilinear elliptic systems with subcritical and critical exponents. The nonlinearities involved have semipositone and positone structures and the existence results are obtained by applying the lower and upper-solution method and variational techniques.     

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L¹-setting.     

Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

In this paper, we study the Dirichlet problem for a class of infinitely degenerate nonlinear elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy's inequality, the existence and regularity of multiple nontrivial solutions have been proved.     

Regularity results for degenerate elliptic systems

We prove regularity results for certain degenerate quasilinear elliptic systems with coefficients which depend on two different weights. By using Sobolev- and Poincaré inequalities due to Chanillo and Wheeden [S. Chanillo, R.L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985) 1191–1226; S. Chanillo, R.L. Wheeden, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986) 1111–1134] we derive a new weak Harnack inequality and adapt an idea due to L. Caffarelli [L.A. Caffarelli, Regularity theorems for weak solutions of some nonlinear systems, Comm. Pure Appl. Math. 35 (1982) 833–838] to prove a priori estimates for bounded weak solutions. For example we show that every bounded weak solution of the system −Dα(aαβ(x,u,∇u)Dβui)=0$-D_{\alpha }(a^{\alpha \beta }(x,u,\mathrm{\nabla }u)D_{\beta }{u}^i)=0$ with |x|²|ξ|²?aαβξαξβ?τ|x||ξ|²${|x|}^2{|\xi |}^2?a^{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }?{|x|}^{\tau }{|\xi |}^2$, |x|<1$|x|<1$, τ∈(1,2)$\tau \in (1,2)$ is Hölder continuous. Furthermore we derive a Liouville theorem for entire solutions of the above systems.     

On a class of critical singular quasilinear elliptic problem with indefinite weights

In this paper, the eigenvalue problem for a class of quasilinear elliptic equations involving critical potential and indefinite weights is investigated. We obtain the simplicity, strict monotonicity and isolation of the first eigenvalue λ₁. Furthermore, because of the isolation of λ₁, we prove the existence of the second eigenvalue λ₂. Then, using the Trudinger-Moser inequality, we obtain the existence of a nontrivial weak solution for a class of quasilinear elliptic equations involving critical singularity and indefinite weights in the case of 0<λ<λ₁ by the Mountain Pass Lemma, and in the case of λ₁≤λ<λ₂ by the Linking Argument Theorem.     

On a new class of elliptic systems with nonlinearities of arbitrary growth

We guarantee the existence of infinitely many different pairs of solutions to the system

     

Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations.     

Regularity for a class of strongly degenerate quasilinear operators

We prove a Harnack inequality and regularity for solutions of a quasilinear strongly degenerate elliptic equation. We assume the coefficients of the structure conditions to belong to suitable Stummel–Kato classes.     

Deterministic homogenization of non-linear non-monotone degenerate elliptic operators

Deterministic homogenization has been till now applied to the study of monotone operators, the determination of the limiting problem being systematically based on the monotonicity of the operator under consideration. Here we mean to show that deterministic homogenization also tackle non-monotone operators. More precisely, under an abstract general hypothesis, we study the homogenization of non-linear non-monotone degenerate elliptic operators. We obtain some general homogenization result, which result is applied to the resolution of several concrete homogenization problems such as the periodic homogenization and the almost periodic homogenization problems. Our main tool is the theory of homogenization structures.     

The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator.     

Nonlinear degenerate elliptic equations and axially symmetric problems

We consider semilinear elliptic equations with a principal part degenerating on a boundary hyperplane. Weak existence, uniqueness and regularity of solutions are established by variational methods and by reduction to uniformly elliptic equations. An important application arises in the mathematical treatment of the rotating star problem in general relativity, where the axial symmetry admits the reduction of one of the Einstein equations to a problem of the above form on a meridian half plane. Received February 12, 1997 / Accepted May 15, 1997     

Partially overdetermined elliptic boundary value problems

We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains.     

On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux

Under some non-degeneracy condition we show that sequences of entropy solutions of a semi-linear elliptic equation are strongly pre-compact in the general case of a Carathéodory flux vector. The proofs are based on localization principles for H-measures corresponding to sequences of measure-valued functions.     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.