Degenerate Quasilinear Elliptic Equations Research Articles

Analysis of degenerate [formula omitted]-Laplacian elliptic equations involving Hardy terms: Existence and numbers of solutions

This article investigates the existence of solutions to quasilinear degenerate elliptic equation with Hardy singular coefficient, in which the weighted function ω(x) is unbounded (singular), then we cannot use the classical space W01,p(Ω), so we have to find another space W01,p(ω,Ω) to deal with the difficulties caused by singularities or degeneracies. New criteria for the existence of at least one and at least two generalized solutions are established via variational methods and critical point theorems provided that the nonlinearity satisfies appropriate hypotheses.

Matrix weights and regularity for degenerate elliptic equations

We prove local boundedness, Harnack inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form. Degeneracy is via a non negative, symmetric, measurable matrix-valued function Q(x) and two suitable non negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev–Poincaré inequalities. Data integrability is close to L1 and it is exploited in terms of suitable version of Stummel-Kato class that in some cases is also necessary to the regularity.

C1,α-regularity for quasilinear degenerate elliptic equation with a drift term on the Heisenberg group

The aim of the paper is to derive the C1,α-regularity of weak solutions to a quasilinear degenerate elliptic equation with a drift term on the Heisenberg group. Establishing a Caccioppoli inequality (i.e. energy estimate) for the horizontal gradient of weak solutions, we use it and the Sobolev embedding theorem on the Heisenberg group and the Moser iteration method to prove the local boundedness of the horizontal gradient of weak solutions. Then we prove an oscillation estimate for the horizontal gradient of weak solutions and combine the iteration Lemma to show that the horizontal gradient of weak solutions is Hölder continuous.

On a class of degenerate quasilinear elliptic equations with zero mass

Existence of radially symmetric positive solutions for a general class of quasi linear elliptic equations in the zero mass situation is proven. Our results include ground-state solutions for both power-type and exponential-type critical growths in the whole space and the class concerned includes, as particular cases, the Laplace, p-Laplace and k-Hessian operators in radial form.

Unique continuation for degenerate quasilinear equations and sum operators

We prove unique continuation property for positive solutions of some quasilinear degenerate elliptic equations.

A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles

<p style="text-indent:20px;">This paper concerns continuous subsonic-sonic potential flows in a two dimensional convergent nozzle, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. It is shown that for a given nozzle which is a perturbation of an straight one, and a given mass flux, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only <inline-formula><tex-math id="M1">\begin{document}$ C^{1/2} $\end{document}</tex-math></inline-formula> Hölder continuous and the acceleration blows up at the sonic state.</p>

Existence of multiple solutions for quasi-linear degenerate elliptic equations

The present paper is concerned with a class of quasi-linear elliptic degenerate equations. The degenerate operator arises from analysis of manifolds with singularities. The variational methods are applied here to verify the existence of infinitely many solutions for the problem.

A degenerate elliptic problem from subsonic-sonic flows in general nozzles

This paper deals with two-dimensional subsonic-sonic potential flows in general nozzles, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. For a large class of nozzles, it is shown that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the inlet and the sonic curve. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only Hölder continuous and the acceleration blows up at the sonic state.

Optimal C1,α estimates for a class of elliptic quasilinear equations

In this paper, we establish sharp [Formula: see text] estimates for weak solutions of singular and degenerate quasilinear elliptic equation [Formula: see text] which includes the standard [Formula: see text]-Laplacian equation with varying coefficients as a special case. The sharp exponent [Formula: see text] is asymptotically optimal and is determined by the Hölder regularity of the coefficients, the exponent [Formula: see text] and the [Formula: see text]-integrability of the source term [Formula: see text].

A Free Boundary Problem of a Degenerate Elliptic Equation and Subsonic-Sonic Flows with General Sonic Curves

This paper concerns a free boundary problem of a quasilinear degenerate elliptic equation, which arises from the studying of subsonic-sonic flows with both exceptional and nonexceptional sonic poin...

An existence and approximation theorem for solutions of degenerate quasilinear elliptic equations

The main result establishes that a weak solution of degenerate quasilinear elliptic equations can be approximated by a sequence of solutions for non-degenerate quasilinear elliptic equations.

Harnack’s Inequality and Applications of Quasilinear Degenerate Elliptic Equations with Rough and Singular Coefficients

We consider the quasilinear degenerate elliptic equation with rough and singular coefficients of the form $$\begin{aligned} div\left( {A(x,u,\nabla u)} \right) = B(x,u,\nabla u) \end{aligned}$$ in an open set $$\Omega \subset {\mathbb {R}^n},$$ where the quadratic form associated with the principle part of this equation allows vanishing and the coefficients in structural conditions on $$A(x,u,\nabla u)$$ and $$B(x,u,\nabla u)$$ require no smoothness but belong to some Stummel–Kato class. We prove a Fefferman–Phong type inequality related to the Stummel–Kato class and then an embedding inequality. Based on these inequalities, the local boundedness and Harnack’s inequality of the weak solutions are derived. As applications, the continuity and Holder continuity for the nonnegative weak solutions are given.

Large solutions and gradient bounds for quasilinear elliptic equations

ABSTRACTWe consider the quasilinear degenerate elliptic equation where Δp is the p-Laplace operator, p > 2, λ ≥0 and Ω is a smooth open bounded subset of ℝN (N ≥ 2). Under suitable structure conditions on the function H, we prove local and global gradient bounds for the solutions. We apply these estimates to study the solvability of the Dirichlet problem, and the existence, uniqueness and asymptotic behavior of maximal solutions blowing up at the boundary. The ergodic limit for those maximal solutions is also studied and the existence and uniqueness of a so-called additive eigenvalue is proved in this context.

On Sonic Curves of Smooth Subsonic-Sonic and Transonic Flows

This paper concerns properties of sonic curves for two-dimensional smooth subsonic-sonic and transonic steady potential flows, which are governed by quasi-linear degenerate elliptic equations and elliptic-hyperbolic mixed-type equations with degenerate free boundaries, respectively. It is shown that a sonic point satisfying the interior subsonic circle condition is exceptional if and only if the governing equation is characteristic degenerate at this point. For a $C^2$ subsonic-sonic flow whose sonic curve ${\mathcal S}$ is a nonisolated $C^2$ simple curve from one solid wall to another one, it is proved that ${\mathcal S}$ is a disjoint union of three connected parts (possibly empty): ${\mathcal S}_{e}$, ${\mathcal S}_{-}$, ${\mathcal S}_{+}$, where ${\mathcal S}_{e}$ is the set of exceptional points, which is empty or a point or a closed segment where the velocity potential equals identically to a constant; ${\mathcal S}_{-}$ and ${\mathcal S}_{+}$ denote the other two connected parts before and after ${\mathcal S}_{e}$ if ${\mathcal S}_{e}\not=\emptyset$, respectively. Moreover, if each interior point of ${\mathcal S}$ is not exceptional, then the velocity potential is strictly monotone along ${\mathcal S}$; otherwise, on ${\mathcal S}_{e}$ the velocity is along the normal direction of ${\mathcal S}$, and the velocity potential is strictly increasing (decreasing) along ${\mathcal S}_{-}$ and strictly decreasing (increasing) along ${\mathcal S}_{+}$ if on ${\mathcal S}_{e}$ the velocity is along the outer (inner) normal direction of ${\mathcal S}$ from the subsonic region. For a $C^2$ transonic flow, it is shown that there exist uniquely two distinct characteristics from each nonexceptional point (except the ones on the walls) pointing into the supersonic region, while two characteristics from an interior point of an exceptional segment coincide with this segment and never approach the supersonic region locally. Furthermore, it is proved that the curvature of the streamline in the physical plane near a nonexceptional point must be nonzero.

Existence of solutions for a class of porous medium type equations with lower order terms

This paper deals with a class of degenerate quasilinear elliptic equations of the form $-\operatorname{div}(a(x,u,\nabla u))+F(x,u,\nabla u)=f $ , where $a(x,u,\nabla u)$ is allowed to degenerate with the unknown u. Under some hypothesis on a, F, and f, we obtain the existence of bounded solutions $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ . For the case $f\in L^{1}(\Omega)$ , we also prove that there exists at least one renormalized solution.

Regularity of weak solutions for degenerate quasilinear elliptic equations involving operator curl

In this paper we consider the regularity of weak solutions of a quasilinear elliptic system involving operator curl. The paper is an extended version of the regularity theory for p-curl system which is an approximation for Bean's critical-state model for type II superconductors. It will be shown that the weak solution is of class C1+α which is optimal.

On Holder property of solutions of degenerate quasilinear elliptic equations

In the paper we consider a class of quasilinear second order elliptic equations of divergent structure with a degeneration. It one of the parts the equation is quasilinear in other one uniformly degenerates in small parameter e. It is shown that any solution is Holder continuous not depending on e.

Existence of Solutions for Dirichlet Problem of Some Degenerate Quasilinear Elliptic Equations

In this article we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate quasilinear elliptic equations in the setting of the Weighted Sobolev Spaces .

Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R N

We establish a result on the existence of a positive solution for the following class of degenerate quasilinear elliptic problems: $$(P)\quad \quad \left\{\begin{array}{ll}{-\Delta_{ap}u + V(x)|x|^{-ap^*} |u|^{p-2} u=K(x)f(x, u), {\rm in} \, R^N,}\\ {u > 0, {\rm in} \, R^N , \, u \in \mathcal{D}^{1,p}_a}{(R^N)},\end{array}\right. $$ denotes the Hardy-Sobolev’s \({{-\Delta_{ap}u = - div(|x|^{-ap}|\nabla u|^{p-2} \nabla u), 1 < p < N, -\infty < a < \frac{N-p}{p}, a \leq e \leq a+1, d=1+a-e}}\), and \({{p^* := p^*(a,e)=\frac{Np}{N-dp}}}\) denotes the Hardy-Sobolev’s critical exponent, V and K are bounded nonnegative continuous potentials, K vanishes at infinity, and f has a subcritical growth at infinity. The technique used here is the variational approach.

Radon measure-valued solutions for some quasilinear degenerate elliptic equations

We prove the existence of suitably defined weak Radon measure-valued solutions for a class of quasilinear elliptic equations. Moreover, we prove uniqueness results introducing the notion of “weak entropy solutions.”