Quasilinear Elliptic Differential Equations Research Articles

A strong maximum principle for quasilinear elliptic differential equations in a variable exponent Sobolev space

A strong maximum principle is an important property for elliptic and parabolic equations. In this paper, we consider a strong maximum principle for a class of quasilinear elliptic equations containing p(⋅)-Laplacian and mean curvature operator.

Quasilinear elliptic equations with sub-natural growth terms in bounded domains

We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type $$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta _{p, w} u = \sigma u^{q} &{} \text {in }\Omega ,\\ u = 0 &{} \text {on }\partial \Omega \end{array}\right. } \end{aligned}$$ in the sub-natural growth case $$0< q < p - 1$$ , where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^{n}$$ , $$\Delta _{p, w}$$ is a weighted p-Laplacian, and $$\sigma $$ is a nonnegative (locally finite) Radon measure on $$\Omega $$ . We give criteria for the existence problem. For the proof, we investigate various properties of p-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data.

Arithmetic three-spheres theorems for quasilinear Riccati type inequalities

We consider arithmetic three-spheres inequalities for solutions of certain second order quasilinear elliptic differential equations and inequalities with a Riccati-type drift term.

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.

On a class of quasilinear elliptic problems with critical exponential growth on the whole space

In this paper we prove a kind of weighted Trudinger-Moser inequality which is employed to establish sufficient conditions for the existence of solutions to a large class of quasilinear elliptic differential equations with critical exponential growth. The class of operators considered includes, as particular cases, the Laplace, $p$-Laplace and $k$-Hessian operators when acting on radially symmetric functions.

Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces

Critical point theory is used to show the existence of weak solutions to a quasilinear elliptic differential equation under the functional framework of the Musielak–Sobolev spaces in a bounded smooth domain with Dirichlet boundary condition.

S. Bernstein’s idea for bounding the gradient of solutions to the quasi-linear Dirichlet problem

S. Bernstein’s idea is outlined to estimate \({\| u_{x} \|_{0,0,\Omega}}\) of a solution u(x) to the quasi-linear elliptic differential equation, especially in the case of n > 2. Up to about 1956, investigations of nonlinear problems have been dealing with the case of n = 2. A large number of methods were developed, but none of them were applicable for n > 2. The maximum-minimum principle had been a powerful tool to find bounds, however in the case of n > 2 this tool is not available for the gradient of the solution. In 1956, Cordes [Math. Ann. 130 (1956), 278–312] gave estimates for the case n > 2. Later, Ladyzhenskaya and Ural’tseva [Linear and Quasilinear Elliptic Equations. Academic Press, New York, 1968] established estimates in the Sobolev space \({W^{2}_{2} (\Omega)}\) . In their line of reasoning they used the idea of S. Bernstein and investigated the transformed function υ(x) with u(x) = ϕ(υ(x)). In this paper, we give a more simpler proof of those results in the classical Banach space C2,α(Ω).

Solution of the Dirichlet problem for G-minimal graphs with a continuity and approximation method

We consider the Dirichlet problem for so-called G-minimal graphs in two dimensions. These are immersions of minimal surface type which can be presented as graphs over a planar domain. With the aid of a weight matrix G we derive a quasilinear elliptic and homogeneous differential equation for this height function. Then we solve the Dirichlet problem over convex domains Ω without differentiability assumptions and continuous boundary data with a constructive continuity and approximation method. We firstly establish an a priori C 1+α -estimate up to the boundary of the solution as we take the dense problem class of strictly convex C 2+α -domains and C 2+α -boundary data. By proving theorems on stability and compactness of graphs we solve this boundary value problem with a nonlinear continuity method. Then we introduce weighted conformal parameters in the graph and consider the parametric problem on the unit disc. Finally, we solve the original Dirichlet problem by using an approximation argument and the important parametric compactness theorem.

Removable sets for Hölder continuous solutions of quasilinear elliptic equations with lower order terms

We consider quasi-linear second order elliptic differential equations with lower order terms and study removable sets for Holder continuous solutions of the equation.

Constructing the grid orthogonal and having the given nodal clustering near the boundary of a two-dimensional domain

A problem of constructing a structured countable grid in a two-dimensional domain by mapping a parametric domain with the given quadratic grid onto this two-dimensional domain is considered. The Dirichlet problem is solved for a system of quasi-linear elliptic differential second-order equations to find mapping functions. An additional local mapping that gives the control metric is used to control coordinate lines of the grid. The additional mapping helps obtain the grid with the grid lines of one family orthogonal near the boundary of the domain and having the given clustering of the grid lines of another family towards the boundary of the domain. An example that constructs the grid around the airfoil is given.

On J. Schauder’s method to solve elliptic differential equations

The aim of this paper is to give complete proofs of the existence of classical \({C_{2,\alpha}\left(\bar{\Omega}\right)}\) solutions of linear and quasi-linear elliptic differential equations.

A priori estimates, existence and non-existence for quasilinear cooperative elliptic systems

Let be a real number and let , , be a connected smooth domain. Consider the system of quasi-linear elliptic differential equations where , , and are real functions. Relations between the Liouville non-existence and a priori estimates and existence on bounded domains are studied. Under appropriate conditions, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established.Bibliography: 11 titles.

Generalized G-convergence for quasilinear elliptic differential operators

Generalized G-convergence for a quasilinear elliptic differential equation is defined and studied. The equation describes heat conduction in the cores of large electric transformers. The coefficients of the equation depend on temperature and the corresponding differential operator is neither potential nor monotone. A theory which generalizes the classical G-convergence is proposed. The theory is applied to the homogenization of the quasilinear elliptic differential equation with periodic coefficients.

Quasi-linear elliptic differential equations for mappings of manifolds, II

We study questions related to the orientability of the infinite-dimensional moduli spaces formed by solutions of elliptic equations for mappings of manifolds. The principal result states that the first Stiefel–Whitney class of such a moduli space is given by the ℤ2-spectral flow of the families of linearised operators. Under an additional compactness hypotheses, we develop elements of Morse–Bott theory and express the algebraic number of solutions of a non-homogeneous equation with a generic right-hand side in terms of the Euler characteristic of the space of solutions corresponding to the homogeneous equation. The applications of this include estimates for the number of homotopic maps with prescribed tension field and for the number of the perturbed pseudoholomorphic tori, sharpening some known results.

Oscillation Theorems for Quasilinear Elliptic Differential Equations

By using vector Riccati transformation and averaging technique, some oscillation criteria for the quasilinear elliptic di.erential equation of second order, $$ \sum\limits_{i,j = 1}^N {D_i \left[ {\Psi \left( y \right)A_{ij} \left( x \right)\left| {Dy} \right|^{p - 2} D_j y} \right] + p\left( x \right)f\left( y \right)} = 0, $$ are obtained. These theorems extend and include earlier results for the semilinear elliptic equation and PDE with p-Laplacian.

Blow-up estimates of positive radial solutions of a class first-order evolution systems

We derive the nonexistence of radially positive solutions for a system of quasilinear elliptic differential equations, and establish blow-up estimates for a class of first-order evolution systems (in time). The main results are new and extend previous known results.

Positive Solutions of Quasilinear Elliptic Equations

This paper is concerned with existence theorems for positive solutions of the Dirichlet problem for quasilinear elliptic differential equation containing a gradient term. Using the shooting method and the a priori estimates for the first zero, we obtain sufficient conditions for the existence of classical positive solutions of the problem in the ball.

Nonexistence of solutions to Hele-Shaw equations

We study nonexistence of the solutions to quasilinear elliptic differential equations arising from nonisothermal, non-Newtonian Hele-Shaw flows. The proof is based on the trial function method developed by Pohozaev without recourse to comparison theorems and to the maximum principle.

Quasilinear elliptic differential equations for mappings between manifolds, I

Quasilinear elliptic differential equations for mappings between manifolds, I

Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a non-Newtonian filtration system

The prior estimate and decay property of positive solutions are derived for a system of quasilinear elliptic differential equations first. Then the result of nonexistence for a differential equation system of radially nonincreasing positive solutions is implied. By using this nonexistence result, blow-up estimates for a class of quasilinear reaction-diffusion systems (non-Newtonian filtration systems) are established to extend the result of semilinear reaction-diffusion (Fujita type) systems.