Quasilinear Elliptic Equations Research Articles

On positive solutions for a class of quasilinear elliptic equations

This paper studies the existence, nonexistence and uniqueness of positive solutions for a class of quasilinear equations. We also analyze the behavior of these solutions with respect to two parameters $$\kappa $$ and $$\lambda $$ that appear in the equation. The proof of our main results relies on bifurcation techniques, the sub- and supersolution method and a construction of an appropriate large solution.

Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and updating basis functions in selected iterations.

Homogenization of some quasi-linear elliptic equations with gradient constraints

We prove a homogenization formula for quasi-linear elliptic equations with gradient constraints on a disperse set, within the framework of monotonic operator theory and compensated compactness methods.

Infinitely many solutions for critical quasilinear equations involving Hardy potential

In this paper, we consider the following quasilinear elliptic equation with critical growth and a Hardy term: [Formula: see text]where [Formula: see text], [Formula: see text] is a constant, [Formula: see text][Formula: see text] is the Sobolev critical exponent. And [Formula: see text] is an open bounded domain which contains the origin. We will study the existence of infinitely many solutions for (P). To achieve this goal, we first perform various kinds of change of variables to overcome the difficulties caused by the unboundedness of [Formula: see text] ([Formula: see text] for large [Formula: see text]) and the lack of a global monotone condition [Formula: see text] (see below) on [Formula: see text], then combining the idea of regularization approach and subcritical approximation we prove the existence of infinitely many solutions for (P). Our results show that under some suitable assumptions on [Formula: see text], without the perturbation of the lower term [Formula: see text] we can still obtain the existence of infinitely many solutions for (P).

Quasilinear elliptic equations with natural growth and quasilinear elliptic equations with singular drift

We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of “natural constraint”.

Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence

The existence of solutions of opposite constant sign is proved for a Dirichlet problem driven by the weighted (p,q)-Laplacian with q<p and exhibiting a (q−1)-order term as well as a convection term. The approach is based on the method of sub–supersolution. Extremal solutions in relevant ordered intervals are obtained as well.

A Nontrivial Solution of a Quasilinear Elliptic Equation Via Dual Approach

In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in ℝN which includes the so-called modified nonlinear Schrodinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.

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].

Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth

We investigate the problem \left\{ \begin{array}{ll} -\Delta_p u = g(u)|\nabla u|^p + f(x,u) &amp; \text{in } \Omega, \\ u&gt;0 &amp;\text{in } \Omega, \\ u = 0 &amp;\text{on } \partial\Omega, \end{array} \right. in a bounded smooth domain \Omega \subset \mathbb{R}^N . Using a Kazdan–Kramer change of variable we reduce this problem to a quasilinear one without gradient term and therefore approachable by variational methods. In this way we come to some new and interesting problems for quasilinear elliptic equations which are motivated by the need to solve (P) . Among other results, we investigate the validity of the Ambrosetti–Rabinowitz condition according to the behavior of g and f . Existence and multiplicity results for (P) are established in several situations.

Operator compact exponential approximation for the solution of the system of 2D second order quasilinear elliptic partial differential equations

In this paper, we suggest a new exponential implicit method based on full step discretization of order four for the solution of quasilinear elliptic partial differential equation of the form A ( x,y,z ) z_{xx} +C ( x,y,z ) z_{yy} =k ( x,y,z, z_{x}, z_{y} ), 0< x,y<1. In this method a single compact cell consisting of nine nodal points is used. Convergence analysis of the said method is discussed in detail. The developed method is successfully applied to solving problems in polar coordinates. The method for scalar equation is eventually applied to solving the system of quasilinear elliptic equations. To measure the rationality and precision, the method is applied to solving several noteworthy problems and numerical results are provided to exhibit the effectiveness of the method.

Solutions for critical quasilinear elliptic equations in RN

In this paper, we consider the following critical quasilinear problems in RN: −∑i,j=1NDj(bij(v)Div)+12∑i,j=1Nbij′(v)DivDjv+a(x)v=μ|v|qs−2v+|v|2*s−2v,inRN,v(x)→0 as|x| →∞, where N ≥ 4, bij(v) ∼ 1 + |v|2s−2, s ≥ 1, μ &amp;gt; 0 is a constant, q &amp;lt; 2*, 2*=2NN−2 is the Sobolev critical exponent for the embedding of H1(RN) into Lp(RN), and a(x) is a potential function which is finite and positive and satisfies some decay assumptions. We first use the truncation to overcome the difficulties caused by both the unbounded domain and the critical exponents. Then we perform a change of variables to overcome the difficulty caused by the unboundness of bij. Finally a regularization approach combined with a compact argument helps us to prove the existence of infinitely many solutions.

A p(x)-Laplacian extension of the Díaz-Saa inequality and some applications

AbstractThe main result of this work is a new extension of the well-known inequality by Díaz and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian, $\Delta _{p(x)}u\equiv {\rm div}( \vert \nabla u \vert ^{p(x)-2}\nabla u)$. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.

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.

Adams’ type inequality and application to a quasilinear nonhomogeneous equation with singular and vanishing radial potentials in $$ \mathbb {R}^4 $$ R 4

In this paper, we establish some Adams’ type inequality for weighted second-order Sobolev spaces in four dimensions. The weights are radial and can have a singular or decaying behavior. This inequality is used to study some nonhomogeneous quasilinear elliptic equation.

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...

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

Regularity and weak comparison principles for double phase quasilinear elliptic equations

We consider the Euler equation of functionals involving a term of the form \begin{document}$ \int_{\Omega}(| \nabla u|^p+a(x)| \nabla u|^q) \,{ {\rm{d}}} x, $\end{document} with \begin{document}$ 1 and \begin{document}$ a(x)\geq 0 $\end{document} . We prove weak comparison principle and summability results for the second derivatives of solutions.

Hardy-Sobolev type inequality and supercritical extremal problem

This paper deals with Hardy-Sobolev type inequalities involving variable exponents. Our approach also enables us to prove existence results for a wide class of quasilinear elliptic equations with supercritical power-type nonlinearity with variable exponent.

Existence of bounded weak solutions of the Robin problem for a quasi-linear elliptic equation with p(x)-Laplacian

Existence of bounded weak solutions of the Robin problem for a quasi-linear elliptic equation with p(x)-Laplacian

СЕТОЧНЫЙ МЕТОД РЕШЕНИЯ КВАЗИЛИНЕЙНЫХ ЭЛЛИПТИЧЕСКИХ УРАВНЕНИЙ ЧЕТВЕРТОГО ПОРЯДКА

СЕТОЧНЫЙ МЕТОД РЕШЕНИЯ КВАЗИЛИНЕЙНЫХ ЭЛЛИПТИЧЕСКИХ УРАВНЕНИЙ ЧЕТВЕРТОГО ПОРЯДКА