Class Of Quasilinear Elliptic Problems Research Articles

Existence and regularity for a general class of quasilinear elliptic problems involving the Hardy potential

In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and L1 data. In the same setting we study sharp (local and global) integral estimates for the second derivatives of the solutions.

Symmetry via the moving plane method for a class of quasilinear elliptic problems involving the Hardy potential

We consider positive solutions to a class of quasilinear elliptic problems involving the Hardy potential under zero Dirichlet boundary condition. Via moving plane method, proving a weak comparison principle, we prove symmetry and monotonicity properties for the solutions defined on strictly convex symmetric domains.

A class of quasilinear elliptic equations with singular and exponential terms

We use a scheme of approximation together with Brouwer's theorem to establish the existence of positive solutions for a class of quasilinear elliptic problems. More precisely, we study the class of equations: (1) { − Div ( a ( | ∇u | p ) | ∇u | p − 2 ∇u ) = η u − γ + λ u s + f ( u ) in Ω , u = 0 on ∂Ω , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary, 2 ≤ p < N , 0 < γ < 1 , 0<s<N−1, η and λ are positive parameters, a : R + → R + is a function of class C 1 ( R + ) . The nonlinear term f involves three different types of Trudinger–Moser growth: subcritical, critical or supercritical. We also study the asymptotic behavior of the solutions with respect to the parameters.

A Hybrid High-Order Method for Quasilinear Elliptic Problems of Nonmonotone Type

In this paper, we design and analyze a Hybrid-High Order (HHO) approximation for a class of quasilinear elliptic problems of nonmonotone type. The proposed method has several advantages, for instance, it supports arbitrary order of approximation and general polytopal meshes. The key ingredients involve local reconstruction and high-order stabilization terms. Existence and uniqueness of the discrete solution are shown by Brouwer's fixed point theorem and contraction result. A priori error estimate is shown in discrete energy norm that shows optimal order convergence rate. Numerical experiments are performed to substantiate the theoretical results.

On fractional p-Laplacian type equations with general nonlinearities

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: \begin{equation*} \left\{ \begin{array}{l} (-\Delta)_p^s u= f(x,u) \quad \hfill \textrm{in} \ \Omega,\\ \quad u=0 \ \hfill \textrm{in} \ R^N \setminus \Omega, \end{array} \right.\\ \end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, $\Omega$ is an open bounded subset of $R^N$ with Lipschitz boundary and $f:\Omega \times R \to R$ is a generic Carath\'eodory function satisfying either a $p-$sublinear or a $p-$superlinear growth condition.

Convergence estimates of finite elements for a class of quasilinear elliptic problems

This paper is concerned with conforming finite element discretizations for quasilinear elliptic problems in divergence form, of a class that generalizes the p-Laplace equation and allows to show existence and uniqueness of the continuous and discrete problems. We derive discretization error estimates under general regularity assumptions for the solution and using high order polynomial spaces, resulting in convergence rates that are then verified numerically. A key idea of this error analysis is to consider carefully the relation between the natural W1,p-seminorm and a specific quasinorm introduced in the literature. In particular, we are able to derive interpolation estimates in this quasinorm from known interpolation estimates in the W1,p-seminorm. We also give a simplified proof of known near-best approximation results in W1,p-seminorm starting from the corresponding result in the mentioned quasinorm.

Existence and behavior of positive solutions for a class of quasilinear elliptic problems with discontinuous nonlinearity

Existence and behavior of positive solutions for a class of quasilinear elliptic problems with discontinuous nonlinearity

A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space

We establish a version of the Trudinger-Moser inequality involving unbounded or decaying radial weights in weighted Sobolev spaces. In the light of this inequality and using a minimax procedure we also study existence of solutions for a class of quasilinear elliptic problems involving exponential critical growth.

Existence of solution for a class of quasilinear elliptic problem without Δ2-condition

The existence and multiplicity of solutions for a class of quasilinear elliptic problems are established for the type [Formula: see text] where [Formula: see text], [Formula: see text], is a smooth bounded domain. The nonlinear term [Formula: see text] is a continuous function which is superlinear at the origin and infinity. The function [Formula: see text] is an [Formula: see text]-function where the well-known [Formula: see text]-condition is not assumed. Then the Orlicz–Sobolev space [Formula: see text] may be non-reflexive. As a main model, we have the function [Formula: see text]. Here, we consider some situations where it is possible to work with global minimization, local minimization and mountain pass theorem. However, some estimates employed here are not standard for this type of problem taking into account the modular given by the [Formula: see text]-function [Formula: see text].

Existence of positive solutions for a class of quasilinear elliptic problems with exponential growth via the Nehari manifold method

In this paper we will be concerned with the problem $$\begin{aligned} -\,\text{ div }(a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u)= f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$ where $$\Omega \subset \mathbb {R}^{N}$$ is bounded, $$1{<}p{<}N$$ , $$f:\mathbb {R}\rightarrow \mathbb {R}$$ is a superlinear continuous function with exponential subcritical or exponential critical growth and the function a is $$C^{1}$$ . We use as a main tool the Nehari manifold method and our results include a large class of problems.

Two-grid finite element method and its a posteriori error estimates for a nonmonotone quasilinear elliptic problem under minimal regularity of data

In this paper, we propose and analyze a two-grid finite element method for a class of quasilinear elliptic problems under minimal regularity of data in a bounded convex polygonal Ω⊂R2, which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasilinear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasilinear elliptic problem on a coarse space. Convergence estimates in the H1-norm are derived to justify the efficiency of the proposed two-grid algorithm. Moreover, we propose a natural and computationally efficient residual-based a posteriori error estimator of the two-grid finite element method for this nonmonotone quasilinear elliptic problem and derive the global upper and lower bounds on the error in the H1-norm. Numerical experiments are provided to confirm our theoretical findings.

Homogenization of a class of singular elliptic problems in perforated domains

In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by ε-periodic holes of size ε. The quasilinear equations present a nonlinear singular lower order term fζ(uε), where uε is the solution of the problem at ε-level, ζ is a continuous function singular in zero and f a function whose summability depends on the growth of ζ near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Ω and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new.The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of uε behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence {uε}, as ε→0. This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem.The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as fζ(uε). This plays an essential role in order to get an almost everywhere convergence of the solution uε, needed in the study the asymptotic behavior of the problem.

Weak Galerkin finite element method for a class of quasilinear elliptic problems

The purpose of this paper is to study the weak Galerkin finite element method for a class of quasilinear elliptic problems. The weak Galerkin finite element scheme is proved to have a unique solution with the assumption that guarantees the corresponding operator to be strongly monotone and Lipschitz-continuous. An optimal error estimate in a mesh-dependent energy norm is established. Some numerical results are presented to confirm the theoretical analysis.

Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L1 data

In the present paper, we study the homogenization of a class of quasilinear elliptic problems in a periodically perforated domain Ωε, with L1 data and nonlinear Robin conditions on the boundary of the holes. Considering that we deal with L1 data, we cannot have solutions in H1(Ωε). Therefore, we use here the convenient notion of renormalized solutions. For the homogenization, we use the periodic unfolding method but we can only apply it to the truncated solutions, which are in H1(Ωε). Hence, as a main difficulty, we have to carefully describe the limits of the truncated unfolded solutions and of their gradients. This allow us to prove that we obtain at the limit an unfolded renormalized problem, as well as a (renormalized) homogenized problem in Ω.

Quasilinear elliptic equations on noncompact Riemannian manifolds

The existence of solutions to a class of quasilinear elliptic problems on noncompact Riemannian manifolds, with finite volume, is investigated. Boundary value problems, with homogeneous Neumann conditions, in possibly irregular Euclidean domains are included as a special instance. A nontrivial solution is shown to exist under an unconventional growth condition on the right-hand side, which depends on the geometry of the underlying manifold. The identification of the critical growth is a crucial step in our analysis, and entails the use of the isocapacitary function of the manifold. A condition involving its isoperimetric function is also provided.

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.

Existence of solutions for fractional p-Laplacian problems via Leray-Schauder’s nonlinear alternative

In this paper, we are concerned with the existence of solutions for a class of quasilinear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem: $$\begin{aligned}& (-\Delta)_{p}^{s} u =f (x,u )\quad \mbox{in } \Omega, \\& u=0 \quad \mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{aligned}$$ where $(-\Delta)_{p}^{s}$ is the fractional p-Laplace operator, Ω is an open bounded subset of $\mathbb{R}^{N}$ with Lipschitz boundary, and $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function. The existence of nonnegative solutions is obtained by using Leray-Schauder’s nonlinear alternative.

On the existence of solutions for quasilinear elliptic problems with radial potentials on exterior ball

In this paper, we are concerned with a class of quasilinear elliptic problems with radial potentials and a mixed nonlinear boundary condition on exterior ball domain. Based on a compact embedding from a weighted Sobolev space to a weighted Ls space, the existence of nontrivial solutions is obtained via variational methods.

Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents

We study the following elliptic equations with variable exponents $$-\text{div}(\phi(x,|\nabla u|)\nabla u)=\lambda f(x,u)\quad\text{in}\Omega$$ which is subject to Dirichlet boundary condition. Under suitable conditions on \({\phi}\) and f, employing the variational methods, we show the existence of nontrivial solutions of a class of quasilinear elliptic problems with variable exponents. Also we show the existence of positivity of the infimum of all eigenvalues for the above problem and then give some examples to demonstrate our main result.

Superlinear critical resonant problems with small forcing term

We prove the existence of solutions of a class of quasilinear elliptic problems with Dirichlet boundary conditions of the following form $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} Lu= g(u) -f &{} \hbox { in } \quad \Omega ,\\ u\in X, \end{array} \right. \end{aligned}$$ where \(\Omega \subset \mathbb R^N\) is a bounded domain, \(N\ge 2\), the differential operator is \(Lu= -\hbox {div}( |\nabla u|^{p-2}\nabla u )-\lambda _1 |u|^{p-2}u\) with \(X=W^{1,p}_0(\Omega )\) or \(Lu= \Delta ^2u -\lambda _1 u\) with \(X=H^2_0(\Omega )\), the nonlinearity is given by \(g(u)=(u^+ )^q\) or \(g(u)=|u|^{q-1}u\) i.e. it is a superlinear, at most critical, term and \(f\) is a small reaction term. We give an abstract formulation for which solutions are found by minimization on an appropriate subset of the Nehari manifold associated to our problem. Our method can be also applied to other related problems involving indefinite weights.