Quasilinear Elliptic Problems Research Articles

Existence of bound states for quasilinear elliptic problems involving critical growth and frequency

Existence of bound states for quasilinear elliptic problems involving critical growth and frequency

Multiplicity results for some quasilinear elliptic problems with concave nonlinearities

Multiplicity results for some quasilinear elliptic problems with concave nonlinearities

TWO-GRID VIRTUAL ELEMENT DISCRETIZATION OF QUASILINEAR ELLIPTIC PROBLEM

In this paper a two grid algorithm for quasilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a quasilinear elliptic problem on a fine grid is reduced to the solution of a quasilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimate in H1 norm is derived. Numerical experiments are carried out to illustrate the theoretical findings.

Gradient bounds for non-uniformly quasilinear elliptic two-sided obstacle problems with variable exponents

We are concerned with a class of quasilinear elliptic variational inequalities, the so-called two-obstacle problems, that are driven by the double-phase operators with variable exponents. In this paper, by using the terminology of fractional maximal distribution functions, we prove decay estimates for level sets of the gradient of weak solutions in turn implying regularity estimates in some generalized weighted function spaces.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Solvability of Parametric Elliptic Systems with Variable Exponents

Abstract In this paper, we study the solvability to the left of the positive infimum of all eigenvalues for some non-resonant quasilinear elliptic problems involving variable exponents. We first prove the existence of at least a weak solution for some non-variational systems by using a surjectivity result for pseudomonotone operators. Furthermore, under additional conditions, we show that the solution is unique and provide examples. Second, we deal with non-resonant gradient-type systems and obtain existence by using a variational approach.

Existence of nontrivial solution for quasilinear equations involving the 1-biharmonic operator

In this paper, we study the existence results of a quasilinear elliptic problem involving the 1-biharmonic operator in R N , whose nonlinearity satisfies appropriate conditions. The existence theorem is proved through a new version of the Mountain Pass Theorem to locally Lipschitz functionals, where it is considered the Cerami compactness condition rather than the Palais–Smale one.

A class of quasilinear and noncoercive elliptic problem in a two-component domain with $$L^{1}$$ data

A class of quasilinear and noncoercive elliptic problem in a two-component domain with $$L^{1}$$ data

Convergence and quasi-optimality of an adaptive finite element method for nonmonotone quasi-linear elliptic problems on L2 errors

Convergence and quasi-optimality of an adaptive finite element method for nonmonotone quasi-linear elliptic problems on L2 errors

A priori estimates for non-coercive Dirichlet problems with subquadratic gradient terms

We deal with some quasilinear elliptic problems posed in a bounded smooth convex domain Ω⊂RN (N≥3), namely{−Δu=λu+μ(x)|∇u|q+f(x),x∈Ω,u=0,x∈∂Ω, where the data satisfyμ∈L∞(Ω),μ⪈0;f∈Lp(Ω),p>N,f⪈0 and 1<q≤2. We provide sufficient conditions on f,μ (allowing μ to vanish on ∂Ω) that yield the sharp estimate λ‖u‖L∞(Ω)≤C for any bounded solution u with λ∈(0,λ1), which is the non-coercive regime. The estimate leads to remarkable consequences such as a multiplicity result and a precise asymptotic behavior of the bounded but blowing up solutions as λ→0+.

Some existence results of bounded variation solutions for a 1‐biharmonic problem with vanishing potentials

In this work, we study a quasilinear elliptic problem in the whole space involving the 1‐biharmonic operator with potentials that can vanish at infinity. We consider two different geometrical assumptions in the nonlinearity and use variational methods to obtain nontrivial bounded variation solutions.

A Variational Approach to Quasilinear Elliptic Problems with Gradient Dependence

A Variational Approach to Quasilinear Elliptic Problems with Gradient Dependence

Homogenization of a class of singular elliptic problems in two-component domains

This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ε γ . We prove an homogenization result for − 1 &lt; γ &lt; 1 by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 (2008) 1585–1620 and The Periodic Unfolding Method (2018) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011) 891–927). One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence { u ε } of the solutions, as ε → 0. More precisely, our result shows that the gradient of u ε behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Multiple solutions for perturbed quasilinear elliptic problems

We investigate the existence of multiple solutions for the $(p,q)$-quasilinear elliptic problem \[ \begin{cases} -\Delta_p u -\Delta_q u\ =\ g(x, u) + \varepsilon\ h(x,u) &amp; \mbox{in } \Omega,\\ u=0 &amp; \mbox{on } \partial\Omega,\\ \end{cases} \] where $1&lt; p&lt; q&lt; +\infty$, $\Omega$ is an open bounded domain of ${\mathbb R}^N$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$, $\varepsilon\in{\mathbb R}$ and $h\in C(\overline\Omega\times{\mathbb R},{\mathbb R})$. In spite of the possible lack of a variational structure of this problem, from suitable assumptions on $g(x,u)$ and appropriate procedures and estimates, the existence of multiple solutions can be proved for small perturbations.

Existence of entropy solutions for some quasilinear anisotropic elliptic unilateral problems with variable exponents

In this paper, we shall be concerned with the study of the following quasilinear anisotropic elliptic Dirichlet problems of the type \begin{document}$\begin{equation} -\mbox{div }\;a(x,u,\nabla u) = f -\mbox{div } F \qquad \mbox{in}\quad \Omega,\quad\quad\quad\quad\quad(1) \end{equation}$ \end{document} where $ f\in L^{1}(\Omega) $ and $ F \in \prod_{i = 1}^{N} L^{p'_{i}(\cdot)}(\Omega), $ and $ a_{i}(x,u,\xi) $ are Carathéodory functions from $ \Omega \times I\!\!R \times I\!\!R^{N} $ into $ I\!\!R $, which satisfy assumptions of growth, coercivity and strict monotonicity. We prove the existence of entropy solutions for the quasilinear elliptic equation associated to the unilateral problem by relying on the penalization method, in the anisotropic variable exponent Sobolev spaces. Our approach is also based on the techniques of monotone operators in Banach spaces, the existence of weak solutions, and some approximations methods. The problems of the type $ (1) $ are very interesting from the purely mathematical point of view. On the other hand, such equations $ (1) $ appear in different contexts, in particular, the mathematical description of motions of the non-newtonien fluids; we quote for instance the electro-rheological fluids; the deformation of membrane constrained by an obstacle, the image processing and other various physical applications.

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Weak Solutions for a Quasilinear Elliptic and Parabolic Problems Involving the (p(x), q(x))-Laplacian Operator

Weak Solutions for a Quasilinear Elliptic and Parabolic Problems Involving the (p(x), q(x))-Laplacian Operator

Gradient estimates in generalized Orlicz spaces for quasilinear elliptic equations via extrapolation

&lt;abstract&gt;&lt;p&gt;The gradient estimates in the generalized Orlicz space for weak solutions of a class of quasi-linear elliptic boundary value problems are obtained using the modern technique of extrapolation. The coefficients are assumed to have small BMO seminorms, and the boundary of the domain is sufficiently flat in the sense of Reifenberg. As a corollary, we apply our results to the variable Lebesgue spaces.&lt;/p&gt;&lt;/abstract&gt;

Problems involving the fractional g-Laplacian with lack of compactness

In this paper, we prove compact embedding of a subspace of the fractional Orlicz–Sobolev space Ws,GRN consisting of radial functions; our target embedding spaces are of Orlicz type. In addition, we prove a Lions and Lieb type results for Ws,GRN that works together in a particular way to get a sequence whose weak limit is non-trivial. As an application, we study the existence of solutions to quasilinear elliptic problems in the whole space RN involving the fractional g-Laplacian operator, where the conjugated function G̃ of G does not satisfy the Δ2-condition.