Nonlinear Elliptic Problems Research Articles

On a class of infinite semipositone problems for ( p , q ) Laplace operator

We analyze a non-linear elliptic boundary value problem that involves ( p , q ) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in ( 0 , ∞ ) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving ( p , q ) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.

Normalized Multi-peak Solutions to Nonlinear Elliptic Problems

Normalized Multi-peak Solutions to Nonlinear Elliptic Problems

On some Nonlinear elliptic problems with boundary conditions in generalized Sobolev spaces

On some Nonlinear elliptic problems with boundary conditions in generalized Sobolev spaces

Calderón-Zygmund estimates for nonlinear elliptic obstacle problems with log-BMO matrix weights

Calderón-Zygmund estimates for nonlinear elliptic obstacle problems with log-BMO matrix weights

Existence of solutions for nonlinear degenerate elliptic problems

Existence of solutions for nonlinear degenerate elliptic problems

Existence of an entropic solution of a nonlinear elliptic problem in an unbounded domain

Existence of an entropic solution of a nonlinear elliptic problem in an unbounded domain

Existence and multiplicity of triple weak solutions for a nonlinear elliptic problem with fourth-order operator and Hardy potential

<abstract><p>This study investigates the existence of triple weak solutions for a system of nonlinear elliptic equations with a fourth-order operator. The problem arises in the mathematical modeling of complex physical phenomena.</p></abstract>

Existence and regularity of solutions of nonlinear anisotropic elliptic problem with Hardy potential

In this paper, we are interested in the existence and regularity of solutions for some anisotropic elliptic equations with Hardy potential and L m ( Ω ) data in appropriate anisotropic Sobolev spaces. The aim of this work is to get natural conditions to show the existence and regularity results for the solutions, that is related to an anisotropic Hardy inequality.

A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

Effective numerical computation of p(x)–Laplace equations in 2D

In this article we implement a method for the computation of a nonlinear elliptic problem with nonstandard growth driven by the Laplacian operator. Our implementation is based in the decomposition–coordination method that allows us, via an iterative process, to solve in each step a linear differential equation and a nonlinear algebraic equation. Our code is implemented in MatLab in two dimensions and turns out to be extremely efficient from the computational point of view.

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

This research work deals with two spectral matrix collocation algorithms based on (novel) clique functions to solve two classes of nonlinear nonlocal elliptic two-points boundary value problems (BVPs) arising in diverse physical models. The nonlinearity together with nonlocality makes the models so difficult to solve numerically. In both matrix methods by expanding the unknown solutions in terms of clique polynomials the nonlocality in the models is handled. In the first and direct technique, the clique coefficients are determined in an accurate way through solving an algebraic system of nonlinear equations. To get rid of the nonlinearity of the models and in order to gain efficacy, the quasilinearization method (QLM) is utilized in the second approach. In the latter algorithm named QLM-clique procedure, the first directly clique collocation method is applied to a family of linearized equations in an iterative manner. In particular, we show that the series expansion of clique polynomials is convergent exponentially in a weighted L2 norm. Numerous numerical simulations validate the performance of two matrix collocation schemes and demonstrate the accuracy as well as the gain in computational efficiency in terms of elapsed CPU time. The proposed matrix algorithms for computation are easy to implement and straightforward, and provide more accuracy compared to other available computational values in the literature.

Regularity of viscosity solutions to fully nonlinear elliptic transmission problems

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the C0,α, C1,α and C2,α regularity of viscosity solutions up to the transmission surface for the case of curved interfaces.

On a discontinuous nonlinear elliptic problem of nonlocal-type with Neumann boundary condition

On a discontinuous nonlinear elliptic problem of nonlocal-type with Neumann boundary condition

Multiplicity of solutions to a nonlinear elliptic problem on a Riemannian orbifold

We employ the photography method to estimate the number of solutions to a nonlinear elliptic problem on a Riemannian orbifold in function of the Lusternik–Schnirelmann category of its submanifold of points with largest local group.

Nonlinear elliptic boundary value problems with convection term and Hardy potential

Abstract In this paper, we deal with a nonlinear elliptic problems that incorporate a Hardy potential and a nonlinear convection term. We establish the existence and regularity of solutions under various assumptions concerning the summability of the source term f.

Solvability of nonlinear anisotropic elliptic unilateral problems with variable exponent and measure data

Solvability of nonlinear anisotropic elliptic unilateral problems with variable exponent and measure data

Singular elliptic problem involving a Hardy potential and lower order term

We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in Ω u > 0 in Ω u = 0 on ∂ Ω , where 1 < p < N; Ω ⊂ R N is a bounded regular domain containing the origin and 0 < γ < 1, a ⩾ 0 , b > 0 , 0 ⩽ f ∈ L m ( Ω ) and 1 < m < N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.

Parameter estimates and a uniqueness result for double phase problem with a singular nonlinearity

We consider the boundary value problem −Δpuλ−Δquλ=λg(x)uλ−β in Ω, uλ=0 on ∂Ω with uλ>0 in Ω. We assume Ω is a bounded open set in RN with smooth boundary, 1<p<q<∞, β∈[0,1), g is a positive weight function and λ is a positive parameter. We derive an estimate for uλ which describes its exact behavior when the parameter λ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems. Here, as an application of this estimate, we obtain a uniqueness result for a nonlinear elliptic boundary value problem with a singular nonlinearity.

Nodal discrete duality numerical scheme for nonlinear diffusion problems on general meshes

Abstract Discrete duality finite volume (DDFV) schemes are known for their ability to approximate nonlinear and linear anisotropic diffusion operators on general meshes, but they possess several drawbacks. The most important drawback of DDFV is the simultaneous use of the cell and the node unknowns. We propose a discretization approach that incorporates DDFV ideas and the associated analysis techniques, but allows for a rapid elimination of the cell unknowns. Further, unlike the DDFV scheme, the new ‘Nodal Discrete Duality’ (NDD) scheme does not require specific adaptation in presence of discontinuities of the diffusion tensor along cell boundaries. We describe in detail the 2D NDD framework and its two 3D variants, focusing on the consistency properties of the discrete gradient and discrete divergence operators and on the key structural property of discrete duality. For the 2D scheme, convergence analysis is carried out and a series of numerical tests are provided for a large family of nonlinear anisotropic elliptic problems with zero Dirichlet boundary condition.

Strongly nonlinear unilateral anisotropic elliptic problem with L^1-data

Strongly nonlinear unilateral anisotropic elliptic problem with L^1-data