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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. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for the nonlinear p(.) degenerate problems involving nonlinear operators of the form - div a(x, ∇u) = f(x, u, ∇u) where a and fare Caratheodoryfunctions satisfying some nonstandard growth conditions. [ABSTRACT FROM AUTHOR]

We are concerned with the nonlinear elliptic equation with Dirichlet boundary condition: , u>0 in Ω, u=0 on where Ω is a smooth bounded domain in and is a -positive function in . Under some suitable conditions on and using dynamical method involving a partial description of the lack of compactness of the associated variational problem, we prove an existence result. [ABSTRACT FROM AUTHOR]

We prove the existence of positive solutions for the equation − ∑ i = 1 N ∂ ∂ x i a i ( x , u , ∇ u ) = f ( x , u , ∇ u ) in Ω under the Dirichlet boundary condition, where the essential point is the dependence of the terms of the elliptic equation on the solution u and its gradient ∇ u . We develop an approach based on approximate solutions and on a new strong maximum principle. [ABSTRACT FROM AUTHOR]

Some existence results for a parametric Dirichlet problem defined on the Sierpiński fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well-determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution. [ABSTRACT FROM AUTHOR]

Abstract: An inverse coefficient problem related to identification of the plasticity function from a given torque is studied for a circular section bar. Within the deformation theory of plasticity the mathematical model of torsion leads to the nonlinear Dirichlet problem , ; , . For determination of the unknown coefficient , an integral of the function over the domain , i.e. the measured torque , is assumed to be given as an additional data. This data , depending on the angle of twist , is obtained during the quasi-static elastic–plastic torsional deformation. It is proved that for a circular section bar, the coefficient-to-torque (i.e. input–output) map is uniquely invertible. Moreover, an explicit formula relating the plasticity function and the torque is derived. The well-known formula between the elastic shear modulus and the torque is obtained from this explicit formula, for pure elastic torsion. The proposed approach permits one to predict some elastic–plastic torsional effects arising in the hardening bar, depending on the angle of twist. [Copyright &y& Elsevier]

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Hardy–Hénon equation with and and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space . It has been conjectured that this property is true if (and only if) , where is the Hardy–Sobolev exponent, given by . However, when , the conjecture had up to now been proved only for . Indeed the case seems more difficult, due to . In this paper, we prove the conjecture for in dimension , in the case of bounded solutions. Next, for the conjecture in the case , and for related estimates near isolated singularities and at infinity, we give new proofs – based in particular on doubling-rescaling arguments – and we provide some extensions of these estimates. These proofs are significantly simpler than the previously known ones. Finally, we clarify some of the previous results on a priori estimates for the related Dirichlet problem. [Copyright &y& Elsevier]

Abstract: This paper deals with both Dirichlet and Neumann problems for a class of nonlinear degenerate elliptic equations with general growth in the gradient. First, we give an existence result of a spherically symmetric solution to the “symmetrized” problems with data depending only on the radials. Second, we prove that the solutions of the original problems can be compared, under a rearrangement, with the solutions of the “symmetrized” problems. [Copyright &y& Elsevier]

Abstract: Here we establish the existence of infinitely many nonradial solutions for a superlinear Dirichlet problem in annulii. Our proof relies on estimating the number of radial solutions having a prescribed number of nodal regions. We prove that, for large, there exist exactly two radial solutions with k nodal regions (connected components of ). The problem need not be homogeneous. [Copyright &y& Elsevier]

We prove the existence of infinitely many solutions to a semilinear Dirichlet boundary value problem in a ball for a nonlinearity g(u) that grows subcritically for u positive and supercritically for u negative. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Dirichlet problem for nonlinear elliptic second-order equations in divergence form with an -right-hand side f. By the function f we define other function with the values which are equal to the integrals of the function over the sets , . Under rather general integral conditions on the latter function we prove that the entropy solution of the problem under consideration belongs to some limit Lebesgue and Sobolev spaces. [Copyright &y& Elsevier]

The work deals with the Dirichlet problem for elliptic equations with nonhomogeneous anisotropic degeneracy in a possibly unbounded domain of multidimensional Euclidean space. The existence of weak solutions is proved. Some conditions are established connecting the character of nonlinearity of the equation and the geometric characteristics of the domain which guarantee the one-dimensional localization (vanishing) of weak solutions. The equation with anisotropic degeneracy is shown to admit localized solutions even in the absence of absorption. [ABSTRACT FROM AUTHOR]

We establish the existence of a weak solution to the Dirichlet problem belonging to a Sobolev space for nonlinear elliptic equations of second order with right-hand sides from a wide class of functions defined in terms of the logarithmic function. [ABSTRACT FROM AUTHOR]