A Numerical Study of the Homogeneous Elliptic Equation with Fractional Boundary Conditions

We consider the homogeneous equation

Regularity of the Very Weak Solutions for Double Obstacle Problems

The apply of harmonic eauation is know to all. Our interesting is to get the regularity os their solution, recently for obstacle problems. Many interesting results have been obtained for the solutions of harmonic equation ande their obstacle problems, however the double obstacle problems about the definition and regularity results for non-homogeneous elliptic equation. In this paper , the basic tool for the Young inequality,Hölder inequality, Minkowski inequality, Poincaré inequality and a basic inequality. The definition of very weak solutions for double obstacle problems associated with non-homogeneous elliptic equation is given, and the local integrability result is obtained by using the technique of Hodge decomposition.

Non-homogeneous elliptic equations in exterior domains

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

Space of Solutions of Homogeneous Elliptic Equations

This is the continuation of our paper [1] and includes the results promised there. As in [1], we consider a homogeneous elliptic equation in two variables. In [1] we showed that all solutions of such equations can be written in a specific form, viz. in the form of an infinite series in certain specific polynomials. Here we first establish that a common solution of any two positive powers of any two linearly independent, linear elliptic polynomials can be expressed as a polynomial (Lemma 2).