Ahmed Mohammed

On the existence of solutions to the Monge-Ampère equation with infinite boundary values

Given a positive and an increasing nonlinearity f that satisfies an appropriate growth condition at infinity, we provide a condition on g ∈ C ∞ (Ω) for which the Monge-Ampere equation det D 2 u = g f(u) admits a solution with infinite boundary value on a strictly convex domain Ω. Sufficient conditions for the nonexistence of such solutions will also be given.

Entire Large Solutions to Elliptic Equations of Power Non-linearities with Variable Exponents

Abstract We give conditions on the variable exponent q that ensure the existence and nonexistence of a positive solution to the elliptic equation Δu = uq(x) on ℝN (N ≥ 3) which satisfies lim|x|→∞ u(x) = ∞. The nonnegative function q is required to be locally Hölder continuous on ℝN. We prove existence for q > 1 provided q(x) decays to unity rapidly as |x| → ∞. We treat the case q ≤ 1 as a special case of q − 1 changing signs and show that a solution exists provided q is asymptotically radial. In addition, we give an example to show that our results are nearly optimal.

On Ground State Solutions to Mixed Type Singular Semi-Linear Elliptic Equations

Abstract The purpose of the paper is to establish the existence of ground state solutions to -Δu = ηa(x)f(u)+γb(x)g(u), where the locally Hölder continuous non-negative functions a and b and the non-linearities f and g satisfy some general conditions.

A Harnack Inequality for Ordinary Differential Equations

is true for all x and y in [a, b] and for each u in H that is not identically zero. In other words, the ratios of the values of the nonzero functions in H at any two points of [a, b] all lie in the fixed interval [1/C, C] that is independent of the functions. (We will see in section 4 that C does, in general, depend on A and B as well as on a and b.) Alex Harnack (1851-1888) first introduced inequality (1.2) in his book Grundlagen der Theorie des Logarithmischen Potentials (Leipzig, 1887). In this book, the inequality was stated for positive harmonic functions, and Harnack used it to show that a bounded sequence of harmonic functions on an open connected set in Euclidean space contains a subsequence that converges uniformly on compacta to a harmonic function. This fact is useful, in turn, for establishing the existence of a harmonic function on an open set 2 that takes prescribed continuous data on the boundary 302 [6, pp. 2326](i.e., for solving the classical Dirichlet problem). Recall that a harmonic function on an open set i2 in R is a function u in C2 (2) that satisfies Laplaces equation:

Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness

Abstract The primary objective of the paper is to study the existence, asymptotic boundary estimates and uniqueness of large solutions to fully nonlinear equations H u2062 ( x , u , D u2062 u , D 2 u2062 u ) = f u2062 ( u ) + h u2062 ( x ) {H(x,u,Du,D^{2}u)=f(u)+h(x)} in bounded C 2 {C^{2}} domains Ω ⊆ ℝ n {Omegasubseteqmathbb{R}^{n}} . Here H is a fully nonlinear uniformly elliptic differential operator, f is a non-decreasing function that satisfies appropriate growth conditions at infinity, and h is a continuous function on Ω that could be unbounded either from above or from below. The results contained herein provide substantial generalizations and improvements of results known in the literature.