We study the existence and non-existence of positive singular solutions of second-order non-divergence type elliptic inequalities of the form $\sum\limits_{i,j = 1}^N {a_{ij} (x)\frac{{\partial ^2 u}} {{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^N {b_i (x)\frac{{\partial u}} {{\partial x_i }} \geqslant K(x)u^p ,} - \infty < p - \infty , $ with measurable coefficients in a punctured ball B R \{0} of ℝ N , N ≥ 1. We prove the existence of a critical value p* which separates the existence region from the non-existence region. We show that in the critical case p = p*, the existence of a singular solution depends on the rate at which the coefficients (a i j ) and (b i ) stabilize at zero, and we provide some optimal conditions in this setting.

The positivity preserving property for the biharmonic operator with Dirichlet boundary condition is investigated. We discuss here the case where the domain is an ellipse (that may degenerate to a strip) and the data is a
polynomial function. We provide various conditions for which the positivity is preserved.

We study the semilinear elliptic system... where Ω⊂R^N(N≥1) is a smooth and bounded domain, p,q,r,s>0. Under suitable ranges of exponents we obtain various results regarding the well posedness of our system.

This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in RN to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory.