Generalized Harnack inequality for semilinear elliptic equations

Boundary Harnack principle and elliptic Harnack inequality

We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that our assumptions are robust to time changes of the corresponding diffusions. In particular, we do not assume volume doubling property for the symmetric measure.

Semilinear Elliptic Equations with Generalized Cubic Nonlinearities

A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.

Stability of Elliptic Harnack inequality

We prove that the elliptic Harnack inequality (on a manifold, graph, or suitably regular metric measure space) is stable under bounded perturbations, as well as rough isometries.

Liouville type theorems and Harnack type inequalities for semilinear elliptic equations

Some remarks on the elliptic Harnack inequality

In this note we give three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of points in balls, and the second that the EHI is equivalent to an annulus type Harnack inequality for Green’s functions. The third result uses the lamplighter group to give a counterexample concer...