We consider Kolmogorov operator $$-\nabla \cdot a \nabla + b $$ with measurable uniformly elliptic matrix and prove Gaussian lower upper bounds on its heat kernel under minimal assumptions the vector field divergence $$\mathrm{div\,}b$$ . More precisely, we prove: (1) bound, provided that $$\mathrm{div\,}b \ge 0$$ , is in class of form-bounded fields (containing e.g. $$L^d$$ weak class, as well...

Abrtract The ethanolamine-containing glycerophospholipids, choline-containing glycerophospholipids, and phosphatidylinmitol fractions are major sources of arachidonic acid in murine mastocytoma P-815 cloned cells. The choline-linked fraction contained high arachidonic acid contents in 1-0-alkyl-2-acyl(18%) and 1,2-diacyl-sn-glycero-3-phosphocholine (ll%), with smaller amounts in l-O-alk-l'-enyl...

We study the existence of sign changing solutions to following problem(0.1){Δu+|u|p−1u=0inΩε;u=0on∂Ωε, where p=n+2n−2 is critical Sobolev exponent and Ωε a bounded smooth domain in Rn, n≥3, form Ωε=Ω\B(0,ε). Here Ω containing origin 0 B(0,ε) denotes ball centered at with radius ε>0. construct new type sign-changing high energy problem (0.1), when parameter ε small enough.