On the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass

For semilinear elliptic equations $-\Delta u=\lambda|u|^{p-2}u-|u|^{q-2}u$, boundary value problems in bounded and unbounded domains are considered. In the plane of exponents $p\times q$, the so-called curves of critical exponents are defined that divide this plane into domains with qualitatively different properties of the boundary value problems and the corresponding parabolic equations. New solvability conditions for boundary value problems, conditions for the stability and instability of stationary solutions, and conditions for the existence of global solutions to parabolic equations are found.