On critical exponents curve for nonlinear elliptic equations in zero mass case

TL;DR

This paper investigates the critical exponents curve for a class of semilinear elliptic equations, identifying conditions for solvability, stability, and global solutions in various domains.

Contribution

It introduces the critical exponents curve in the p-q plane and derives necessary conditions for solutions and stability, advancing understanding of nonlinear elliptic equations.

Findings

Critical exponents curve separates domains with different solution properties.

Necessary conditions for solvability and stability are established.

Existence of global solutions for associated parabolic problems is demonstrated.

Abstract

Semilinear elliptic equations of the form $-\Delta u =\lambda|u|^{p-2}u- |u|^{q-2}u$ in bounded and unbounded domains are considered. In the plane of exponents $p\times q$, the so-called critical exponents curve is introduced which separates domains with qualitatively distinctive properties of the considered equations and the associated parabolic problems. Necessary conditions for solvability to equations, stable and unstable stationary solutions, an existence of global solutions for parabolic problems in the whole space are obtained.