Multiple Solutions for a Henon-Like Equation on the Annulus

TL;DR

This paper proves the existence of multiple solutions for a Henon-like equation on an annulus, revealing symmetry-breaking phenomena and solutions near critical Sobolev exponents, with implications for nonlinear PDEs.

Contribution

It introduces new existence results for multiple solutions of a Henon-like PDE on an annulus, including symmetry-breaking and solutions near critical exponents.

Findings

Existence of two solutions for large alpha.

Additional solutions near the critical Sobolev exponent.

Symmetry-breaking solutions that are non-radial.

Abstract

For the equation (-\Delta u = | |x|-2 |^\alpha u^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (\alpha) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A symmetry--breaking phenomenon appears, showing that the least--energy solutions cannot be radial functions.