Entire solutions for a class of fourth order semilinear elliptic equations with weights

TL;DR

This paper studies entire solutions of a class of fourth order semilinear elliptic equations with weights, revealing existence, symmetry breaking, and asymptotic behaviors related to critical growth and parameters.

Contribution

It establishes existence of solutions as extremals of Rellich-Sobolev inequalities and analyzes symmetry breaking phenomena for near-critical nonlinearities.

Findings

Existence of entire solutions as extremal functions.

Symmetry breaking occurs for large parameter values.

Asymptotic behavior of solutions is characterized near criticality.

Abstract

We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non compact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter {\lambda}. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter {\lambda} is large, symmetry breaking phenomena occur and in some cases the asymptotic behavior of radial and non radial ground states can be somehow described.