On supercritical elliptic problems: existence, multiplicity of positive and symmetry breaking solutions

TL;DR

This paper develops a variational approach to establish the existence of multiple positive solutions, including symmetry-breaking solutions, for supercritical elliptic PDEs, using a minimax principle and new Sobolev embeddings.

Contribution

It introduces a novel variational framework and minimax principle to find multiple solutions, including symmetry-breaking ones, for supercritical elliptic problems.

Findings

Existence of multiple positive solutions including symmetry-breaking solutions.

Development of a new minimax principle on convex subsets.

New Sobolev embeddings for monotonic functions.

Abstract

The main thrust of our current work is to exploit very specific characteristics of a given problem in order to acquire improved compactness for supercritical problems and to prove existence of new types of solutions. To this end, we shall develop a variational machinery in order to construct a new type of classical solutions for a large class of supercritical elliptic partial differential equations.\\ The issue of symmetry and symmetry breaking is challenging and fundamental in mathematics and physics. Symmetry breaking is the source of many interesting phenomena namely phase transitions, instabilities, segregation, etc. As a consequence of our results we shall establish the existence of several symmetry breaking solutions when the underlying problem is fully symmetric. Our methodology is variational, and we are not seeking non symmetric solutions which bifurcate from the symmetric one. Instead, we construct many new positive solutions by developing a minimax principle for general semilinear elliptic problems restricted to a given convex subset instead of the whole space. As a byproduct of our investigation, several new Sobolev embeddings are established for functions having a mild monotonicity on symmetric monotonic domains.