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

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 first introduce an efficient tool in the context of variational methods 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 utilizing 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.