| Abstract: |
| In this talk, we shall describe some supercritical variants of first and second-order Sobolev and Hardy--Sobolev-type embeddings and related elliptic problems that incorporate power nonlinearities with strongly supercritical variable exponents. In
particular, we shall describe a variational approach to derive such inequalities. Then, we will discuss how to obtain some existence results for the Dirichlet problem to a class of supercritical elliptic problems with variable exponents in the unit ball, which contrasts with non-existence results for the classical constant exponent case. We will also discuss complimentary non-existence results in both ball domains and the whole space. |
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