| Abstract: |
| In this talk, I will present an existence result for the Dirichlet problem associated with the elliptic equation
\[
-\Delta u + u = a(x)|u|^{p-2}u
\]
set in an annulus of $\mathbb R^N$. Here $p>2$ is allowed to be supercritical in the sense of Sobolev embeddings, and $a(x)$ is a positive weight with additional symmetry and monotonicity properties, which are shared by the solution that we construct.
For this problem, we find a new type of positive, axially symmetric solution. Moreover, in the case where the weight $a(x)$ is constant, we detect a condition, depending only on the exponent $p$ and on the inner radius of the annulus, that ensures that the solution is nonradial. In this setting, the major difficulty to overcome is the lack of compactness in a nonradial framework. The proofs rely on a combination of variational methods and dynamical system techniques. |
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