| Abstract: |
| In this talk, we will discuss the existence and variational characterization of two distinct non-constant, radial, radially non-decreasing solutions to the supercritical equation
\[
-\Delta_p u+u^{p-1}=u^{q-1}
\]
under Neumann boundary conditions, in the unit ball of $\mathbb R^N$. Here $p\in(1,2)$ and $q$ is large. Using a variational approach in an invariant cone, we can distinguish the two solutions on their energy: one has minimal energy inside a Nehari-type set and the other is obtained via a mountain pass argument inside the same set. In the talk, we will also highlight the differences with the cases $p=2$ and $p>2$. We will show that for $p\in(1,2)$, the constant solution 1 is a local minimizer on the Nehari set: this is a peculiarity of this case and is responsible for the appearance of the second (higher-energy) solution. Finally, we will detect the limit profiles of the two solutions as $q\to\infty$. |
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