Two solutions on a Nehari set in an invariant cone*

Abstract: In this talk, we will discuss the existence and variational characterization of two distinct non-constant, radial, radially nondecreasing solutions to a $p$-Laplacian problem set in a ball of $\bf{R}^N,$ under Neumann boundary conditions. The problem involves a power nonlinearity, which is supercritical in the sense of Sobolev embeddings, and the power $p$, appearing in the operator, is between $1$ and $2$. Moreover, the problem admits a unique constant solution 1. To find the two non-constant solutions, we use a variational approach in an invariant cone. These solutions are distinguished upon their energy: one has minimal energy inside a Nehari-type set, the other is obtained via a mountain pass argument inside the same set. We will also highlight the differences with the semilinear case ($p=2$) and with the possibly degenerate case ($p>2$). We will show that for $1<p<2,$ 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 the power in the nonlinearity goes to infinity. This talk is based on joint work with Benedetta Noris and Gianmaria Verzini.

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