# On qualitative properties of solutions for elliptic problems with the $p$-Laplacian through domain perturbations

## TL;DR

This paper investigates how the critical energy levels of solutions to a p-Laplacian elliptic problem depend on domain shape perturbations, establishing formulas and geometric properties that influence solution symmetry and energy levels.

## Contribution

It introduces Hadamard-type formulas for critical levels and characterizes domain shapes that maximize these levels, revealing nonradial solutions in symmetric domains.

## Key findings

- Critical levels attain maximum in concentric spherical annuli.
- Least energy nodal solutions are nonradial in balls and concentric annuli.
- Domain shape significantly influences solution properties.

## Abstract

We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem $-\Delta_p u = f(u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ upon domain perturbations. Assuming that the nonlinearity $f$ is superlinear and subcritical, we establish Hadamard-type formulas for such critical levels. As an application, we show that among all (generally eccentric) spherical annuli $\Omega$ least nontrivial critical levels attain maximum if and only if $\Omega$ is concentric. As a consequence of this fact, we prove the nonradiality of least energy nodal solutions whenever $\Omega$ is a ball or concentric annulus.