## Differential and Integral Equations

- Differential Integral Equations
- Volume 31, Number 3/4 (2018), 257-272.

### Nodal solutions for Lane-Emden problems in almost-annular domains

Anna Lisa Amadori, Francesca Gladiali, and Massimo Grossi

#### Abstract

In this paper, we prove an existence result to the problem $$\left\{\begin{array}{ll} -\Delta u = |u|^{p-1} u \qquad & \text{ in } \Omega , \\ u= 0 & \text{ on } \partial\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in $\mathbb R^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1 < p_2 < \cdots$ with $\lim\limits_{k\rightarrow+\infty}p_k=+\infty$ such that for any real number $p > 1$ and $p\ne p_k$ there exist at least one solution with $m$ nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze the asymptotic behavior as $p\to 1$.

#### Article information

**Source**

Differential Integral Equations, Volume 31, Number 3/4 (2018), 257-272.

**Dates**

First available in Project Euclid: 19 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1513652426

**Mathematical Reviews number (MathSciNet)**

MR3738198

**Zentralblatt MATH identifier**

06837097

**Subjects**

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

#### Citation

Amadori, Anna Lisa; Gladiali, Francesca; Grossi, Massimo. Nodal solutions for Lane-Emden problems in almost-annular domains. Differential Integral Equations 31 (2018), no. 3/4, 257--272. https://projecteuclid.org/euclid.die/1513652426