## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 9/10 (2009), 1047-1074.

### Asymptotics and symmetries of ground-state and least energy nodal solutions for boundary-value problems with slowly growing superlinearities

Denis Bonheure, Vincent Bouchez, and Christopher Grumiau

#### Abstract

We study the problems $$ \begin{array}{lcr} -\Delta u= f_{\theta}(u)\text{ in }\Omega, & & u=0\text{ on }\partial\Omega,\\ -\Delta u + u= f_\theta(u)\text{ in }\Omega, & & \partial_{\nu} u=0\text{ on }\partial\Omega, \end{array} $$ where $f_{\theta}$ is a slowly superlinearly growing nonlinearity, and $\Omega$ is a bounded domain. Namely, we are interested in generalizing the results obtained in [4], where the model nonlinearity $f_\theta(u)= | {u} | ^{\theta-2} u$ was considered in the case of Dirichlet boundary conditions. We derive the asymptotic behaviour of ground state and least energy nodal solutions when $\theta\to 2$, leading to symmetry results for $\theta$ small. Our assumptions permit us to study some typical nonlinearities such as a superlinear perturbation of a small pure power or the sum of small powers and slowly exponentialy growing nonlinearities in dimension $2$.

#### Article information

**Source**

Differential Integral Equations Volume 22, Number 9/10 (2009), 1047-1074.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019522

**Mathematical Reviews number (MathSciNet)**

MR2553070

**Zentralblatt MATH identifier**

1240.35198

**Subjects**

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian

Secondary: 35B06: Symmetries, invariants, etc. 35B25: Singular perturbations 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations

#### Citation

Bonheure, Denis; Bouchez, Vincent; Grumiau, Christopher. Asymptotics and symmetries of ground-state and least energy nodal solutions for boundary-value problems with slowly growing superlinearities. Differential Integral Equations 22 (2009), no. 9/10, 1047--1074.https://projecteuclid.org/euclid.die/1356019522