## Differential and Integral Equations

- Differential Integral Equations
- Volume 17, Number 11-12 (2004), 1201-1212.

### On the uniqueness of solutions for a semilinear elliptic problem in convex domains

#### Abstract

We exhibit a class of convex and nonsymmetric domains $\Omega$ in $\mathbb R^N,$ $N\ge4,$ such that the slightly subcritical problem $$ \begin{cases} -\Delta u=u^{{N+2\over N-2}-{\varepsilon}} & \text{ in $\Omega,$ } \\ u>0 & \text{ in $\Omega,$ } \\ u=0 & \text{ on $\partial\Omega $} \end{cases} $$ does not have any solutions blowing up at more than one point in $\Omega$ as ${\varepsilon}$ goes to zero. Moreover if $\Omega$ is a small perturbation of a convex and symmetric domain, we prove that such a problem has a unique solution provided ${\varepsilon}$ is small enough.

#### Article information

**Source**

Differential Integral Equations Volume 17, Number 11-12 (2004), 1201-1212.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060241

**Mathematical Reviews number (MathSciNet)**

MR2100022

**Zentralblatt MATH identifier**

1150.35418

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B33: Critical exponents 35J25: Boundary value problems for second-order elliptic equations

#### Citation

Pistoia, Angela. On the uniqueness of solutions for a semilinear elliptic problem in convex domains. Differential Integral Equations 17 (2004), no. 11-12, 1201--1212.https://projecteuclid.org/euclid.die/1356060241