## Differential and Integral Equations

### Multiplicity of solutions to a nonhomogeneous elliptic equation in $\Bbb R^2$

#### Abstract

In this article, we consider the following problem $$(P)\hspace{3cm}\left\{ \begin{array}{cllll}\left. \begin{array}{rllll} -\Delta u & = & \mu u|u|^{p}e^{u^{2}} +\lambda h(x)\\ u & > & 0 \end{array}\right\} \;\; \text{in} \; \Omega, \\ u\;\;=\;\; 0 \;\;\text{on}\;\; \partial \Omega,\hspace{1.355cm} \end{array}\right.\hspace{2cm}$$ where $0\le p<\infty, \mu,\lambda >0, \Omega\subset \mathbb R^{2}$ is a bounded domain and $h \geq 0$ in $\Omega$ with $\| h \|_{L^{2}(\Omega)}=1.$ We show that there exist real numbers $0<\lambda_{*} \le \lambda^{*}$ such that the above problem admits at least two solutions for all $\lambda\in (0,\lambda_{*})$ and no solution for $\lambda > \lambda^{*}$.

#### Article information

Source
Differential Integral Equations, Volume 18, Number 6 (2005), 681-698.

Dates
First available in Project Euclid: 21 December 2012

Prashanth, S.; Sreenadh, K. Multiplicity of solutions to a nonhomogeneous elliptic equation in $\Bbb R^2$. Differential Integral Equations 18 (2005), no. 6, 681--698. https://projecteuclid.org/euclid.die/1356060176