## Differential and Integral Equations

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

### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060176

**Mathematical Reviews number (MathSciNet)**

MR2136705

**Zentralblatt MATH identifier**

1212.35154

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 35J20: Variational methods for second-order elliptic equations 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

#### Citation

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