Differential and Integral Equations

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

S. Prashanth and K. Sreenadh

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)


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

Export citation