Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 152, pp. 1-19.

Title: Bifurcation and multiplicity results for a
       nonhomogeneous semilinear elliptic problem
   
Author: Kuan-Ju Chen (Naval Academy, Zuoying, Taiwan)

Abstract:
 In this article we consider the  problem
 $$\displaylines{
 -\Delta u(x)+u(x)=\lambda (a(x)u^{p}+h(x))\quad\hbox{in }\mathbb{R}^N, \cr
 u\in H^{1}(\mathbb{R}^N),\quad u>0\quad\hbox{in }\mathbb{R}^N,
 }$$
 where $\lambda$ is a positive parameter.
 We assume there exist $\mu >2$ and $C>0$ such that
 $a(x)-1\geq -Ce^{-\mu |x|}$  for all $x\in \mathbb{R}^N$.
 We prove that there exists a positive $\lambda^*$
 such that there are at least two positive solutions for
 $\lambda\in (0,\lambda^*)$ and a unique positive solution
 for $\lambda =\lambda^*$. Also we show that
 $(\lambda ^{*},u(\lambda^*))$ is a bifurcation point in
 $C^{2,\alpha}(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$.
  
Submitted May 13, 2008. Published November 04, 2008.
Math Subject Classifications: 35A15, 35J20, 35J25, 35J65.
Key Words: Nonhomogeneous semilinear elliptic problems; bifurcation.