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.