Electron. J. Diff. Eqns., Vol. 2008(2008), No. 152, pp. 1-19.

Bifurcation and multiplicity results for a nonhomogeneous semilinear elliptic problem

Kuan-Ju Chen

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.

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Kuan-Ju Chen
Department of Applied Science, Naval Academy
P.O. BOX 90175 Zuoying, Taiwan
email: kuanju@mail.cna.edu.tw

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