Bifurcation and multiplicity results for a nonhomogeneous semilinear elliptic problem

oleh: Kuan-Ju Chen

Format: Article
Diterbitkan: Texas State University 2008-11-01

Deskripsi

In this article we consider the problem $$displaylines{ -Delta u(x)+u(x)=lambda (a(x)u^{p}+h(x))quadhbox{in }mathbb{R}^N, cr uin H^{1}(mathbb{R}^N),quad u>0quadhbox{in }mathbb{R}^N, }$$ where $lambda$ is a positive parameter. We assume there exist $mu >2$ and $C>0$ such that $a(x)-1geq -Ce^{-mu |x|}$ for all $xin mathbb{R}^N$. We prove that there exists a positive $lambda^*$ such that there are at least two positive solutions for $lambdain (0,lambda^*)$ and a unique positive solution for $lambda =lambda^*$. Also we show that $(lambda ^{*},u(lambda^*))$ is a bifurcation point in $C^{2,alpha}(mathbb{R}^N)cap H^{2}(mathbb{R}^N)$.