Existence and bifurcation of positive global solutions for parametrized nonhomogeneous elliptic equations involving critical exponents (Q2792462)

This paper deals with the semilinear critical problem: NEWLINE\[NEWLINE-\Delta u+u=u^{2^{\ast }-1}+\mu f\quad \text{in}\, \mathbb{R}^{N};\qquad u>0\quad \text{in}\, \mathbb{R}^{N}. \tag{1}NEWLINE\]NEWLINE Here \(\mu >0\) is a parameter, \(f\in H^{-1}(\mathbb{R}^{N})\), \(f\geq 0\) and \( f\not \equiv 0\). In the literature, there are some works studying the existence of multiple solutions and bifurcation phenomena under some assumption on the decay rate of \(f\).NEWLINENEWLINEIn this paper, the author does not assume the decay rate on \(f\) but assumes only uniform boundedness of \(f\). The main result is: There exists a positive constant \(\mu ^{\ast}\) such that (1) possesses at least two positive solutions for \(0<\mu <\mu ^{\ast}\), a unique solution for \(\mu =\mu ^{\ast}\) and no positive solution for \(\mu >\mu ^{\ast}\).