Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear fractional Laplacian problems

Abstract

In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned}$$

where $\lambda >0$ and $\lim _{|x|\rightarrow \infty }f(x,u)=\overline{f}(u)$ uniformly on any compact subset of $[0,\infty )$. We prove that under suitable conditions on f and h, there exists $0<\lambda ^*<+\infty$ such that the problem has at least two positive solutions if $\lambda \in (0,\lambda ^*)$, a unique positive solution if $\lambda =\lambda ^*$, and no solution if $\lambda >\lambda ^*$. We also obtain the bifurcation of positive solutions for the problem at $(\lambda ^*,u^*)$ and further analyse the set of positive solutions.