Eigenvalues, global bifurcation and positive solutions for a class of nonlocal elliptic equations. (English) Zbl 1368.35094

Dai, Guowei

Topol. Methods Nonlinear Anal. 48, No. 1, 213-233 (2016).

Summary: In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem:
\[ \begin{cases} -\bigg(a+b\int_\Omega | \nabla u|^2 dx\bigg)\Delta u=\lambda u+h(x,u,\lambda)\quad&\text{in}\quad \Omega,\\ u=0&\text{on}\quad\Omega. \end{cases} \]
Under some natural hypotheses on \(h\), we show that \((a\lambda_1,0)\) is a bifurcation point of the above problem. As an application of the above result we shall determine the interval of \(\lambda\), in which there exist positive solutions for the above problem with \(h(x,u;\lambda)=\lambda f(x,u)-\lambda u\), where \(f\) is asymptotically linear at zero and asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we provide a positive answer to an open problem involving the case \(a=0\).

MSC:

35J25

Boundary value problems for second-order elliptic equations

Keywords:

Kirchhoff type equation; positive solutions; bifurcation