Global bifurcation and positive solutions for a nonlocal eigenvalue problem
DOI:
https://doi.org/10.12775/TMNA.2025.070Keywords
Bifurcation, positive solutions, nonlocal problemAbstract
We consider the following nonlocal problem \begin{equation}\label{abstract-equation}\tag{$*$} \begin{cases} \displaystyle -\Delta u+\alpha \int_{\Omega} u dx=\lambda f(u) & \text {in } \Omega,\\ u=0 & \text {on } \partial \Omega . \end{cases} \end{equation} which is a nonlocal operator consisting of a perturbation of the standard Dirichlet Laplacian by an integral of the unknown function. By employing bifurcation and topological techniques, we establish the existence of positive solutions. Furthermore, under certain appropriate conditions on the function $f$, we demonstrate that \eqref{abstract-equation} possesses two positive solutions. Additionally, we present several results concerning the nonexistence of solutions.References
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