Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition
Using some topological methods and nonlinear analytic techniques, the authors prove a global bifurcation result from infinity for the \(p\)-Laplacian problem \[ -(\varphi_p(u'))'=\lambda a(t)\varphi_p(u)+g(t,u;\lambda),\,\,\,t\in (0,1);\,\,\,u(0)=u(1)=0, \] where \(\varphi_p(s)=|s|^{p-2}s\), and \(a:[0,1]\to [0,+\infty)\) and \(f:\mathbb{R}\to\mathbb{R}\) are continuous functions, which satisfy some additional assumptions. As an application, they investigate the global behavior of the components of nodal solutions of the nonlinear eigenvalue problem \[ (\varphi_p(u'))'+\lambda a(t)f(u)=0,\,\,\,t\in (0,1);\,\,\,u(0)=u(1)=0. \] The intervals for the parameter \(\lambda\) which ensure the existence of multiple nodal solutions if \(f_0\in (0,\infty)\) and \(f_{\infty}\in (0,\infty)\) are also established, where \(f_0=\lim_{s\to 0}f(s)/\varphi_p(s)\) and \(f_{\infty}=\lim_{|s|\to\infty}f(s)/\varphi_p(s)\).
scientific article; zbMATH DE number 6099585
scientific article