A note on a nonlinear elliptic problem with a nonlocal coefficient
The author studies a nonlocal and nonlinear elliptic problem \[ \begin{cases} -a(\int_\Omega| \nabla u|^2 dx) \Delta u=\lambda u+u^p & \text{in } \Omega,\\ u=0 & \text{on }\partial\Omega \end{cases} \tag{P} \] where \(\Omega\subset \mathbb R^N\), \(N\geq 3\) is a bounded domain with smooth boundary \(\partial \Omega\), \(a\) is a nondegenerate continuous function, \(p>1\) and \(\lambda\in \mathbb R\). There are obtained several effects of the nonlocal coefficient \(a\) on the structure of the solution set of (P). Introducing a scaling observation and describing the solution set by using that of an associated semilinear problem the author shows unbounded continua of solutions \((\lambda, u)\). A rich variety of new bifurcation and multiplicity results are observed. It is also proved that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. There are given some remarks from the variational point of view.
scientific article; zbMATH DE number 6522740
scientific article