Some examples on solutions structures for weakly nonlinear elliptic equations (Q556488)

The author deals with a bifurcation problem for a weakly nonlinear elliptic equation, that is: \[ -\Delta u=\lambda f(u)\quad\text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega\tag{1} \] where \(f: \mathbb R\to\mathbb R\) is given and \(C^1\). Here the author produces a number of superlinear problems for which one has complete information on the shape of the bifurcation curves. The examples are where \(f\) is of the form \(u^p+\alpha u\) where \(p\) is close to \(1\) or a number of similar nonlinearities. The author shows that if \(p\) is close to \(1\), all positive solutions of (1) are nondegenerate. Moreover, he shows that in a number of cases one can obtain rather complete information about the structure of the set of positive solutions.