# Article

**Keywords:**

$p$-Laplacian; nonlinear Fredholm alternative; bifurcation from infinity; existence and multiplicity results

**Summary:**

This lecture follows joint result of the speaker, Petr Girg, Peter Takáč and Michael Ulm. We concentrate on the Fredholm alternative for the $p$-Laplacian at the first eigenvalue. In contrast with the linear case ($p=2$), the nonlinear case ($p\ne2$) appears to be completely different not only concerning the methods (which cannot benefit from the linear structure of the problem and the Hilbert structure of the function spaces) but also from the point of view of the results which seem to be rather surprizing. In particular, the difference between the cases $1<p<2$ and $p>2$ is quite interesting. The main tool to prove existence and multiplicity results is “the bifurcation from infinity” argument.