A bifurcation problem for a one-dimensional \(p\)-Laplace elliptic problem with non-odd absorption (Q6048938)

The authors investigate a bifurcation problem for the one-dimensional \(p\)-Laplace elliptic problem \[ \begin{cases} -\left( \left\vert \phi_{x}\right\vert ^{p-2}\phi_{x}\right)_{x} = \lambda\left( \left\vert \phi\right\vert ^{q-2}\phi-f\left( \phi\right)\right) \text{ in } \left(0,1\right),\\ \qquad \phi\left( 0\right) =\phi\left( 1\right) =0,\end{cases} \] where \(p,q>1,\) \(\lambda>0\) is a real parameter and \(f\in C\left(\mathbb{R}\right)\) is a function. They study how the structure of set \(E_{\lambda}\) of the solutions of the problem depends on \(\lambda.\) Unlike studies in the literature, they consider more general functions \(f\) which are not necessarily a power, not necessarily odd and \(p\) is not necessarily equal to \(q.\) Depending on these, they obtain a sequence of bifurcations.