On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting
The existence of principal eigenvalues (i.e., having a positive eigenfunction) for some quasilinear elliptic operators generalizing the \(p\)-Laplacian is proved. More precisely, this result is proved for the eigenvalue problem \(-\text{div} (A(|\nabla u|^2 \nabla u)= \lambda\psi(u)\) in \(D\), \(u=0\) on \(\partial D\), where \(D\) is a smooth bounded domain and the nonlinearities \(A\) and \(\psi\) correspond to the case of very fast-growing nonlinear functions, as for example \(e^{s^2}-1\). It is shown that under suitable assumptions on the nonlinearities the problem can be solved in the framework of eigenvalues to variational inequalities and then a theorem by Kubrusky on the existence of Lagrange multipliers is applied. Some existence results for an associated nonlinear problem are also obtained.
scientific article; zbMATH DE number 1310757