Spectral asymptotics of nonlinear elliptic two-parameter problems
Spectral asymptotics is derived for a nonlinear two-parameter problem connected with the Laplace operator on three- and higher-dimensional domains, namely \(\Delta u+ \mu f(u)= \lambda g(u)\). The approach is different from that used by the author and by others for linear two-parameter problems. Under some specified assumptions on the nonlinear functions \(f\) and \(g\) the author first proves the existence of variational eigenvalues. Then the optimal estimates of \(\lambda(\mu)\) from above and below are derived. Finally, the author proves the fundamental theorem which gives the asymptotic formula showing the dependence of parameter \(\lambda\) on \(\mu\).
scientific article; zbMATH DE number 1539292
scientific article