an:01667109
Zbl 1011.35062
Liu, Shibo
Existence of solutions to a superlinear \(p\)-Laplacian equation
EN
Electron. J. Differ. Equ. 2001, Paper No. 66, 6 p. (2001).
00062468
2001
j
35J65 35B34 35A07 49J35
Morse theory; subcritical growth; first eigenvalue; second eigenvalue; Dirichlet problem; corresponding variational functional
Existence of nontrivial solutions for the Dirichlet problem \(-\Delta_p u= f(x,u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) is studied. Here, \(-\Delta_p u\) is the \(p\)-Laplacian, \(p>1\), and \(f\) is a Caratheodory function, which has ``superlinear'' and subcritical growth for large \(|u|\). For small \(|u|\), \(f\) is assumed to behave like \(\lambda|u|^{p-2}u\), where \(\lambda\) is between the first and the second Dirichlet eigenvalue of the \(p\)-Laplacian. The corresponding variational functional is studied by means of Morse theory.
Hans-Christoph Grunau (Bayreuth)