Existence of solutions to a superlinear \(p\)-Laplacian equation. (English) Zbl 1011.35062

Liu, Shibo

Electron. J. Differ. Equ. 2001, Paper No. 66, 6 p. (2001).

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.

Reviewer: Hans-Christoph Grunau (Bayreuth)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B34	Resonance in context of PDEs
35A07	Local existence and uniqueness theorems (PDE) (MSC2000)
49J35	Existence of solutions for minimax problems

Keywords:

Morse theory; subcritical growth; first eigenvalue; second eigenvalue; Dirichlet problem; corresponding variational functional