Nonlinear eigenvalue problem for a modified capillary surface equation. (English) Zbl 0807.35044

The authors study the eigenvalue problem \[ -p \text{div} \left( {| \nabla u |^{2p - 2} \nabla u \over \sqrt {1+ | \nabla u |^{2p}}} \right) = \lambda f(x,y),\;u \geq 0, \quad \text{in} \quad \Omega, \qquad u = 0 \quad \text{on} \quad \partial \Omega \] in a bounded domain \(\Omega \subset \mathbb{R}^ n\), \(n\geq 2\), \(\lambda > 0\), \(p > 1\). Using direct variational methods, monotone operator theory and the Ambrosetti-Rabinowitz form of the mountain pass lemma (without Palais- Smale condition) they obtain a number of results on existence and limiting behavior of eigenfunctions. The particular case \(f = qu^{q - 1}\) is illustrative. For this case, setting \(Mu_ \lambda = \| \nabla u_ \lambda \|_{L^ p (\Omega)}\), they show: (i) if \(1<q<p\) then there is a nonzero solution \(u_ \lambda\) for any \(\lambda > 0\); it satisfies \(Mu_ \lambda \to 0\) as \(\lambda \to 0\), \(Mu_ \lambda \to \infty\) as \(\lambda \to \infty\). (ii) If \(p < q < 2p\), then there exists \(\lambda_ *>0\) such that for any \(0 < \lambda < \lambda_ *\), there are two nonzero solutions \(u_ \lambda\), \(v_ \lambda\) for which \(Mu_ \lambda \to 0\), \(Mv_ \lambda \to \infty\) as \(\lambda \to 0\). (iii) if \(q > 2p\), then there is a nonzero solution \(u_ \lambda\) that satisfies \(Mu_ \lambda \to \infty\) as \(\lambda \to 0\), \(Mu_ \lambda \to 0\) as \(\lambda \to \infty\).
Reviewer: R.Finn (Stanford)


35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
76B45 Capillarity (surface tension) for incompressible inviscid fluids
35J20 Variational methods for second-order elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs