## 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)

### MSC:

 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

### Keywords:

capillary surface equation; mountain pass lemma