##
**Nonlinear eigenvalue problem associated to capillary surfaces.**
*(English)*
Zbl 0702.76024

We study a nonlinear eigenvalue problem for the model equation appearing in capillary surfaces. Consider a membrane which is framed horizontally and filled up with fluid. Let the frame be given by a circle with radius R and denote by \(u=(u^ 1(x),u^ 2(x),u^ 3(x))\), \(x\in S_ R\equiv \{x\in {\mathbb{R}}^ 2:| x| <R\}\), the deformed surface in equilibrium filled up with fluid. Then u satisfies the equation
\[
(1)\;TH[u](x)=mg u^ 3(x)\text{ in } S_ R
\]
with the boundary condition
\[
(2)\;(u^ 1(x),u^ 2(x))\in \partial S_ R,\quad u^ 3(x)=0\text{ on } \partial S_ R
\]
and the constraint (3) \(u^ 3(x)<0\) in \(S_ R.\) Here we denote by H[u](x) the mean curvature of the surfaces \(u=(u^ 1(x),u^ 2(x),u^ 3(x))\), \(x\in S_ R\), at u(x), and T, m and g are positive constants which are a tension of membrane, density of fluid and acceleration of gravity, respectively. As for the reduction of the equation (1) and physical meanings, see e.g. Chapter 1 in the book by R. Finn [Equilibrium capillary surfaces (1986; Zbl 0583.35002)].

Put \(\lambda =mg/T\) and let us consider the problem to give a relations between solutions of (1)\(\sim (3)\) and the parameter \(\lambda >0\). It is easy to see that a solution bifurcates at the first eigenvalue \(\lambda =\lambda_ 0\) of the Laplacian -\(\Delta\) with Dirichlet boundary condition from the trivial solution \(u(x)=(x_ 1,x_ 2,0)\) and further the branch of solutions is subcritical. But it seems to be difficult to analyze the global behavior of this branch. By P. Concus and R. Finn [Philos. Trans. R. Soc. Lond., A 292, 307-340 (1979; Zbl 0436.76073)], it is shown that there is a constant \(\delta <2.888\) such that there are no solutions of (1), (2), (3) for \(\lambda\) greater than \({\bar \lambda}\equiv \delta^ 2/R^ 2\). Compared with this, we have

Theorem. - There exists a positive constant \({\underline \lambda}\) such that there are no solutions of (1), (2), (3) for \(0<\lambda <{\underline \lambda}\), that is, the branch of solutions does not approach asymptotically to \(\lambda =0.\)

It is our aim in this paper to give the proof of this theorem.

Put \(\lambda =mg/T\) and let us consider the problem to give a relations between solutions of (1)\(\sim (3)\) and the parameter \(\lambda >0\). It is easy to see that a solution bifurcates at the first eigenvalue \(\lambda =\lambda_ 0\) of the Laplacian -\(\Delta\) with Dirichlet boundary condition from the trivial solution \(u(x)=(x_ 1,x_ 2,0)\) and further the branch of solutions is subcritical. But it seems to be difficult to analyze the global behavior of this branch. By P. Concus and R. Finn [Philos. Trans. R. Soc. Lond., A 292, 307-340 (1979; Zbl 0436.76073)], it is shown that there is a constant \(\delta <2.888\) such that there are no solutions of (1), (2), (3) for \(\lambda\) greater than \({\bar \lambda}\equiv \delta^ 2/R^ 2\). Compared with this, we have

Theorem. - There exists a positive constant \({\underline \lambda}\) such that there are no solutions of (1), (2), (3) for \(0<\lambda <{\underline \lambda}\), that is, the branch of solutions does not approach asymptotically to \(\lambda =0.\)

It is our aim in this paper to give the proof of this theorem.

### MSC:

76B99 | Incompressible inviscid fluids |