## 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.

### MSC:

 76B99 Incompressible inviscid fluids

### Citations:

Zbl 0583.35002; Zbl 0436.76073