Nonlinear eigenvalue problem for a model equation of an elastic surface
The existence of nonzero weak solutions of the boundary value problem \[ - \gamma \text{div} \left[ {\bigl( \sqrt {1 + |\nabla u |^2} - 1 \bigr)^{\gamma - 1} \over \sqrt {1 + |\nabla u |^2}} \nabla u \right] = \lambda f(x,u) \text{ in } \Omega \tag{1} \] \[ (2) \quad u \geq 0 \text{ in } \Omega \qquad (3) \quad u = 0 \text{ on } \partial \Omega \] is discussed for \(\gamma > 1\) and \(\lambda > 0\). In (1), \(\Omega\) is a bounded domain in \(\mathbb{R}^n\). For \(n = 1\) the derivation of (1) is given. In this case (1) describes the equilibrium configuration of a nonlinear elastic string and \(\lambda\) depends on the tension force in the string. The critical points of the functional \[ I_\lambda [u] = \int_\Omega \Bigl( \sqrt {1 + |\nabla u |^2} - 1 \Bigr)^\gamma dx - \lambda \int_{\Omega} F(x,u) dx \tag{4} \] where \[ F(x,u) = \int^u_0 f(x, \xi) d \xi \tag{5} \] are the solutions of (1), (3). Under some conditions on \(F(x,u)\) the existence of the local minimizer of \(I_\lambda [u]\) is proven. The second critical point of \(I_\lambda [u]\) is obtained and the asymptotic behavior of the secondary solution is analysed. The special case, when \(f(x,u) = qu^{q - 1}\), is investigated in detail.
scientific article; zbMATH DE number 774753