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Global bifurcation result for the \(p\)-biharmonic operator. (English) Zbl 0983.35099

The authors prove that the nonlinear eigenvalue problem \[ \Delta(|\Delta u|^{p-2}\Delta u)= \lambda|u|^{p- 2}u\quad\text{in }\Omega\subset \mathbb{R}^n,\quad p\in (1,+\infty), \]
\[ u= \Delta u= 0\quad\text{on }\partial\Omega, \] has a principal positive eigenvalue \(\lambda_1(p)\), which is simple and isolated. Exists a strictly positive eigenfunction \(u_1(p)\) in \(\Omega\) associated with \(\lambda_1(p)\) and satisfying \((\partial u_1/\partial n)< 0\) on \(\partial\Omega\). \(\lambda_1(p)\) is a bifurcation point of \[ \Delta(|\Delta|^{p- 2}\Delta u)= \lambda|u|^{p-2} u+g(x,\lambda,u)\quad \text{in }\Omega, \]
\[ u= \Delta u= 0\quad\text{on }\partial\Omega, \] from which a global continuum of nontrivial solutions emanates. The one-dimensional case (\(n=1\), \(\Omega= (0,1)\)) is investgated in detail.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
37G10 Bifurcations of singular points in dynamical systems
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