## 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

### Keywords:

principal eigenvalue; nonlinear eigenvalue problem
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