Solutions to some singular nonlinear boundary value problems. (English) Zbl 1293.35324

Medak, Beata; Tret’yakov, Alexey A.; Żołądek, Henryk

Topol. Methods Nonlinear Anal. 41, No. 2, 255-265 (2013).

The \(p\)-regularity theory is applied to some equations of mathematical physics. The first one is a homogeneous Dirichlet boundary value problem for an equation of rod bending, i.e., \[ \frac{d^2u}{dx^2} + (1 + \epsilon)(u + u^2) = 0, \quad u(0) = u(\pi) = 0. \] The application of this theory gives existence and uniqueness of a nonzero solution for sufficiently small \(|\epsilon|\) together with its asymptotical estimate (according to \(\epsilon\)).
The second one is a problem for a nonlinear membrane equation \[ \Delta u + (10 + \epsilon)\phi(u) = 0, \quad u|_{\partial\Omega} = 0, \] where \(\Omega = [0,\pi]^2\), \(\Delta\) stands for the Laplace operator, \(\epsilon\) is a small real parameter, and the function \(\phi\) satisfies \(\phi(0) = 0\), \(\phi'(0) = 1\) and \(10\phi''(0) = a\neq 0\). The existence of three nonzero solutions is obtained together with their estimates.

Reviewer: Jan Tomeček (Olomouc)

MSC:

35Q74	PDEs in connection with mechanics of deformable solids
34B15	Nonlinear boundary value problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations

Keywords:

\(p\)-regularity; bifurcation; nonlinear boundary value problem