Solutions to some singular nonlinear boundary value problems (Q378461)

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., NEWLINE\[NEWLINE \frac{d^2u}{dx^2} + (1 + \epsilon)(u + u^2) = 0, \quad u(0) = u(\pi) = 0. NEWLINE\]NEWLINE The application of this theory gives existence and uniqueness of a nonzero solution for sufficiently small \(|\epsilon|\) together with its asymptotical estimate (according to \(\epsilon\)).NEWLINENEWLINEThe second one is a problem for a nonlinear membrane equation NEWLINE\[NEWLINE \Delta u + (10 + \epsilon)\phi(u) = 0, \quad u|_{\partial\Omega} = 0, NEWLINE\]NEWLINE 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.