The existence of solutions to a class of semilinear differential equations. (English) Zbl 0818.34013

The paper deals with the problem of the existence of solutions to the differential equation \(-(\alpha^{p- 1}(t) | u'|^{p- 2} u')'= f(t, u)+ h(t)\), \(t\in [0, 1]\) fulfilling periodic, Dirichlet, Neumann or mixed boundary conditions. It is supposed that \(1< p< \infty\), \(\alpha\in W^{1, \infty}(\mathbb{R})\) and \(0< m\leq \alpha(t)\leq M\).
Conditions which guarantee the existence of a solution to the given problem for all \(h\in L^ 1(0, 1)\), provided the asymptotes of \(f(t, u)\) as \(u\to \pm\infty\) interfere properly with the Fučík spectrum are given. To this aim, a detailed investigation of the eigenvalues and of the Fučík spectrum of the corresponding differential operators is done.
Reviewer: M.Tvrdý (Praha)


34B15 Nonlinear boundary value problems for ordinary differential equations