On Semilinear Problems with Nonlinearities Depending Only on Derivatives
A. Cañada, Pavel Drábek · SIAM Journal on Mathematical Analysis · 1996
In this paper we deal with the semilinear boundary value problems (BVPs) \[ u''(t) + \lambda _1 u(t) + g\left( {t,u'(t)} \right) = f(t),\quad t \in I,\quad (Bu)(t) = 0,\quad t \in \partial I, \] where $I = [0,\pi ]$, B denotes either the Dirichlet or the Neumann or the periodic boundary conditions, respectively, and $\lambda _1 $ is the first eigenvalue of the corresponding linear problem \[ u'' + \lambda u(t) = 0,\quad t \in I,\quad (Bu)(t) = 0,\quad t \in \partial I. \] This kind of problem is very important in applications where the quantity $g(t,u')$ may be regarded as a nonlinear damping term. The nonlinear function g is supposed to be bounded and, in some cases, satisfies additional differentiability assumptions and asymptotic conditions. We emphasize the dependence of g on the derivative of the solution $u'(t)$ in order to show the qualitative difference of this case and the “classical” Landesman–Lazer-type problem in which the nonlinearity g depends only on the solution $u(t)$.