Bifurcation Theory and Application to Semilinear Problems near the Resonance Parameter
David Arcoya, José Luis Gámez · Birkhäuser Boston eBooks · 2001
In this note, we present without proof some of the main results obtained in [6] about Bifurcation Theory and related problems: resonance and the antimaximum principle, and we prove also some new applications about existence of solutions for semilinear elliptic problems near the resonance parameter. Specifically, we consider here the semilinear elliptic boundary value problem 1.1 $$ \begin{gathered} - \Delta u(x) = \lambda m(x)u + g(\lambda ,x,u) if x \in \Omega , \hfill \\ u(x) = 0 if x \in \partial \Omega , \hfill \\ \end{gathered} $$ for a bounded domain S2 C RN with sufficiently smooth boundary as-2 and A E R, under the hypothesis:(H) There exists rEN,[+oo] such that mE Lr (S2)with m+ =max{m, 0} 0 0and g: R x S2 x R R is a continuous function.