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A PRIORI BOUNDS AND MULTIPLE SOLUTIONS FOR SUPERLINEAR INDEFINITE ELLIPTIC PROBLEMS
 

Summary: A PRIORI BOUNDS AND MULTIPLE SOLUTIONS FOR
SUPERLINEAR INDEFINITE ELLIPTIC PROBLEMS
H. Amann and J. L opez-G omez
Institute for Mathematics, University of Zurich
Winterthurerstr. 190, CH-8057 Zurich, Switzerland
Departamento de Matematica Aplicada
Universidad Complutense, E-28040 Madrid, Spain
Abstract. In this work we study existence and multiplicity questions for positive
solutions of second order semilinear elliptic boundary value problems, where the
nonlinearity is multiplied by a weight function which is allowed to change sign and
vanish on sets of positive measure. We do not impose a variational structure so
that techniques from the calculus of variations are not applicable. Under various
qualitative assumptions on the nonlinearity we establish a priori bounds and employ
bifurcation and xed point index theory to prove existence and multiplicity results
for positive solutions. In an appendix we derive interior Lp -estimates for general
elliptic systems of arbitrary order under minimal smoothness hypotheses. Special
instances of these results are used in the derivation of a priori bounds.
1 Introduction. In this paper we analyze existence and multiplicity questions for
positive solutions of
Au = u + a(x)f(x; u)u

  

Source: Amann, Herbert - Institut für Mathematik, Universität Zürich

 

Collections: Mathematics