Compact components of positive solutions for superlinear indefinite elliptic problems of mixed type

Abstract

In this paper we construct an example of superlinear indefinite weighted elliptic mixed boundary value problem exhibiting a mushroom shaped compact component of positive solutions emanating from the trivial solution curve at two simple eigenvalues of a related linear weighted boundary value problem. To perform such construction we have to adapt to our general setting some of the rescaling arguments of H. Amann and J. López-Gómez [Section 4, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), 336–374] to get a priori bounds for the positive solutions. Then, using the theory of [H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math. 45 (1983), 225–254], [S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. 49 (2002), 361–430] and [S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations 178 (2002), 123–211], we give some sufficient conditions on the nonlinearity and the several potentials of our model setting so that the set of values of the parameter for which the problem possesses a positive solution is bounded. Finally, the existence of the component of positive solutions emanating from the trivial curve follows from the unilateral results of P. H. Rabinowitz ([Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513], [J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, 2001]). Monotonicity methods, re-scaling arguments, Liouville type theorems, local bifurcation and global continuation are among the main technical tools used to carry out our analysis.