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, < i> A priori bounds and multiple solutions for superlinear indefinite elliptic problems< /i> , J. Differential Equations < b> 146< /b> (1998), 336–374] to get a priori bounds for the positive solutions. Then, using the theory of [H. Amann, < i> Dual semigroups and second order linear elliptic boundary value problems< /i> , Israel J. Math. < b> 45< /b> (1983), 225–254], [S. Cano-Casanova, < i> Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems< /i> , Nonlinear Anal. < b> 49< /b> (2002), 361–430] and [S. Cano-Casanova and J. López-Gómez, < i> Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems< /i> , J. Differential Equations < b> 178< /b> (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 ([< i> Some global results for nonlinear eigenvalue problems< /i> , J. Funct. Anal. < b> 7< /b> (1971), 487–513], [J. López-Gómez, < i> Spectral Theory and Nonlinear Functional Analysis< /i> , 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.
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