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 [2, Section 4] to get a priori bounds for the positive solutions. Then, using the theory of [1], [4] and [5], 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 ([18], [14]). 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. 2000 Mathematics Subject Classification. 35B50, 35B32, 35B60, 35B45, 35J65.

Local and global bifurcation results for a semilinear boundary value problem

“…Such results were first obtained in [2] and these have been strengthened in [1,3,4,13] and [9]. Suppose…”

“…The author would like to thank the referee of an earlier version of this paper for bringing to his attention recent work on the problem of closed loops of solutions, viz., [4,5,7] which has led to a major revision of the work.…”

“…Such results have been established in [4][5][6][7]14]. In particular an abstract theorem has been proved in [7] which gives sufficient conditions for two principal eigenvalues to be joined by a continuum of positive solutions.…”

Some indefinite nonlinear eigenvalue problems

“…Problem (12) has attracted a great deal of attention during last years (see for example [1], [3], [5], [6], [18] and [24]) when m ≡ 1 in the first term on the right-hand side of (12) and in [11], [12] and [13] with the right-hand side of the form µh(x)u + g(x)u p and restrictive conditions on h and g which are not satisfied in our case. In [16] was proved (see Fig.…”

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

“…The proof of Theorem 1.1 is based on the abstract result of the global bifurcation theory proposed by López-Gómez [13,Theorem 6.4.3]. For more recent works on the global bifurcation analysis, we refer to López-Gómez and Molina-Meyer [14], Dancer [9], Cano-Casanova [5], and Brown [3].…”