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

Santiago Cano-Casanova

DOI: http://dx.doi.org/10.12775/TMNA.2004.003

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.

Keywords


Principal eigenvalue; maximum principle; positive solutions; compact solution components; bifurcation theory; a priori bounds

Full Text:

FULL TEXT

Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism