### 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,

< 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.

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

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