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May  2018, 17(3): 1255-1269. doi: 10.3934/cpaa.2018060

## A loop type component in the non-negative solutions set of an indefinite elliptic problem

 1 Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile 2 Department of Mathematics, Faculty of Education, Ibaraki University, Mito 310-8512, Japan

* Corresponding author

Received  July 2017 Revised  September 2017 Published  January 2018

Fund Project: The first author was supported by the FONDECYT grants 1161635, 1171532 and 1171691. The second author was supported by JSPS KAKENHI Grant Number 15K04945.

We prove the existence of a loop type component of non-negative solutions for an indefinite elliptic equation with a homogeneous Neumann boundary condition. This result complements our previous results obtained in [12], where the existence of another loop type component was established in a different situation. Our proof combines local and global bifurcation theory, rescaling and regularizing arguments, a priori bounds, and Whyburn's topological method. A further investigation of the loop type component established in [12] is also provided.

Citation: Humberto Ramos Quoirin, Kenichiro Umezu. A loop type component in the non-negative solutions set of an indefinite elliptic problem. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1255-1269. doi: 10.3934/cpaa.2018060
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.   Google Scholar [2] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.   Google Scholar [3] K. J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.   Google Scholar [4] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.   Google Scholar [5] S. Cano-Casanova, Compact components of positive solutions for superlinear indefinite elliptic problems of mixed type, Topol. Methods Nonlinear Anal., 23 (2004), 45-72.   Google Scholar [6] S. Cingolani and J. L. Gámez, Positive solutions of a semilinear elliptic equation on $\mathbf{R}^ N$ with indefinite nonlinearity, Adv. Differential Equations, 1 (1996), 773-791.   Google Scholar [7] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.   Google Scholar [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar [9] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC, Boca Raton, FL, 2001.  Google Scholar [10] J. López-Gómez and M. Molina-Meyer, Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas, J. Differential Equations, 209 (2005), 416-441.   Google Scholar [11] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.   Google Scholar [12] H. Ramos Quoirin and K. Umezu, An indefinite concave-convex equation under a Neumann boundary condition I, Israel J. Math., 220 (2017), 103-160.   Google Scholar [13] H. Ramos Quoirin and K. Umezu, An indefinite concave-convex equation under a Neumann boundary condition II, Topol. Methods Nonlinear Anal., 49 (2017), 739-756.   Google Scholar [14] K. Umezu, Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions, Nonlinear Differential Equations Appl. NoDEA, 17 (2010), 323-336.   Google Scholar [15] G. T. Whyburn, Topological Analysis, Second, revised edition, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1964.  Google Scholar

show all references

##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.   Google Scholar [2] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.   Google Scholar [3] K. J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.   Google Scholar [4] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.   Google Scholar [5] S. Cano-Casanova, Compact components of positive solutions for superlinear indefinite elliptic problems of mixed type, Topol. Methods Nonlinear Anal., 23 (2004), 45-72.   Google Scholar [6] S. Cingolani and J. L. Gámez, Positive solutions of a semilinear elliptic equation on $\mathbf{R}^ N$ with indefinite nonlinearity, Adv. Differential Equations, 1 (1996), 773-791.   Google Scholar [7] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.   Google Scholar [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar [9] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC, Boca Raton, FL, 2001.  Google Scholar [10] J. López-Gómez and M. Molina-Meyer, Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas, J. Differential Equations, 209 (2005), 416-441.   Google Scholar [11] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.   Google Scholar [12] H. Ramos Quoirin and K. Umezu, An indefinite concave-convex equation under a Neumann boundary condition I, Israel J. Math., 220 (2017), 103-160.   Google Scholar [13] H. Ramos Quoirin and K. Umezu, An indefinite concave-convex equation under a Neumann boundary condition II, Topol. Methods Nonlinear Anal., 49 (2017), 739-756.   Google Scholar [14] K. Umezu, Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions, Nonlinear Differential Equations Appl. NoDEA, 17 (2010), 323-336.   Google Scholar [15] G. T. Whyburn, Topological Analysis, Second, revised edition, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1964.  Google Scholar
Loop type components of nontrivial non-negative solutions of $(P_\lambda)$.
Possible bifurcation diagrams for $\mathcal{C}_\epsilon$: the case $\int_\Omega a > 0$.
Possible bifurcation diagram for $\mathcal{C}_\epsilon$: the case $\int_\Omega a = 0$.
Three possibilities for the bounded component $\mathcal{C}_{\epsilon, \rho}$.
Possible bifurcation diagrams for $\mathcal{C}_0^\prime$ when $\int_\Omega a \geq 0$.
A bifurcation diagram for $\mathcal{C}_0$ at $(0, 0)$: the case $\int_\Omega a < 0$.
The behaviors of $\Sigma_{\epsilon}^\pm$.
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