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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

Figure 1.  Loop type components of nontrivial non-negative solutions of $(P_\lambda)$.
Figure 2.  Possible bifurcation diagrams for $\mathcal{C}_\epsilon$: the case $\int_\Omega a > 0$.
Figure 3.  Possible bifurcation diagram for $\mathcal{C}_\epsilon$: the case $\int_\Omega a = 0$.
Figure 4.  Three possibilities for the bounded component $\mathcal{C}_{\epsilon, \rho}$.
Figure 5.  Possible bifurcation diagrams for $\mathcal{C}_0^\prime$ when $\int_\Omega a \geq 0$.
Figure 6.  A bifurcation diagram for $\mathcal{C}_0$ at $(0, 0)$: the case $\int_\Omega a < 0$.
Figure 7.  The behaviors of $\Sigma_{\epsilon}^\pm$.
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