Figure 1.
Loop type components of nontrivial non-negative solutions of $(P_\lambda)$ .
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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 |
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 [
Keywords:
Semilinear elliptic problem,
non-negative solution,
loop type component,
bifurcation,
topological method.
Mathematics Subject Classification:
Primary: 35J25, 35J61; Secondary: 35B32.
Citation:
Humberto Ramos Quoirin, Kenichiro Umezu. A loop type component in the non-negative solutions set of an indefinite elliptic problem. Communications on Pure and Applied Analysis,
2018, 17
(3)
: 1255-1269.
doi: 10.3934/cpaa.2018060
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References:
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H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.
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| [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.
|
| [3] |
K. J. Brown,
Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.
|
| [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.
|
| [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.
|
| [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.
|
| [7] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
|
| [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. |
| [9] |
J. López-Gómez,
Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC, Boca Raton, FL, 2001. |
| [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.
|
| [11] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
|
| [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.
|
| [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.
|
| [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.
|
| [15] |
G. T. Whyburn,
Topological Analysis, Second, revised edition, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1964. |
show all references
References:
| [1] |
H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.
|
| [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.
|
| [3] |
K. J. Brown,
Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.
|
| [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.
|
| [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.
|
| [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.
|
| [7] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
|
| [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. |
| [9] |
J. López-Gómez,
Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC, Boca Raton, FL, 2001. |
| [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.
|
| [11] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
|
| [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.
|
| [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.
|
| [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.
|
| [15] |
G. T. Whyburn,
Topological Analysis, Second, revised edition, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1964. |
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 7.
The behaviors of $\Sigma_{\epsilon}^\pm$ .
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