October  2016, 21(8): 2649-2662. doi: 10.3934/dcdsb.2016066

On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities

1. 

Department of Mathematics, Northwest Normal University, Lanzhou, 730070

2. 

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received  October 2015 Revised  June 2016 Published  September 2016

Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(\beta) = \beta, \ f(s) < s$ for $s\in (0,\beta), \ f(s) > s$ for $s \in (\beta, \infty)$ and $f'(\beta)>\lambda^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$ -\Delta u+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_\nu u=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $\beta$ provided that $f'(\beta) >\lambda^r_k$ for $k>2$, where $\lambda^r_k$ is the $k$-th radial eigenvalue of $\Delta + I$ in the unit ball with Neumann boundary conditions. In this paper, we show that the answer is yes in the case of linearly bounded nonlinearities.
Citation: Ruyun Ma, Tianlan Chen, Yanqiong Lu. On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2649-2662. doi: 10.3934/dcdsb.2016066
References:
[1]

D. Bonheure, B. Noris and T. Weth, Increasing radial solutions for Neumann problems without growth restrictions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 573-588. doi: 10.1016/j.anihpc.2012.02.002.

[2]

D. Bonheure, E. Serra and P. Tilli, Radial positive solutions of elliptic systems with Neumann boundary conditions, J. Funct. Anal., 265 (2013), 375-398. doi: 10.1016/j.jfa.2013.05.027.

[3]

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468. doi: 10.1016/j.jde.2011.09.026.

[4]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. Lond. Math. Soc., 34 (2002), 533-538. doi: 10.1112/S002460930200108X.

[5]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1973/74), 1069-1076.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[7]

A. C. Lazer and P. J. Mckenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648-655. MR1264537 doi: 10.1006/jmaa.1994.1049.

[8]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, in: Research Notes in Mathematics, vol. 426, Chapman & Hall/CRC, Boca Raton, Florida, 2001.

[9]

R. Ma and B. Thompson, Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal., 59 (2004), 707-718. doi: 10.1016/S0362-546X(04)00280-9.

[10]

A. Miciano and R. Shivaji, Multiple positive solutions for a class of semipositone Neumann two point boundary value problems, J. Math. Anal. Appl., 178 (1993), 102-115. doi: 10.1006/jmaa.1993.1294.

[11]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 81 (1983), 181-197. doi: 10.1007/BF00250651.

[12]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.

[13]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[14]

E. Serra and P. Tilli, Monotonicity constraints and supercritical Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 63-74. doi: 10.1016/j.anihpc.2010.10.003.

[15]

W. Walter, Ordinary Differential Equations, Translated from the sixth German (1996) edition by Russell Thompson. Graduate Texts in Mathematics 182. Springer-Verlag, New York, 1998.

show all references

References:
[1]

D. Bonheure, B. Noris and T. Weth, Increasing radial solutions for Neumann problems without growth restrictions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 573-588. doi: 10.1016/j.anihpc.2012.02.002.

[2]

D. Bonheure, E. Serra and P. Tilli, Radial positive solutions of elliptic systems with Neumann boundary conditions, J. Funct. Anal., 265 (2013), 375-398. doi: 10.1016/j.jfa.2013.05.027.

[3]

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468. doi: 10.1016/j.jde.2011.09.026.

[4]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. Lond. Math. Soc., 34 (2002), 533-538. doi: 10.1112/S002460930200108X.

[5]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1973/74), 1069-1076.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[7]

A. C. Lazer and P. J. Mckenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648-655. MR1264537 doi: 10.1006/jmaa.1994.1049.

[8]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, in: Research Notes in Mathematics, vol. 426, Chapman & Hall/CRC, Boca Raton, Florida, 2001.

[9]

R. Ma and B. Thompson, Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal., 59 (2004), 707-718. doi: 10.1016/S0362-546X(04)00280-9.

[10]

A. Miciano and R. Shivaji, Multiple positive solutions for a class of semipositone Neumann two point boundary value problems, J. Math. Anal. Appl., 178 (1993), 102-115. doi: 10.1006/jmaa.1993.1294.

[11]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 81 (1983), 181-197. doi: 10.1007/BF00250651.

[12]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.

[13]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[14]

E. Serra and P. Tilli, Monotonicity constraints and supercritical Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 63-74. doi: 10.1016/j.anihpc.2010.10.003.

[15]

W. Walter, Ordinary Differential Equations, Translated from the sixth German (1996) edition by Russell Thompson. Graduate Texts in Mathematics 182. Springer-Verlag, New York, 1998.

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