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 & 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. doi: 10.1016/j.anihpc.2012.02.002. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems,, Indiana Univ. Math. J., 23 (): 1069. Google Scholar

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998). Google Scholar

[7]

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

[8]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, in: Research Notes in Mathematics, (2001). Google Scholar

[9]

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

[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. doi: 10.1006/jmaa.1993.1294. Google Scholar

[11]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbbR^n$,, Arch. Rational Mech. Anal., 81 (1983), 181. doi: 10.1007/BF00250651. Google Scholar

[12]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984). Google Scholar

[13]

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

[14]

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

[15]

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

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. doi: 10.1016/j.anihpc.2012.02.002. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems,, Indiana Univ. Math. J., 23 (): 1069. Google Scholar

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998). Google Scholar

[7]

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

[8]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, in: Research Notes in Mathematics, (2001). Google Scholar

[9]

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

[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. doi: 10.1006/jmaa.1993.1294. Google Scholar

[11]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbbR^n$,, Arch. Rational Mech. Anal., 81 (1983), 181. doi: 10.1007/BF00250651. Google Scholar

[12]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984). Google Scholar

[13]

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

[14]

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

[15]

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

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