# American Institute of Mathematical Sciences

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.

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##### 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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