March  2011, 10(2): 507-525. doi: 10.3934/cpaa.2011.10.507

Bifurcations of some elliptic problems with a singular nonlinearity via Morse index

1. 

Department of Mathematics, Henan Normal University, Xinxiang, 453007, China

2. 

Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

3. 

Department of Mathematics, East China Normal University, Shanghai 200062

Received  January 2010 Revised  May 2010 Published  December 2010

We study the boundary value problem

$\Delta u=\lambda |x|^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1)

where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function satisfying $\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$, there is a critical power $p_c (\alpha)>0$, which is decreasing in $\alpha$, such that the branch of positive solutions possesses infinitely many bifurcation points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on the shape of the domain $\Omega$. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that $p>p_c (\alpha)$ and the Morse index of any radial solution (regular or singular) in this branch is finite provided that $0 < p \leq p_c (\alpha)$. This implies that the structure of the radial solution branch of (1) changes for $0 < p \leq p_c (\alpha)$ and $p > p_c (\alpha)$.

Citation: Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure & Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507
References:
[1]

Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489. doi: doi:10.1090/S0002-9939-01-06132-9.

[2]

A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations,, Comm. Pure Appl. Math., 51 (1998), 625. doi: doi:10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.

[3]

A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations,, Indiana Univ. Math. J., 49 (2000), 1323. doi: doi:10.1512/iumj.2000.49.1887.

[4]

J. P. Burelbach, S. G. Bankoff and S. H. Davis, Nonlinear stability of evaporating/condensing liquid films,, J. Fluid Mech., 195 (1988), 463. doi: doi:10.1017/S0022112088002484.

[5]

B. Buffoni, E. N. Dancer and J. Toland, The sub-harmonic bifurcation of Stokes waves,, Arch. Ration. Mech. Anal., 152 (2000), 241. doi: doi:10.1007/s002050000087.

[6]

E. N. Dancer, Infinitely many turning points for some supercritical problems,, Ann. Mat. Pura Appl., 178 (2000), 225. doi: doi:10.1007/BF02505896.

[7]

Y. H. Du and Z. M. Guo, Positive solutions of an elliptic equation with negative exponent: Stability and critical power,, J. Differential Equations, 246 (2009), 2387. doi: doi:10.1016/j.jde.2008.08.008.

[8]

P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity,, Commun. Contemp. Math., 10 (2008), 17. doi: doi:10.1142/S0219199708002697.

[9]

P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity,, Comm. Pure Appl. Math., 60 (2007), 1731. doi: doi:10.1002/cpa.20189.

[10]

G. Flores, G. A. Mercado and J. A. Pelesko, Analysis of the dynamics and touchdown in a model of electrostatic MEMS,, SIAM J. Appl. Math., 67 (): 434. doi: doi:10.1137/060648866.

[11]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case,, SIAM J. Math. Anal., 38 (): 1423. doi: doi:10.1137/050647803.

[12]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 115. doi: doi:10.1007/s00030-007-6004-1.

[13]

Y. Guo, On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior,, J. Differential Equations, 244 (2008), 2277. doi: doi:10.1016/j.jde.2008.02.005.

[14]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: doi:10.1007/BF01221125.

[15]

Z. M. Guo and X. F. Bai, On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity,, Commun. Pure Appl. Anal., 7 (2008), 1091. doi: doi:10.3934/cpaa.2008.7.1091.

[16]

Z. M. Guo and J. C. Wei, Hausdorff domension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity,, Manuscripta Math., 120 (2006), 193. doi: doi:10.1007/s00229-006-0001-2.

[17]

Z. M. Guo and J. C. Wei, Asymptotic behavior of touch down solutions and global bifurcations for an elliptic problem with a singular nonlinearity,, Commun. Pure Appl. Anal., 7 (2008), 765. doi: doi:10.3934/cpaa.2008.7.765.

[18]

Z. M. Guo and J. C. Wei, Infinitely many turning points for an elliptic problem with a singular non-linearity,, J. Lond. Math. Soc., 78 (2008), 21. doi: doi:10.1112/jlms/jdm121.

[19]

Z. M. Guo and X. Z. Peng, On the structure of positive solutions to an elliptic problem with a singular nonlinearity,, J. Math. Anal. Appl., 354 (2009), 134. doi: doi:10.1016/j.jmaa.2009.01.001.

[20]

Z. M. Guo, D. Ye and F. Zhou, Existence of singular positive solutions for some semilinear elliptic equations,, Pacific J. Math., 236 (2008), 57. doi: doi:10.2140/pjm.2008.236.57.

[21]

Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties,, SIAM J. Appl. Math., 66 (2005), 309. doi: doi:10.1137/040613391.

[22]

C. C. Hwang, C. K. Lin and W. Y. Uen, A nonlinear three-dimensional rupture theory of thin liquid films,, J. Colloid Interf. Sci., 190 (1997), 250. doi: doi:10.1006/jcis.1997.4867.

[23]

H. Q. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations,, European J. Appl. Math., 18 (2007), 153. doi: doi:10.1017/S0956792507006936.

[24]

R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations,, European J. Appl. Math., 11 (2000), 293. doi: doi:10.1017/S0956792599003794.

[25]

R. S. Laugesen and M. C. Pugh, Energy levels of steady-states for thin-film-type equations,, J. Differential Equations, 182 (2002), 377. doi: doi:10.1006/jdeq.2001.4108.

[26]

R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, Arch. Ration. Mech. Anal., 154 (2000), 3. doi: doi:10.1007/PL00004234.

[27]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties,, SIAM J. Appl. Math., 62 (): 888. doi: doi:10.1137/S0036139900381079.

[28]

J. A. Pelesko and D. H. Bernstein, "Modeling MEMS and NEMS,'', Chapman & Hall/CRC, (2003), 1.

[29]

D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 37 (2010), 259. doi: doi:10.1007/s00526-009-0262-1.

show all references

References:
[1]

Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489. doi: doi:10.1090/S0002-9939-01-06132-9.

[2]

A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations,, Comm. Pure Appl. Math., 51 (1998), 625. doi: doi:10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.

[3]

A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations,, Indiana Univ. Math. J., 49 (2000), 1323. doi: doi:10.1512/iumj.2000.49.1887.

[4]

J. P. Burelbach, S. G. Bankoff and S. H. Davis, Nonlinear stability of evaporating/condensing liquid films,, J. Fluid Mech., 195 (1988), 463. doi: doi:10.1017/S0022112088002484.

[5]

B. Buffoni, E. N. Dancer and J. Toland, The sub-harmonic bifurcation of Stokes waves,, Arch. Ration. Mech. Anal., 152 (2000), 241. doi: doi:10.1007/s002050000087.

[6]

E. N. Dancer, Infinitely many turning points for some supercritical problems,, Ann. Mat. Pura Appl., 178 (2000), 225. doi: doi:10.1007/BF02505896.

[7]

Y. H. Du and Z. M. Guo, Positive solutions of an elliptic equation with negative exponent: Stability and critical power,, J. Differential Equations, 246 (2009), 2387. doi: doi:10.1016/j.jde.2008.08.008.

[8]

P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity,, Commun. Contemp. Math., 10 (2008), 17. doi: doi:10.1142/S0219199708002697.

[9]

P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity,, Comm. Pure Appl. Math., 60 (2007), 1731. doi: doi:10.1002/cpa.20189.

[10]

G. Flores, G. A. Mercado and J. A. Pelesko, Analysis of the dynamics and touchdown in a model of electrostatic MEMS,, SIAM J. Appl. Math., 67 (): 434. doi: doi:10.1137/060648866.

[11]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case,, SIAM J. Math. Anal., 38 (): 1423. doi: doi:10.1137/050647803.

[12]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 115. doi: doi:10.1007/s00030-007-6004-1.

[13]

Y. Guo, On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior,, J. Differential Equations, 244 (2008), 2277. doi: doi:10.1016/j.jde.2008.02.005.

[14]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: doi:10.1007/BF01221125.

[15]

Z. M. Guo and X. F. Bai, On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity,, Commun. Pure Appl. Anal., 7 (2008), 1091. doi: doi:10.3934/cpaa.2008.7.1091.

[16]

Z. M. Guo and J. C. Wei, Hausdorff domension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity,, Manuscripta Math., 120 (2006), 193. doi: doi:10.1007/s00229-006-0001-2.

[17]

Z. M. Guo and J. C. Wei, Asymptotic behavior of touch down solutions and global bifurcations for an elliptic problem with a singular nonlinearity,, Commun. Pure Appl. Anal., 7 (2008), 765. doi: doi:10.3934/cpaa.2008.7.765.

[18]

Z. M. Guo and J. C. Wei, Infinitely many turning points for an elliptic problem with a singular non-linearity,, J. Lond. Math. Soc., 78 (2008), 21. doi: doi:10.1112/jlms/jdm121.

[19]

Z. M. Guo and X. Z. Peng, On the structure of positive solutions to an elliptic problem with a singular nonlinearity,, J. Math. Anal. Appl., 354 (2009), 134. doi: doi:10.1016/j.jmaa.2009.01.001.

[20]

Z. M. Guo, D. Ye and F. Zhou, Existence of singular positive solutions for some semilinear elliptic equations,, Pacific J. Math., 236 (2008), 57. doi: doi:10.2140/pjm.2008.236.57.

[21]

Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties,, SIAM J. Appl. Math., 66 (2005), 309. doi: doi:10.1137/040613391.

[22]

C. C. Hwang, C. K. Lin and W. Y. Uen, A nonlinear three-dimensional rupture theory of thin liquid films,, J. Colloid Interf. Sci., 190 (1997), 250. doi: doi:10.1006/jcis.1997.4867.

[23]

H. Q. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations,, European J. Appl. Math., 18 (2007), 153. doi: doi:10.1017/S0956792507006936.

[24]

R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations,, European J. Appl. Math., 11 (2000), 293. doi: doi:10.1017/S0956792599003794.

[25]

R. S. Laugesen and M. C. Pugh, Energy levels of steady-states for thin-film-type equations,, J. Differential Equations, 182 (2002), 377. doi: doi:10.1006/jdeq.2001.4108.

[26]

R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, Arch. Ration. Mech. Anal., 154 (2000), 3. doi: doi:10.1007/PL00004234.

[27]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties,, SIAM J. Appl. Math., 62 (): 888. doi: doi:10.1137/S0036139900381079.

[28]

J. A. Pelesko and D. H. Bernstein, "Modeling MEMS and NEMS,'', Chapman & Hall/CRC, (2003), 1.

[29]

D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 37 (2010), 259. doi: doi:10.1007/s00526-009-0262-1.

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