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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

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