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Radial single point rupture solutions for a general MEMS model

Abstract

We study the initial value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} r^{-(\gamma -1)}\left( r^{\alpha }|u'|^{\beta -1}u'\right) '=\frac{1}{f(u)} &{} \text {for}\ 0<r<r_0,\\ u(r)>0 &{} \text {for}\ 0<r<r_0,\\ u(0)=0, \end{array}\right. } \end{aligned}$$

for \(\gamma>\alpha >\beta \ge 1\) and \(f\in C[0,{{\bar{u}}})\cap C^2(0,{{\bar{u}}})\), \(f(0)=0\), \(f(u)>0\) on \((0, {{\bar{u}}})\) and f satisfies certain assumptions which include the standard case of pure power nonlinearities encountered in the study of Micro-Electromechanical Systems (MEMS). We obtain the existence and uniqueness of a solution \(u^*\) to the above problem, the rate at which it approaches the value zero at the origin and the intersection number of points with the corresponding regular solutions \(u(\,\cdot \,,a)\) (with \(u(0,a)=a\)) as \(a\rightarrow 0\). In particular, these results yield the uniqueness of a radial single point rupture solution and other qualitative properties for MEMS models. The bifurcation diagram is also investigated.

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References

  1. Cassani, D., Fatorusso, L., Tarsi, A.: Global existence for nonlocal MEMS. Nonlinear Anal. 74, 5722–5726 (2011)

    MathSciNet  Article  Google Scholar 

  2. Castorina, D., Esposito, P., Sciunzi, B.: \(p\)-MEMS equation on a ball. Methods Appl. Anal. 15, 277–284 (2008)

    MathSciNet  Article  Google Scholar 

  3. Dancer, E.: Infinitely many turning points for some supercritical problems. Ann. Mat. Pura Appl. 178, 225–233 (2000)

    MathSciNet  Article  Google Scholar 

  4. Davila, J., Wei, J.: Point ruptures for a MEMS equation with fringing field. Comm. Part. Diff. Equ. 37, 1462–1493 (2012)

    MathSciNet  Article  Google Scholar 

  5. Davila, J., Wang, K., Wei, J.: Qualitative analysis of rupture solutions for a MEMS problem. Ann. Inst. H. Poincaré, Anal. Non Linéaire 33, 221–242 (2016)

    MathSciNet  Article  Google Scholar 

  6. do Ó, J., da Silva, E.: Some results for a class of quasilinear elliptic equations with singular nonlinearity. Nonlinear Anal. 148, 1–29 (2017)

  7. Esposito, P., Ghoussoub, N.: Uniqueness of solutions for an elliptic equation modeling MEMS. Methods Appl. Anal. 15, 341–354 (2008)

    MathSciNet  Article  Google Scholar 

  8. Esposito, P., Ghoussoub, N., Guo, Y.: Mathematical Analysis of Partial Differential Equations Modelling Electrostatic MEMS, pp. 1-318, Courant Lecture Notes in Maths 20 (2010). CIMS/AMS

  9. Esteve, C., Souplet, P.: Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity. Nonlinearity 31, 4883–4934 (2018)

    MathSciNet  Article  Google Scholar 

  10. Esteve, C., Souplet, P.: No touchdown at points of small permittivity and nontrivial touchdown sets for the MEMS problem. Adv. Diff. Equ. 24, 465–500 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Ghergu, M., Goubet, O.: Singular solutions of elliptic equations with iterated exponentials. J. Geometric Anal. 30, 1755–1773 (2020)

    MathSciNet  Article  Google Scholar 

  12. Guo, J.-S., Souplet, Ph.: No touchdown at zero points of the permittivity profile for the MEMS problem. SIAM J. Math. Anal. 47, 614–625 (2015)

    MathSciNet  Article  Google Scholar 

  13. Guo, Z., Liu, Z., Wei, J., Zhou, F.: Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Commun. Pure Appl. Anal. 10, 507–525 (2011)

    MathSciNet  Article  Google Scholar 

  14. Guo, Z., Wei, J.: Infinitely many turning points for an elliptic problem with a singular non-linearity. J. Lond. Math. Soc. 78, 21–35 (2008)

    MathSciNet  Article  Google Scholar 

  15. Guo, Z., Wei, J.: On solutions with point ruptures for a semilinear elliptic problem with singularity. Methods Appl. Anal. 15, 377–390 (2008)

    MathSciNet  Article  Google Scholar 

  16. Guo, Z., Wei, J.: Asymptotic Behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Comm. Pure Appl. Anal. 7, 765–786 (2008)

    MathSciNet  Article  Google Scholar 

  17. Guo, Z., Wei, J.: Rupture solutions of an elliptic equation with a singular nonlinearity. Proc. Roy. Soc. Edin. A 144, 905–924 (2014)

    MathSciNet  Article  Google Scholar 

  18. Guo, Y., Pan, Z., Ward, M.J.: Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. SIAM J. Appl. Math. 66, 309–338 (2005)

    MathSciNet  Article  Google Scholar 

  19. Korman, P.: Solution curves for semilinear equations on a ball. Proc. Amer. Math. Soc. 125, 1997–2005 (1997)

    MathSciNet  Article  Google Scholar 

  20. Korman, P.: Infinitely many solutions for three classes of self-similar equations with p-Laplace operator: Gelfand, Joseph-Lundgren and MEMS problems. Proc. Roy. Soc. Edinburgh Sect. A 148, 341–356 (2018)

    MathSciNet  Article  Google Scholar 

  21. Laurençot, Ph., Walker, C.: A fourth-order model for MEMS with clamped boundary conditions. Proc. Lond. Math. Soc. 109, 1435–1464 (2014)

    MathSciNet  Article  Google Scholar 

  22. Laurençot, Ph., Walker, C.: Some singular equations modelling MEMS. Bull. AMS 54, 437–479 (2017)

    Article  Google Scholar 

  23. Li, K., Guo, H., Guo, Z.: Positive single rupture solutions to a semilinear elliptic equation. Appl. Math. Letters 18, 1177–1183 (2005)

    MathSciNet  Article  Google Scholar 

  24. Lindsay, A., Ward, M.: Asymptotics of some nonlinear eigenvalue problems modelling a MEMS capacitor Part II multiple solutions and singular asymptotics. Eur. J. Appl. Math. 22, 83–123 (2011)

    MathSciNet  Article  Google Scholar 

  25. Liu, Y., Li, Y., Deng, Y.: Separation property of solutions for a semilinear elliptic equation. J. Diff. Equ. 163, 381–406 (2000)

    MathSciNet  Article  Google Scholar 

  26. Miyamoto, Y.: Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations. NoDEA Nonlinear Diff. Equ. Appl. 23, 16–24 (2016)

    MathSciNet  Article  Google Scholar 

  27. Miyamoto, Y.: A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth. J. Diff. Equ. 264, 2684–2707 (2018)

    MathSciNet  Article  Google Scholar 

  28. Miyamoto, Y., Naito, Y.: Singular extremal solutions for supercritical elliptic equations in a ball. J. Diff. Equ. 265, 2842–2885 (2018)

    MathSciNet  Article  Google Scholar 

  29. Miyamoto, Y., Naito, Y.: Fundamental properties and asymptotic shapes of the singular and classical radial solutions for supercritical semilinear elliptic equations. NoDEA Nonlinear Diff. Equ. Appl., 27, Article 52, 25 pages (2020)

  30. Pelesko, J.A., Bernstein, D.H.: Modelling MEMS and NEMS. Chapman & Hall CRC, Boca Raton, FL (2003)

    MATH  Google Scholar 

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Correspondence to Yasuhito Miyamoto.

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Communicated by Manuel del Pino.

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The second author was supported by JSPS KAKENHI Grant Numbers 19H01797, 19H05599.

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Ghergu, M., Miyamoto, Y. Radial single point rupture solutions for a general MEMS model. Calc. Var. 61, 47 (2022). https://doi.org/10.1007/s00526-021-02158-4

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  • DOI: https://doi.org/10.1007/s00526-021-02158-4

Mathematics Subject Classification

  • primary 34A12
  • 35B40
  • secondary 35B32
  • 35J62