doi: 10.3934/dcds.2021110
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A semilinear problem with a gradient term in the nonlinearity

Universidad de Santiago de Chile (USACH), Facultad de Ciencia, Departamento de Matemática y Ciencia de la Computación, Las Sophoras 173, Estación Central, Santiago, Chile

Received  February 2021 Revised  June 2021 Early access August 2021

Fund Project: Ignacio Guerra was supported by Proyecto Fondecyt Regular 1180628 and by Proyecto DICYT 061733GB, Universidad de Santiago de Chile

We consider the following semilinear problem with a gradient term in the nonlinearity
$ \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u>0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $
where
$ \lambda,p,q>0 $
and
$ \Omega $
be a bounded, smooth domain in
$ {\mathbb R}^N $
. We prove that when
$ \Omega $
is a unit ball and
$ p = 1 $
for
$ q\in (0,q^*(N)) $
with
$ q^*(N)\in (1,2) $
, we have infinitely many radial solutions for
$ 2\leq N<2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $
and
$ \lambda = \tilde \lambda $
. On the other hand, for
$ N>2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $
there exists a unique radial solution for
$ 0<\lambda<\tilde \lambda $
.
Citation: Ignacio Guerra. A semilinear problem with a gradient term in the nonlinearity. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021110
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, U. S. Government Printing Office, Washington, D. C., 1964  Google Scholar

[2]

J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field, Comm. Partial Differential Equations, 37 (2012), 1462-1493.  doi: 10.1080/03605302.2012.679990.  Google Scholar

[3]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, American Mathematical Society, 20 (2010). doi: 10.1090/cln/020.  Google Scholar

[4]

M. Ghergu and Y. Miyamoto, Radial regular and rupture solutions for a MEMS model with fringing field, submitted, preprint, arXiv: 2007.01406. Google Scholar

[5]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal, 38 (2006/07), 1423-1449.  doi: 10.1137/050647803.  Google Scholar

[6]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA: Nonlinear Differential Equations and Applications, 15 (2008), 115-145.  doi: 10.1007/s00030-007-6004-1.  Google Scholar

[7]

Y. GuoZ. 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), 309-338.  doi: 10.1137/040613391.  Google Scholar

[8]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Archive for Rational Mechanics and Analysis, 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[9]

A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor: Part I: Fold point asymptotics,, Methods Appl. Anal., 15 (2008), 297-325.  doi: 10.4310/MAA.2008.v15.n3.a4.  Google Scholar

[10]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2001/02), 888-908.  doi: 10.1137/S0036139900381079.  Google Scholar

[11]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[12]

J. A. Pelesko and T. A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models, J. Eng. Math., 53 (2005), 239-252.  doi: 10.1007/s10665-005-9013-2.  Google Scholar

[13]

J. Wei and D. Ye, On MEMS equation with fringing field, Proc. Amer. Math. Soc., 138 (2010), 1693-1699.  doi: 10.1090/S0002-9939-09-10226-5.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, U. S. Government Printing Office, Washington, D. C., 1964  Google Scholar

[2]

J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field, Comm. Partial Differential Equations, 37 (2012), 1462-1493.  doi: 10.1080/03605302.2012.679990.  Google Scholar

[3]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, American Mathematical Society, 20 (2010). doi: 10.1090/cln/020.  Google Scholar

[4]

M. Ghergu and Y. Miyamoto, Radial regular and rupture solutions for a MEMS model with fringing field, submitted, preprint, arXiv: 2007.01406. Google Scholar

[5]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal, 38 (2006/07), 1423-1449.  doi: 10.1137/050647803.  Google Scholar

[6]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA: Nonlinear Differential Equations and Applications, 15 (2008), 115-145.  doi: 10.1007/s00030-007-6004-1.  Google Scholar

[7]

Y. GuoZ. 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), 309-338.  doi: 10.1137/040613391.  Google Scholar

[8]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Archive for Rational Mechanics and Analysis, 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[9]

A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor: Part I: Fold point asymptotics,, Methods Appl. Anal., 15 (2008), 297-325.  doi: 10.4310/MAA.2008.v15.n3.a4.  Google Scholar

[10]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2001/02), 888-908.  doi: 10.1137/S0036139900381079.  Google Scholar

[11]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[12]

J. A. Pelesko and T. A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models, J. Eng. Math., 53 (2005), 239-252.  doi: 10.1007/s10665-005-9013-2.  Google Scholar

[13]

J. Wei and D. Ye, On MEMS equation with fringing field, Proc. Amer. Math. Soc., 138 (2010), 1693-1699.  doi: 10.1090/S0002-9939-09-10226-5.  Google Scholar

Figure 1.  Bifurcation diagram for problem (1) in the case $ N = 3 $ and $ q = 1 $
Figure 2.  Bifurcation diagram for problem (1) in the case $ N = 22 $ and $ q = 1 $
Figure 3.  Phase diagram for problem in the case $ N = 3 $ and $ q = 1.3 $. Thicker lines describe the trapping region
Figure 4.  Phase diagram for problem in the case $ N = 7 $ and $ q = 1.4 $. Thicker lines describe the trapping region
Figure 5.  Diagram of curves $ q = N/(N-1) $, $ q^*(N), $ $ \tilde q(N) $, and $ \hat q(N) $.
Figure 6.  Bifurcation diagram for problem in the case $ N = 3 $ and $ q = 2.3 $. Numerically we observe that for $ 1.126\ldots<\lambda<1.127\ldots $ we have six solutions, four solutions to the left of $ \lambda = 1.126\ldots $ and two solutions to the right of $ \lambda = 1.127\ldots $. This is due to a global bifurcation occurs at $ \lambda = 1.127\ldots $. We find infinitely many solutions for $ \lambda = \tilde \lambda = 1. $
Figure 7.  Diagram of curves for $ 0\leq q\leq 1 $, $ \hat q(N) = 2\frac{N-2-2\sqrt{N-1}}{N-1} $, $ \bar q(N) = \frac 12\left((N-2)\sqrt{\frac{N-9}{N-1}}+8-N\right) $, and $ q = \frac{N-2-2\sqrt{N-1}}{N-1} $ which is equivalent to $ \sqrt{N-1} = \frac{1+\sqrt{2-q}}{1-q}, $ see (25)
Figure 8.  The shape of the function $ \tilde f(q), $ with $ k = 16 $ and $ \hat b = 0.9 $
Figure 9.  Phase diagram for the case $ N = 9 $ and $ q = 0.1 $. The isoclines are the dashed lines. Here the trapping region is given by the isocline $ y' = 0 $, the two straight lines and $ y = 0 $. We plot for $ b = 1 $, $ k = 16 $ and $ a $ given by (30) the positive root
[1]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002

[2]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solution of the Novikov equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290

[3]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099

[4]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[5]

Brian D. O. Anderson, Shaoshuai Mou, A. Stephen Morse, Uwe Helmke. Decentralized gradient algorithm for solution of a linear equation. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 319-328. doi: 10.3934/naco.2016014

[6]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401

[7]

Tianliang Yang, J. M. McDonough. Solution filtering technique for solving Burgers' equation. Conference Publications, 2003, 2003 (Special) : 951-959. doi: 10.3934/proc.2003.2003.951

[8]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[9]

Ellen Baake, Michael Baake, Majid Salamat. The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63

[10]

Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations & Control Theory, 2020, 9 (2) : 341-358. doi: 10.3934/eect.2020008

[11]

Qi Wang. On some touchdown behaviors of the generalized MEMS device equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2447-2456. doi: 10.3934/cpaa.2016043

[12]

Ken Shirakawa, Hiroshi Watanabe. Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 139-159. doi: 10.3934/dcdss.2014.7.139

[13]

Yi An, Zhuohan Li, Changzhi Wu, Huosheng Hu, Cheng Shao, Bo Li. Earth pressure field modeling for tunnel face stability evaluation of EPB shield machines based on optimization solution. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1721-1741. doi: 10.3934/dcdss.2020101

[14]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895

[15]

Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429

[16]

Ellen Baake, Michael Baake, Majid Salamat. Erratum and addendum to: The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2365-2366. doi: 10.3934/dcds.2016.36.2365

[17]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601

[18]

Jagadeesh R. Sonnad, Chetan T. Goudar. Solution of the Michaelis-Menten equation using the decomposition method. Mathematical Biosciences & Engineering, 2009, 6 (1) : 173-188. doi: 10.3934/mbe.2009.6.173

[19]

Adnan H. Sabuwala, Doreen De Leon. Particular solution to the Euler-Cauchy equation with polynomial non-homegeneities. Conference Publications, 2011, 2011 (Special) : 1271-1278. doi: 10.3934/proc.2011.2011.1271

[20]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (86)
  • HTML views (113)
  • Cited by (0)

Other articles
by authors

[Back to Top]