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January
2022, 42(1): 137-162.
doi: 10.3934/dcds.2021110
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 |
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 $ |
| $ \Omega $ |
| $ {\mathbb R}^N $ |
| $ \Omega $ |
| $ p = 1 $ |
| $ q\in (0,q^*(N)) $ |
| $ q^*(N)\in (1,2) $ |
| $ 2\leq N<2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $ |
| $ \lambda = \tilde \lambda $ |
| $ N>2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $ |
| $ 0<\lambda<\tilde \lambda $ |
Mathematics Subject Classification:
Primary: 34A12, 35B32; Secondary: 35B40, 35J62.
Citation:
Ignacio Guerra. A semilinear problem with a gradient term in the nonlinearity. Discrete and Continuous Dynamical Systems,
2022, 42
(1)
: 137-162.
doi: 10.3934/dcds.2021110
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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 |
| [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. |
| [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. |
| [4] |
M. Ghergu and Y. Miyamoto, Radial regular and rupture solutions for a MEMS model with fringing field, submitted, preprint, arXiv: 2007.01406. |
| [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. |
| [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. |
| [7] |
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), 309-338.
doi: 10.1137/040613391. |
| [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. |
| [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. |
| [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. |
| [11] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. |
| [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. |
| [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. |
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 |
| [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. |
| [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. |
| [4] |
M. Ghergu and Y. Miyamoto, Radial regular and rupture solutions for a MEMS model with fringing field, submitted, preprint, arXiv: 2007.01406. |
| [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. |
| [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. |
| [7] |
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), 309-338.
doi: 10.1137/040613391. |
| [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. |
| [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. |
| [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. |
| [11] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. |
| [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. |
| [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. |
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 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 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
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