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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

Received  February 2021 Revised  June 2021 Published  January 2022 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 and Continuous Dynamical Systems, 2022, 42 (1) : 137-162. 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 [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.

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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.
Bifurcation diagram for problem (1) in the case $N = 3$ and $q = 1$
Bifurcation diagram for problem (1) in the case $N = 22$ and $q = 1$
Phase diagram for problem in the case $N = 3$ and $q = 1.3$. Thicker lines describe the trapping region
Phase diagram for problem in the case $N = 7$ and $q = 1.4$. Thicker lines describe the trapping region
Diagram of curves $q = N/(N-1)$, $q^*(N),$ $\tilde q(N)$, and $\hat q(N)$.
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 $\lambda = \tilde \lambda = 1.$
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)
The shape of the function $\tilde f(q),$ with $k = 16$ and $\hat b = 0.9$
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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