Radial regular and rupture solutions for a PDE problem with gradient term and two parameters

Ghergu, Marius; Miyamoto, Yasuhito

doi:10.1090/proc/15861

Radial regular and rupture solutions for a PDE problem with gradient term and two parameters
HTML articles powered by AMS MathViewer

by Marius Ghergu and Yasuhito Miyamoto PDF

Proc. Amer. Math. Soc. 150 (2022), 1697-1709 Request permission

Abstract:

We investigate radial solutions for the problem \[ \begin {cases} \displaystyle -\Delta U=\frac {\lambda +\delta |\nabla U|^2}{1-U},\; U>0 & \text {in}\ B,\\ U=0 & \text {on}\ \partial B, \end {cases} \] where $B\subset \mathbb {R}^N$ $(N\geq 2)$ denotes the open unit ball and $\lambda , \delta >0$ are real numbers. Two classes of solutions are considered in this work: (i) regular solutions, which satisfy $0<U<1$ in $B$, and (ii) rupture solutions, which satisfy $U(0)=1$, and thus make the equation singular at the origin. Bifurcation with respect to parameter $\lambda >0$ is also discussed.

References

Patricio Aviles, Local behavior of solutions of some elliptic equations, Comm. Math. Phys. 108 (1987), no. 2, 177–192. MR 875297, DOI 10.1007/BF01210610
Adrian Constantin, On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl. (4) 184 (2005), no. 2, 131–138. MR 2149089, DOI 10.1007/s10231-004-0100-1
S. Chandrasekhar, An introduction to the study of stellar structure, Dover Publications, Inc., New York, N.Y., 1957. MR 0092663
Juan Dávila and Juncheng Wei, Point ruptures for a MEMS equation with fringing field, Comm. Partial Differential Equations 37 (2012), no. 8, 1462–1493. MR 2957549, DOI 10.1080/03605302.2012.679990
Filippo Gazzola and Andrea Malchiodi, Some remarks on the equation $-\Delta u=\lambda (1+u)^p$ for varying $\lambda ,\ p$ and varying domains, Comm. Partial Differential Equations 27 (2002), no. 3-4, 809–845. MR 1900564, DOI 10.1081/PDE-120002875
M. Ghergu and Y. Miyamoto, Radial single point rupture solutions for a general MEMS model, Calc. Var. PDE (2022), DOI 10.1007/s00526-021-02158-4.
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879, DOI 10.1007/BF01221125
Mohammed Guedda and Laurent Véron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations 76 (1988), no. 1, 159–189. MR 964617, DOI 10.1016/0022-0396(88)90068-X
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269. MR 340701, DOI 10.1007/BF00250508
Philip Korman and Junping Shi, New exact multiplicity results with an application to a population model, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 5, 1167–1182. MR 1862448, DOI 10.1017/S0308210500001323
A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. I. Fold point asymptotics, Methods Appl. Anal. 15 (2008), no. 3, 297–325. MR 2500849, DOI 10.4310/MAA.2008.v15.n3.a4
Linfeng Mei, Structure of positive radial solutions of a quasilinear elliptic problem with singular nonlinearity, Complex Var. Elliptic Equ. 63 (2018), no. 11, 1595–1603. MR 3847100, DOI 10.1080/17476933.2017.1399367
Fulbert Mignot and Jean-Pierre Puel, Solution radiale singulière de $-\Delta u=\lambda e^u$, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 8, 379–382 (French, with English summary). MR 965802
Yasuhito Miyamoto and Yūki Naito, Singular extremal solutions for supercritical elliptic equations in a ball, J. Differential Equations 265 (2018), no. 7, 2842–2885. MR 3812216, DOI 10.1016/j.jde.2018.04.055
Yasuhito Miyamoto and Yūki Naito, Fundamental properties and asymptotic shapes of the singular and classical radial solutions for supercritical semilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 27 (2020), no. 6, Paper No. 52, 25. MR 4160935, DOI 10.1007/s00030-020-00658-4
John A. Pelesko and David H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1955412
Pavol Quittner and Philippe Souplet, Superlinear parabolic problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states. MR 2346798
James Serrin and Henghui Zou, Classification of positive solutions of quasilinear elliptic equations, Topol. Methods Nonlinear Anal. 3 (1994), no. 1, 1–25. MR 1272885, DOI 10.12775/TMNA.1994.001
Wei-Ming Ni and Paul Sacks, Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc. 287 (1985), no. 2, 657–671. MR 768731, DOI 10.1090/S0002-9947-1985-0768731-8
J. Ignacio Tello, Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation, J. Math. Anal. Appl. 324 (2006), no. 1, 381–396. MR 2262478, DOI 10.1016/j.jmaa.2005.12.011
Juncheng Wei and Dong Ye, On MEMS equation with fringing field, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1693–1699. MR 2587454, DOI 10.1090/S0002-9939-09-10226-5