Abstract
A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with “negative exponent” is established. It is well known that such a monotonicity formula plays an essential role in the study of finite Morse index solutions of equations with “positive exponent”. Unlike the positive exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.
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References
Catrina F, Wang Z Q. On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm Pure Appl Math, 2001, 54: 229–258
Davila J, Ye D. On finite Morse index solutions of two equations with negative exponent. Proc R Soc Edinb Ser A, 2013, 143: 121–128
Du Y H, Guo Z M. Positive solutions of an elliptic equation with negative exponent: Stability and critical power. J Differential Eqnations, 2009, 246: 2387–2414
Du Y H, Guo Z M. Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations. Adv Differential Equations, 2013, 18: 737–768
Du Y H, Guo Z M, Wang K L. Monotonicity formula and ε-regularity of stable solutions to supercritical problems and application to finite Morse index solutions. Calc Var Partial Differential Equations, 2014, 50: 615–638
Esposito P, Ghoussoub N, Guo Y. Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity. Comm Pure Appl Math, 2007, 60: 1731–1768
Ghoussoub N, Guo Y. On the partial differential equations of electrostatic MEMS devices: Stationary case. SIAM J Math Anal, 2007, 38: 1423–1449
Guo Y, Pan Z, Ward M J. Touchdown and pull-in voltage behavior of a MEMS device with vaeying dielectric properties. SIAM J Appl Math, 2006, 166: 309–338
Guo Z M, Wei J C. Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity. Manuscripta Math, 2006, 120: 193–209
Guo Z M, Wei J C. Symmetry of nonnegative solutions of a semilinear elliptic equation with singular nonlinearity. Proc R Soc Edinb Ser A, 2007, 137: 963–994
Guo Z M, Wei J C. Infinitely many turning points for an elliptic problem with a singular nonlinearity. J London Math Soc, 2008, 78: 21–35
Guo Z M, Wei J C. On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity. J Differential Equations, 2007, 240: 279–323
Jiang H Q, Ni W M. On steady states of van der Waals force driven thin film equations. European J Appl Math, 2007, 18: 153–180
Ma L, Wei J C. Properties of positive solutions to an elliptic equation with negative exponent. J Funct Anal, 2008, 254: 1058–1087
Meadows A M. Stable and singular solutions of the equation \(\Delta u = \tfrac{1} {u}\) . Indiana Univ Math J, 2004, 53: 1681–1703
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Guo, Z., Zhou, F. Sub-harmonicity, monotonicity formula and finite Morse index solutions of an elliptic equation with negative exponent. Sci. China Math. 58, 2301–2316 (2015). https://doi.org/10.1007/s11425-015-4988-2
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DOI: https://doi.org/10.1007/s11425-015-4988-2