 Advances in Pure Mathematics, 2011, 1, 99-104 doi:10.4236/apm.2011.13022 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM Liouville Type Theor ems for Lichner owicz Equations and Ginzburg-Landau Equation: Survey Li Ma1 Department of Mat hematics, Henan Normal University, Xinxiang, China Department of Mat he m at i c al Sci e nces, Tsinghua Universi ty, Beijing, China E-mail: lma@math.tsinghua.edu.cn Received April 11, 2011; revised May 10, 2011; accepted May 12, 2011 Abstract In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginz-burg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations in Rn via two approaches from the use of maximum principle and the monotonicity formula. Keywords: Liouville Theorems, Ancient Solution, Ginzburg-Landau Equation, Lichenrowicz Equation 1. Introduction This article is based on the lecture given at March 3rd, 2011 in the international conference “Recent Advances in Nonlinear Partial Differential Equations: Part I” held at the Chinese University of Hong Kong. The initial-value problem of gen eral relativity consists in the resolution of a coupled system of three linear equ-ations and a quasilinear equation which determines the conformity factor, on an initial Riemannian manifold ,Mg. In the case where there are sources, the qua-si-linear equation can be written as 7358=0guRuVu Quu  (1.1) where >0u is the unknown on the Riemannian mani-fold ,Mg, R is the scalar curvature, V, Q,  are functions derived from the Ricci curvature of ,Mg. (1.1) is called the Lichnerowicz equation on ,Mg. Let 2=2 2nn and S be the best So-bolev constant. One interesting result derived from mountain pass lemma is below. Theorem 1.1 Assume that ,nMg is compact, 3n. Consider the following Lichnerowicz equation 21 21=u uBuAu  with >0A and >0maxMB. Assume that there is a positive function >0 and a constant >0Cn such that 12221,>0max nHMMMCnABSB  Then there is a positive solution to Lichnerowicz equa-tion. Hebey-Pacard-Pollack  have applied the mountain pass lemma to the perturbation functionals to get positive approximation solutions and have proved the conver-gence of a subsequence to a positive solution. The Ginzburg-Landau (GL) model is proposed in 50’s in the context of super-co nductivity th eory an d its ener gy density is 22211=1.24eu uu  Here :nkuR R. The stationary E-L equation for GL model is 2=1.uuu Yanyan Li and Z. C. Han (1995) have also studied p-laplacian GL models. Other models with term 221mu, 1m, is also of interesting. The Lichnerowicz equation and the Ginzburg-Landau equations are important models in mathematical physics. The existence of solutions to both can be obtained via variational methods, the monotone method (also called sub-super solution method or barrier method), and per-1The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 2009 0 0 02110019. L. MA Copyright © 2011 SciRes. APM 100 turbation methods. There are huge literatures about G-L models [2,5, 10,14,15,20], but there is not much works about the Lichnerowicz equation [4,12,18]. For the existence re-sults of the Lichnerowicz equation, one may see the works , , and . For the existence results of the Ginzburg-Landau equation, one may find more refer-ences from the work . Our main topic for both equations is about the Liou-ville type results, which are closely related to compact-ness theorems of the solution spaces. The heat flow me-thod to both equations will be an interesting topic for studying. 2. Classical Liouville Type The or e m s a n d Keller-Osserman Theory The famous Liouville theory says that any non-negative harmonic function is constant. One can prove this by Harnack inequality or differential gradient Harnack es-timate. People may extend this to non-negative p-harmonic function or to non-negative solution to the elliptic equa-tion ,==0,in.nij ijijLuauR We now recall the famous Keller-Osserman theory obtained around 1957. Given a domain . Consider the differential inequality ,in nufuR where ft is a positive, continuous, and monotone increasing function for 0tt satisfying the Osgood condition 1200 dd<.tfs st Then any twice continuously differentiable function u can not satisfy >0u on the whole space and ufu outside of some ball. As an application of above theory, J. B. Keller and R. Osserman consider the non-existence result for the Gaussian curvature equation on the plane 2=,in.uuKxe R Professor. Ni, Lin, W. Ding, W. Chen and C. Li, etc., have obtained a lot interesting existence results to this problem. More results and references may be found in my work . It is also interesting to study the non-existence of non-trivial non-negative solutions or energy solutions to 1=,in.pnuKxu uR where 2n and >1p. Another application of the Keller and Osserman theory (H. Brezis, 1984) we have that for 20nvCR sat-isfying ,>1,inpnvvpR we have =0v. In fact, H. Brezis getting upper bound of solutions by using the following boundary blow-up super-solution on the ball RBp, 22=RCRux Rxp with =12p and a suitable constant C inde-pendent of >0R. Then for any fixed point npR, sending R, we get =0v. If the Laplacian is replaced by p-Laplacian, Du and Guo (2002) can ex tend the Keller and Osserman theorem to this case. For more, one may see the works of A. Fa-rina, A. Ratto, M. Rigoli. Parallel result to Keller and Osserman can be done for non-negative ancient solutions to the parabolic inequal-ity: ,>1,in ,0.pntvvp R  See also the interesting work of J. Serrin  for more Liouville type theorems about elliptic and parabolic equ-ations. We have the following Liouville type Theorem. Theorem 2.1 (L. Ma, 2010 ) Let >0u in (1.1) with =nMR, =0Q, =1V, =1. Then =1u. For positive solutions to the general equation 2=,qquu u on nR with >1q, we have the same result. However, H. Brezis  proves that for 0,1q, the same result is not true, but we always have 1u. Similar result is also true for M being a complete Riemannian manifold ,Mg with its Ricci curvature bounded from blow. Here is the argument of Theorem 2.1. Let fs 2=qquu for >0q. For any fixed nxR and >0, consider the new function  2=.uy uyyx Note that uyu y as y. Then the minimum of it can be achieved at some point z. Then, =,uz ux ux which implies that uz ux. Using the monotonicity of f, i .e. 0f, we have fuz fux L. MA Copyright © 2011 SciRes. APM 101At this point z, we have 220Du z,  0=2= 2.uzuzn fuzn  Then we have 02 2fuzn fuxnfux as 0. Recall that =qpfuuu or some >0q and >0p. Then we have 1u. Assume now that >1q. =10vu . Then =1.qvfvv Using the Keller-Osserman theory we then conclude that =0,..=1.vieu This completes the argument of Theorem 2.1. We now give som e remark. Set 3=fuu u for 0u, which is the special case of Ginzurg-Landau equation. We then conclude from the argument above that =1u. In general, we know the below. Theorem 2.2 (Du, Ma , 2001) Assume that 2,nuCRR such that 3=uuu on nR. Then we have 1u. In the papers of Du and Ma (2001-2003), more general logistic equations have been studied. It is there that we are interested in a problem related to Di Giorgi conjec-ture, which has been completely solved by Del Pino, J. Wei, etc., Savin, C. Gui, Ambrosio and Cabre, etc. Interestingly, Du and Guo can obtaine the below. Theorem 2.3 (Du, Guo, 2002) Assume that 2,nuCRR such that 3=,1<,puu up on nR. Then we have 1u. There is a similar De Giorgi conjecture related to the equation above (see the work of L. Caffarelli, etc., A Gradient Bound for Entire Solutions of Quasi-Linear Equations and Its Consequences, Communications on Pure and Applied Mathematics, Volume 47, Issue 11, November 1994, Pages: 1457-1473 ). It would be interesting to study the following evolu-tion equation 3=0, in(0,)tpuuuuT  with suitable initial and boundary conditions. The results above can be extended to nonlinear heat equations. Set =qpfuu u for some >0q and >0p. Consider ancient solutions to the following parabolic equation =,>0,in ,0.ntufuu R  (2.1) For any fixed ,,0nxR and >0, con-sider the new function  22,= ,,,nuytuyyxtyt R  Note that ,uyt as yt. Then the minimum of it can be achieved at some point ,zt . Then at this point 0tu , which implies that 220,2=,2,tuzt nfuztn  which is less than 2,2 ,fux nfux  as 0. Hence 1u (and one may also show that =1u). Theorem 2.4 Let >0u be an ancient solution to (2.1). Then 1u. 3. Results for Ginzburg-Landau Equations Our uniform bound result (due to H. Brezis) is Theorem 3.1 Any smooth solution to GL model is bounded in the sense that 1ux. H. Brezis uses the Kato inequality to prove Theorem 3.1. We shall report here his argument. My argument is different but it is also based on the maximum principle. Before I give the proof, let’s recall the famous Kato inequality. Assume that 1locuL and 1locuL . Firstly we may assume that u is smooth. Note that for >0, we have 22223=.()uuuuuu  The latter is bigger than 2.uuu Hence we have for any 200C we have 22.uuuu  Sending 0, we then get sign ,uuu i.e., sign .uuu We have the parabolic version of the Kato inequality. L. MA Copyright © 2011 SciRes. APM 102 Consider =,uuxt with in addition 1tlocuL. Using 22=,ttuuuu we have  22.ttuuuuu  Then, letting 0, we have ||sign .ttuuuu  With the understanding above, we have that for any non-negative ancient solution to ,>1,in ,0pntvvvpR  is trivial. The argument of this fact is almost the same as Bre-zis’s argument (1984). Therefore, we have the following extension of H. Bre-zis’ theorem. Theorem 3.2 Any ancient solution:,nuR 0kR to 2=1tuu uu must have 1u. Going back to the parabolic version of Lichnerowicz equation on manifold with non-negative Ricci curvature, we have the following result. Proposition 3.1 Any positive ancient solution to =qptuuuu with >1q and >0p must be =1u. We now give the proof o f T h eorem 3.1. Brezis’s argument: Let 2=1Wu . Then we have 22sign1 .Wu u Note that 2222=2 221.uuu uuu  Then we have 22 2221sign12 12.WuuuWW W Using the Keller-Osserman theory we then have 2=0,..1.Wieu In my proof of this bound, I have used the original barrier functions use d by Kel ler -O sserman. It is interesting to know if one can extend the result above to p-Laplacian G-L model. In the following, we present the monotnonicity for-mula method to the Liouville type theorems [1,10,19]. It is our intention here to generalize the Liouville type results to a large class of solutions of a more general non-linear equations/systems 2=0,in,,,nnkuWuRuC RR (3.1) where W is the gradient of the smooth function Wu on kR and 0Wu. We shall use an idea from Professor Hesheng Hu (1980) who introduced it for harmonic maps. Please see the book of Y. L. Xin  for results of harmonic maps. It is also interesting to know if one can extend this kind of result to p-laplacian case. Theorem 3.3 Assume that 0Wu is a non-trivial smooth function. Let :nkuR R, 2n, be a smooth solution to the Ginzburg-Landau system (3.1). Assume that there are a positive constant 0>0R and a positive function r on 0,R such that  022<.lim BBRRRru Wu  and 0d= .Rrrr Then u is a constant. For any smooth mapping from the Riemannian mani-fold ,Mg to kR, we define the Stress-Energy tensor (see the paper of Baird-Eells) by 21=dd,2uSuguu which is 1=2ijjj ikikSuuguu in local frames je on M. By direct computation we know that div=,.ugSudu A consequence of this formula is that if u is harmonic, then div 0uS. Let X be a smooth vector field on M. Define the tensor X by ,= ,.ijejiXee g Xe Then we hav e 21div=div,2,,jjuuX duXuedu XuSX Take any compact domain DM with smooth L. MA Copyright © 2011 SciRes. APM 103boundary D. Choose the local frame je such that =ne be the unit outward normal to the boundary. Then we hav e 21,d,2=div, .DuuDuX uXuSXS X (3.2) Below we let =nMR. To explain the main idea of the proof, we start with the simple case when =0Wu and =1r. That is, =0u and u has finite energy. We take in (3.2) =0RDB , =rXr. Note that =ij ijX. Then we have 22,= ==,2uijijunSXSXtrS u ,=XR and 2d,=ruX uRu on 0RB. Hence 202,=2uBRnSX u (3.3) and by (3.2), we have 22212=.22rBBRRnRuu u  Then we have the following Liouville theorem for harmonic functions. Theorem 3.4 Let :nuR R be a harmonic fun ction with slowly energy divergence, i.e., there are a positive constant 0>0R and a positive function r on 0,R such that 20<.lim BBRRRru  and 0d= .Rrrr Then u is a constant. Here is the idea of proof: Assume that u is not a constant. Then there are some positive constants >0C and 0>0R such that 02022>0.2BRnuC By this we know that 2BRRuC (3.4) for any 0RR. Hence we have 00022d,RRBB RBRRR RruuCr  as R, which gives a contradiction. In fact, the proof of Theorem 3.4 goes below. As sume u is not a constant. By (3.4) we know that  22000d,RBBR BRR rRRrur urCrr  as R, which gives a contradiction. This completes the proof of Theorem 3.4. We now turn to the proof of Theorem 3.3. Assume that Wu is non-trivial. In this case div=, =,uSu duWudu  and div=,=.uXXSXWu uWu Again we take =0RDB . Then by (3.2) we have 21,,2=,.DXuDuX duXuWuSX  (3.5) Simplifying this identity we can derive th e following 21.2BRRuWuC Then using the argument above we obtain Theorem 3.3. We remark that the results above can be extended to complete Riemannian manif olds with bounded Ricci cu r- vature. We remark that for a large class of elliptic equations/ systems (also for parabolic equation/system), the Liou-ville type theorems are equivalent to a local uniform bound of solutions. For this direction, one may see the recent works of P. Polacik, P. Souplet, P. Quittner, H. Zou, etc. However, it is open for Lichnerowicz equation on compact Riemannian manifolds (see also in ). We also make a remark below. In the study of elliptic systems, one may use the Pohozaev type identity (which is a sister of monotonicity formula) and the interpolation inequalities to derive a contraction mapping property about pL norm of the solution. From the contraction mapping property one then get the solution trivial (and the Liouville type theorem). One may see the works of Chen-Li , Souplet, etc. [21,23], for th is kind of results for the Lane-Emden conjecture. We would like to thank Professor H. 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