RAFAEL D. BENGURIA of Chandrasekhar's book is also interesting fromthe mathematical point ofview, beacuse of the many techniques described there to an

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1 The Lane Emden equation revisited Rafael D. Benguria Abstract. Several results concerning existence and nonexistence of radial (but not necessarily positive) solutions to the nonlinear boundary value problem u = u + juj p 1 u; on a (geodesic) ball of R N or S 3, with Dirichlet boundary condition are presented (depending on the possible values of and the exponent p). The dependence on the underlying space is illustrated through a discussion of the boundary value problem on S 3. (1.1) The equation 1. Introduction u = u p for u>in a ball of radius R in R 3, with Dirichlet boundary conditions, is called, in physics, the Lane Emden equation of index p. It was introduced in 1869 by Homer Lane [1], who was interested in computing both the temperature and the density of mass on the surface of the Sun. Unfortunately Stefan's law was unknown at the time (Stefan published his law in 1879). Instead, Lane used some experimental results of Dulong and Petit and Hopkins on the rate of emission of radiant energy by a heated surface, and he got the value of 3, degrees Kelvin for the temperature of the Sun, which istoobigby a factor of 5. Then he used his value of the temperature together with the solution of (1.1) with p = 3=, to estimate the density u near the surface. Equation (1.1), in appropriate units, describes the behavior of the density of a gas sphere in hydrostatic equilibrium. The index p (which is called the polytropic index in astrophysics) is related to the ratio fl of the specific heats of the gas, through p =1=(fl 1). Even though the values obtained by Lane for the temperature and density near the surface of the sun were wrong, the values for both quantities at the interior are very reasonable, and the Lane Emden equation is still in use today to compute the structure of the interior of polytropic stars. A wonderful account onthe history of the subject can be found in the appendix of Chapter IV of the classical book of Chandasekhar [5] (pp ). Chapter IV 1991 Mathematics Subject Classification. Primary 35J6, 35B33, 58J5 ; Secondary 35B3, 58J3. Key words and phrases. nonlinear elliptic boundary value problems; Lane-Emden equation; critical Sobolev exponent. This work was supported in part by Fondecyt (Chile), Project #

2 RAFAEL D. BENGURIA of Chandrasekhar's book is also interesting fromthe mathematical point ofview, beacuse of the many techniques described there to analize the behavior of solutions of (1.1) for different values of p. After the Lane Emden equation was introduced, it was soon realized that it only had bounded solutions vanishing at R if the exponent is below 5.In fact, for 1» p<5 there are bounded solutions, which are decreasing with the distance from the center. In 1883, Sir Arthur Schuster constructed a bounded solution of (1.1) in the whole R 3 vanishing at infinity [15]. Equation (1.1) on the whole R 3,withexponent p =5plays a major role in mathematics. It is the Euler Lagrange equation equation that one obtains when minimizing the quotient R (ru) (1.) dx R u 6 dx 1=3 : This quotient is minimized if u(x) = 1=(jxj + m ) 1=. The minimizer is unique modulo multiplications by a constant, and translations. This function u(x), is precisely the function determined by A. Schuster, up to a multiplicative constant. Inserting this function u back in (1.), gives the classical Sobolev inequality [16], R (ru) dx (1.3) R u 6 dx 1=3 3( ß )4=3 ; for all functions in D 1 (R 3 ) (see, e.g., [11], theorem8.3). In 1983, Brézis and Nirenberg considered the following linear perturbation of the Lane Emden equation, (1.4) u = juj p 1 u + u on a bounded smooth domain Ω of R 3, with Dirichlet boundary conditions [4]. They showed that if the domain is starshaped, there exists a number Λ 1(Ω) depending on the domain Ω, such that there exists a regular, positive solution of (1.4) vanishing at the boundary if and only if Λ 1(Ω) < < 1 (Ω). Here 1 denotes the first Dirichlet eigenvalue of Ω. In the case Ω is a ball, Brézis and Nirenberg proved that Λ 1 = 1 =4. For the case of the ball, the nonexistence of positive solutions for» Λ 1 is based on a refinement of the classical Pohozaev Rellich identity [13, 14], where for 1 it follows immediately by projecting (1.3) along the first Dirichlet eigenfunction, and integrating by parts. The proof of existence of positive solutions follows from a compactness argument due to E. H. Lieb [4]. If one is only interested in radial, not necessarily positive, solutions of (1.4) one can analyze the existence and nonexistence of solutions by using an Emden Fowler transformation [6, 7]. In fact, for radial solutions (1.4) reduces to the ODE (1.5) u N 1 u = u + juj p 1 u; r for < r < 1, with r = jxj, N 3, p > 1 and u(1) =. Here denotes differentiation wit respect to r. Let's denote ffi = u (1) the slope of the solution at the boundary (which can be assumed to be nonnegative, without loss of generality). If one performs the change of variables introduced by Emden[6], i.e., (1.6) with (1.7) u(r) =r =(p 1) w(s); s = log(r)

3 THE LANE EMDEN EQUATION REVISITED 3 then, s :!1, whereas w(s) satisfies the ODE, (1.8) w + aw + bw + jwj p 1 w + e s w =: Now, denotes differentiation with respect to s. Moreover, w()=andw () = ffi. The parameters a and b that appear in (1.8) depend only on the dimension N and the exponent p, and are given explicitly by (1.9) and (1.1) b = a =( N)+ 4 p 1 ; [(p +1) (N 1)(p 1)] : (p 1) As s goes to infinity, the last termin (1.9) vanishes, and it is therefore natural to consider the autonomous system (1.11) y + ay + by + jyj p 1 y =: Interpreting s as time, and y as position, one can readily interpret (1.11) as Newton's equation of motion for one particle in one dimension subject to a conservative force by jyj p 1 y and a viscous force ay. We can consider the conservative force as deriving froma potential V (y) (b=)y + jyj p+1 =(p +1). For a given N, there are two special values of p for which there is a qualitative change in (1.11). The viscosity a will remain positive aslongas1<p<(n +)=(N ), and the motion will be damped, so eventually y (and for that matter w) will converge to the equilibriumpoints of the conservative force. If p =(N +)=(N ) (i.e., for the critical Sobolev exponent), the viscosityvanishes, and the asymptotic autonoumous systemwill have an energy conserved (the precise value of the energy depends on and ffi). Finally, for p > (N +)=(N ), i.e., above the critical Sobolev exponent, the viscosity is negative, and the analysis gets much harder. The value of the coefficient, b, of the linear restoring force changes sign at p = N=(N ). Thus, for 1 <p<n=(n ) there is only one equilibriumpoint, namely y =, while for N=(N ) < p < (N +)=(N ) there are three equlibria (y =, unstable, and y = ±( b) 1=(p 1), stable). This heuristic arguments can be made rigorous and are the basis of the results reviewed in Section. In Section 3, we analyze the Brézis Nirenberg problemon a geodesic cap of S 3 with Dirichlet boundary conditions.. Classification of the solutions of the radial solutions of the Brézis Nirenberg problem in a ball of R N, N 3 The heuristic considerations at the end of the introduction can be put in a rigorous footing, and used to classify all the radial solutions of (1.4) in a ball of R N, N 3. This has been accomplished in [3]. The main results can be embodied in the following theorems. Lemma.1. If a, and w L 1 (; +1) is a solution of the equation (.1) w + aw + bw + jwj p 1 w + e s w = in R + ;

4 4 RAFAEL D. BENGURIA then, there exists a function ~w, solution of (.) such that (.3) w + aw + bw + jwj p 1 w = in R + ; lim s!+1 jw(s) ~w(s)j + jw (s) ~w (s)j =: Moreover, if we define the energy" functional E(w(t)) by E(w(t)) w (t) + V (w(t)); then, E(~w(s)) is constant for all s and the function ~w is periodic. Note that when a 6=, the only periodic solutions of (.) are constant functions, corresponding to the critical points of V (w). As we stressed in the heuristic considerations, there are different behaviors for different values of p. Thus, one must consider the special values 1 < p < N=(N ), p = N=(N ), N=(N ) <p<(n +)=(N ), p =(N +)=(N ), and p>(n +)=(N ), separately. If 1 <p<n=(n ), there is only one equilibriumpoint (i.e., w = ), and going back to the original variables, all the solutions u will be bounded. We have ([3], Theorem4.1) the following result. Theorem. (1 <p<n=(n )). Assume that for some k 1, < k = (kß) (the k-th Dirichlet eigenvalue of the unit ball). Then, all the solutions of (1.5) in D (; 1) are bounded and radial solutions of (1.4) in D (B 1 ). Moreover, for every i k, there exists at least a solution of (1.1), u i, which is bounded and has i 1 zeroes in (; 1). The positive bounded solution, which exists if < 1,is unique. If N=(N ) <p<(n +)=(N ), there are three equilibriumpoints (i.e., w = and w = ±( b) 1=(p 1). Going back to the original variables, the solutions u associated to the w's that converge to will be bounded, while the solutions u associated to the w's that converge to the other equilibria will be unbounded near the origin. We have ([3], Theorem4.) the following result. Theorem.3 (N=(N ) <p<(n +)=(N )). If for some k 1, < k then there exists: ffl abounded and smooth solution of (1.5), u i, with (i 1) zeroes in (; 1), for all i k (if it exists, the bounded positive solution u 1 is unique), ffl an uncountable set of unbounded solutions with a finite number of zeroes in (; 1). All these solutions behave at the origin like a(r)r =(p 1) for some bounded function a(r) which is bounded away from at the origin. Under the assumptions of this theorem, the number of zeroes is finite for all solutions of (1.5) and all the solutions of (1.5) are also solutions of (1.4). For p = N=(N ) the potential V has only one equilibriumpoint (w = ), with zero curvature, and the analysis of the solutions is harder. We have ([3], Theorem 4.3) the following result. Theorem.4 (p = N=(N )). If for some k 1, < k then there exists: ffl abounded and smooth solution of (1.5), u i, with (i 1) zeroes in the interval (; 1), for all i k, and the bounded positive solution is unique if it exists,

5 THE LANE EMDEN EQUATION REVISITED 5 ffl an uncountable set of unbounded solutions with a finite number of zeroes in (; 1). All these solutions behave at the origin like C jln rj 1=(p 1) r =(p 1) with C = (N )= p N. Under the assumptions of this theorem, the number of zeroes is finite for all solutions of (1.5) and all the above solutions of (1.5) are also solutions of (1.4). Before we go onto the result for the critical exponent, we recall the following known results on the branches of nodal bounded solutions (see [3], Proposition 3.1, and the references given there). Theorem.5. For every i 1 there exists a nonnegative constant μ i [; i ) and a continuous curve, C i ρf( ; c) [; 1) g corresponding to the set of bounded solutions u of (1.5) with i 1 zeroes in the interval (; 1) and such that u c () = c. The point ( i ; ) belongs to C i and the half line ( = μ i ;c>) is asymptotic to C i. When i =1, C 1 can be parametrized by c> and the function (c) is decreasing, i.e., for every (μ 1 ; 1 ), there is a unique bounded positive solution of (1.4). Finally, when N =3, μ i = i 1 ß. When N =4; 5, μ 1 =and μ i = i 1 for i, and when N =6;μ 1 =; <μ i < i 1 for all i. Before we give the corresponding result for the critical case, we need some definitions. We say that a point ( ; u) in a branch of solutions of (1.1) in (; +1) L 1 is regular if locally the branch can be parametrized by. Apoint (; 1) is k-regular if there exists a bounded solution with k 1 zeroes u k such that ( ; u k ) is regular. fi We define I k as the projection of C k onto R and denote by J k the set f I k fi is k-regularg. For the critical exponent theviscosity a vanishes and there is conservation of energy for the asymptotic system. The nature of the solutions depends on the asymptotic value of the energy, which in turn depends on and ffi. If, e.g., the asymptotic energy is positive, w, and for that matter u, will have an infinite number of oscillations, it will change sign, and u will be unbounded at the origin. Our main result in the critical case is the following theorem([3], Theorem3.4). Theorem.6 (p =(N +)=(N ).). (i) If R n[ 1 k=1 I k, there isanuncountable number of solutions of (1.5) which are unbounded, oscillating at the origin, with sign changing oscillations. No other solutions to (1.5) exist. (ii) If I k for some k 1, then, there exist ffl abounded solution of (1.5), u k, with k 1 zeroes in (; 1) (this solution is unique at least if k =1), ffl an uncountable number of solutions of (1.5) which are unbounded, oscillating at the origin, with sign changing oscillations, ffl an uncountable number of unbounded solutions of (1.5) with k 1 zeros in (; 1) if J k. Finally, all the above solutions of (1.5) are also solutions of (1.4) in D (B 1 ). If the exponent p is supercritical, i.e., p > (N +)=(N ) the mechanical analog obtained through the Emden Fowler transformations fails. However, one can perform a different change of variables and analize the structure of all the radial solutions in that case as well (see [3], Section 5).

6 6 RAFAEL D. BENGURIA 3. The Brézis Nirenberg Problem on S 3 The existence of solutions of the Brézis Nirenberg problemfor different values of the parameter depends critically on the domain. While for starshaped domains in R N (N 3) there are no classical solutions if is nonpositive, if the domain is an annulus, there are solutions for all negative [9]. It is also interesting to study the range of values of for which there are nontrivial positive solutions when one considers different subyacent manifolds. In particular, the Brézis Nirenberg problemhas been studied recently in spaces of constant curvature (see [1] for the case of S 3 and [17] forthecaseofh n ). Here, we will review our main results for the Brézis Nirenberg problemon S 3. Let D be a geodesic ball on S 3. Consider the problem (3.1) S 3u = u5 + u on D; with u and Dirichlet boundary conditions, i.e., (3.) u = Here S3 is the Laplace-Beltrami operator on S 3, and 5 is the corresponding critical Sobolev exponent. We are interested in determining the range of values of for which there exists a positive solution of (3.1) and (3.). It follows immediatly from the maximumprinciple that no positive solutions exist if 1 where 1 is the first eigenvalue of the Laplace-Beltrami operator with Dirichlet boundary conditions. Our main result is the following theorem. Theorem 3.1. Let D ρ S 3 beaball whose geodesic radius will be denoted by 1. Then the following statements are true. i) If (3.3) ß < < 1 = ß 1 1 ; there is a unique positive solution to (3.1) and (3.). ii) If D is contained in the hemisphere, i.e. if 1» ß=, and if < ß, there is no nontrivial solution to (3.1) and (3.). iii) If D contains the hemisphere, there exists a function ν :( ß ;ß)! ( 3 4 ; 1), ν(t)! 3 4 as t! ß such that for ν( 1) < < ß there is no nontrivial solution to (3.1) and (3.). Numerical computations indicate that there are solutions if <ν( 1 ). Remarks 3.. i) In particular for a hemisphere of S 3 there is a solution of (3.1) and (3.), if and only if < <3. ii) On the other hand, in the limit as the geodesic radius of D, 1 goes to zero, one recovers the Brézis Nirenberg result, i.e., there is a positive solution if and only if 1 =4 < < 1, where 1 is the first Dirichlet eigenvalue of the ball. iii) Except for =and = ß= where no solution exist [], it is not known whether or not for the borderline cases = ß and = ν( 1 ) there exist a solution. iv) The picture for the geodesic balls contained in a hemisphere is not surprising. It was expected after the recent results of Bandle and Peletier [] on best critical constants for the Sobolev embeddings in S 3 in the case =.

7 THE LANE EMDEN EQUATION REVISITED 7 v) An interesting open problem is to prove the existence of solutions in the range <ν( 1 ). Numerical calculations show that in contrast to the solutions in the range ß < < ß 1 their maximumis not at the origin. They become 1 singular at the boundary as tends to ν( 1 ). The proof of Theorem(3.1) has two parts: i) the existence of positive solutions for values of between μ 1 and 1, ii) the nonexistence of solutions for values outside this interval when the geodesic cap is contained in the hemisphere and for >ν( 1 ) (and outside the interval (μ 1, 1 ) for caps beyond the hemisphere). The existence of solutions is proved using a compactness argument due to E. H. Lieb (see [1], theoremfor details). Concerning the nonexistence of solutions, one uses a refinement of the Pohozaev Rellich identity [13, 14]. Of the many results concerning nonexistence, embodied in Theorem3.1, we will give a proof of part of it in the sequel to illustrate the method. For the complete proof we refer to [1]. Lemma 3.3. Assume» ß : i) If 1» ß= then, there isnopositive solution of (3.1), (3.). ii) If 1 ß= and if in addition 3 4 then, there isnopositive solution of (3.1), (3.). Proof. For the proof we use geodesic coordinates. For that purpose, let us choose the north pole of S 3 as the center of the geodesic ball. Given a point on S 3, let be the azimuthal angle of that point (i.e., if we consider rays coming fromthe center of the ball to the north pole and to the given point, respectively, is the angle between those two rays. By a result of Padilla [1] (extending the classical result of Gidas, Ni, and Nirenberg [8] to domains on manifolds of constant curvature), a positive solution u to (3.1) is symmetric, i.e., it only depends on the azimuthal angle. Writing u(x) =u( ), where is the azimuthal angle of x, (3.1) can be written as (3.4) u cot u = u 5 + u; with u() finite, u () = and u( 1 )=. The proof of this lemma is a Pohozaev type argument. We first multiply (3.4) by sin g( )u ( ) andintegrate the result in from to 1. Here, g( ) is a smooth function satisfying g() =,g( ) > for (; 1 ), and otherwise arbitrary. Integrating by parts, using the boundary conditions on u and g() =, we obtain (3.5) (u ) h( ) d 1 u ( 1 )g( 1 )sin ( 1 ) = ( sin cos g( ) + sin g ( ))( 1 6 u6 + 1 u ) d ; where h( ) =(1=)g ( )sin sin cos g( ). Then we multiply (3.4) by h( ) just defined. Thus we obtain, (3.6) u 1 h( ) d u 4 g + g sin d = (u 6 + u )h( ) d :

8 8 RAFAEL D. BENGURIA Substracting (3.6) from(3.5) we get (3.7) u sin 1 4 g + g (1 + ) = 1 u ( 1 )g( 1 ) sin u 6 sin (cos g( ) sin g ( )) d : Fromthis point onwehave to distinguish two cases: i) < 3=4, 1» ß= and ii) 3=4. Case i): < 3=4, 1» ß=. For these values of choose g( ) sin. Then, cos g( ) sin g ( ). Moreover, (3.8) 1 4 g ( ) + (1 + )g ( ) = 1 cos (3 + 4 ) < ; 4 for» < 1» ß=. Since g( 1 ), from(3.7) and (3.8) we getacontradiction. Therefore, there are no solutions of (3.4) in this case. Case ii): 3=4. For these values of,! p 4(1 + ) >. Now choose g( ) = sin(! ), so that (3.9) 1 4 g ( ) + (1 + )g ( ) : Clearly, g( ) > if<» 1 and! is such that the product! 1 is less than ß, i.e., if <(ß 4 1)=4 1. Also, (3.1) F ( ) cos g( ) sin g ( ) =cos sin(! )! sin cos(! ) > ; for < < 1, since F () = and F ( ) =(3 + 4 ) sin sin(! ) >, whenever < < 1,! 1 <ßand >. Using (3.9) and (3.1) in (3.7), noticing that g( 1 ) >, we get a contradiction again, and this lemma is proved. An interesting phenomena occurs when the geodesic cap is larger than the hemisphere. In fact, we have found numerical evidence that for those caps, not only we have the solutions embodied in our Theorem3.1 (i), but also for a given value of 1 > ß=, there are positive solutions for all sufficiently negative values of the parameter. This is somewhat reminiscent of the analogous problem on an annulus in Euclidean space (see [9]). However, the situation is not completely similar, because here, for a fixed value of geodesic radius 1 >ß=, there will be agapbetween the values of for which we have thepositive solutions described by Theorem3.1 (i) and the (sufficiently negative) values of for which these new positive solutions exist. There is no such a gap in the classical example of the annulus in Euclidean space [9]. Unfortunately we do not have at the moment and existence theoremfor values of below the curve ν( 1 ). However, we have performed extensive numerical computations that indicate the existence of positive solutions for values of below this curve, and in fact we can get very close to the curve we determined.

9 THE LANE EMDEN EQUATION REVISITED 9 4. Acknowledgements It is a pleasure to thank the Department of Mathematics of the University of Alabama, Birmingham, for their kind hospitality during the course of the conference, and the organizers for giving me the opportunity to present these results. References [1] C.Bandle, R.D. Benguria, The Brézis Nirenberg Problem on S 3. Journal of Differential Equations 178 (), [] C. Bandle, L.A. Peletier, Best Sobolev constants and Emden equations for the critical exponent in S 3. Math. Ann. 313 (1999) [3] R. D. Benguria, J. Dolbeault, and M.J. Esteban, Classification of the solutions of semilinear elliptic problems in a ball. Journal of Differential Equations 167 (), [4] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical exponents. Comm. Pure Appl. Math. 36 (1983), [5] S. Chandrasekhar, An introduction to the stellar structure. Dover Publications, Inc., NY, [6] V. R. Emden, Gaskugeln. Teubner, Leipzig, 197. [7] R.H. Fowler, Further studies of Emden's and similar differential equations. Q.J. Math. (Oxford Series) (1931), [8] B. Gidas, W. M. Ni, and L. Nirenberg. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68 (1979), [9] J. Kazdan, F. Warner, Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. XXVIII (1975), [1] J. Homer Lane, On the Theoretical Temperature of the Sun under the Hypothesis of a gaseous mass Maintining Its Volume by Its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment. Amer. J. Sci., d ser. 5 (1869), [11] E. H. Lieb, M. Loss, Analysis. nd ed. Graduate Studies in Mathematics. 14, Providence, RI, American Mathematical Society (AMS), 1. [1] P. Padilla, Symmetry properties of positive solutions of elliptic equations on symmetric domains. Appl. Anal. 64 (1997), [13] S. I. Pohozaev, On the eigenfunctions of the equation u + f(u) =. Sov. Math. Dokl. 6 (1965), (Translated from the Russ. Dokl. Akad. Nauk SSSR 165 (1965), 36 39). [14] F. Rellich, Darstellung der Eigenwerte von u + u = durch ein Randintegral. Math. Z. 46 (194), [15] A. Schuster, Brit. Assoc. Rept., p. 47, [16] S. L. Sobolev, On a theorem of functional analysis. Mat. Sb. 4 (1938), (English translation in Amer. Math. Soc, Transl. Ser. 34 (1963), ) [17] S. Stapelkamp, Das Brézis Nirenberg Problem im H n, Existenz und Eindeutigkeit von Lösungen. Diploma Thesis, University of Basel,. Department of Physics, P. Universidad Católica de Chile, Casilla 36, Santiago, Chile address: rbenguri@fis.puc.cl