A Generalized Radial Brezis–Nirenberg Problem Rafael D. Benguria Instituto de F´ ısica, PUC Santiago, Chile The Analysis of Complex Quantum Systems: Large Coulomb Systems and Related Matters CIRM, Luminy, Marseille October 23, 2019.
This is joint work with Soledad Benguria Department of Mathematics University of Wisconsin Madison, WI, USA Talk partially based on the manuscript: RB and S. Benguria, A Generalized Radial Brezis–Nirenberg Problem , preprint, 2019.
Motivation (The Lane–Emden equation): The equation − ∆ u = u p (1) for u > 0 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, 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 30,000 degrees Kelvin for the temperature of the Sun, which is too big by a factor of 5. Then he used his value of the temperature together with the solution of (1) with p = 3 / 2, to estimate the density u near the surface. 5,778 K
Motivation (The Lane–Emden equation): 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 the Lane–Emden equation in the whole R 3 vanishing at infinity. This equation on the whole R 3 , with exponent p = 5 plays a major role in mathematics. It is the Euler–Lagrange equation equation that one obtains when minimizing the quotient ( r u ) 2 dx R � 1 / 3 . (1) u 6 dx �R This quotient is minimized if u ( x ) = 1 / ( | x | 2 + m 2 ) 1 / 2 . 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 con- stant. Inserting this function u back in (1), gives the classical Sobolev inequality (S. Sobolev 1938), ( r u ) 2 dx R � 1 / 3 � 3( π 2 ) 4 / 3 , (2) u 6 dx �R for all functions in D 1 ( R 3 ).
The Brezis–Nirenberg problem on R N In 1983 Brezis and Nirenberg considered the nonlinear eigenvalue problem, − ∆ u = λ u + | u | 4 / ( n − 2) u, with u ∈ H 1 0 ( Ω ), where Ω is bounded smooth domain in R n , with n ≥ 3. Among other results, they proved that if n ≥ 4, there is a positive solution of this problem for all λ ∈ (0 , λ 1 ) where λ 1 ( Ω ) is the first Dirichlet eigenvalue of Ω . They also proved that if n = 3, there is a µ 1 ( Ω ) > 0 such that for any λ ∈ ( µ 1 , λ 1 ), the nonlinear eigenvalue problem has a positive solution. Moreover, if Ω is a ball, µ 1 = λ 1 / 4.
The Brezis–Nirenberg problem on R N For positive radial solutions of this problem in a (unit) ball, one is led to an ODE that still makes sense when n is a real number rather than a natural number. Precisely this problem with 2 ≤ n ≤ 4, was considered by E. Jannelli, The role played by space dimension in elliptic critical problems , J. Di ff erential Equations, 156 (1999), pp. 407–426. Among other things Jannelli proved that this problem has a positive solution if and only if λ is such that √ j − ( n − 2) / 2 , 1 < λ < j +( n − 2) / 2 , 1 , where j ν ,k denotes the k –th positive zero of the Bessel function J ν .
The Brezis–Nirenberg problem on R N π j 0 , 1 = 2 . 4048 . . . π / 2
The Bakry-´ Emery Laplacian. We think of this problem as a further variant of the Brezis-Nirenberg prob- lem, namely we study (1) with the Laplacian replaced by a particular form of a weighted Laplacian, or if one prefers, the drift Laplacian . The interest on weighted Laplacians originated in the early 1980’s for di ff erent reasons coming from physics, geometry, and probability. Depending on the context, a weighted Laplacian is often called the Witten Lapla- cian (Witten, 83) or the Bakry-´ Emery Laplacian (Bakry-Emery, 85).
The Bakry-´ Emery Laplacian. During the past decade there has been a growing interest in studying the spectral properties of weighted Laplacians or drift Laplacians . A Bakry–´ Emery manifold, denoted by the triple ( M, g, φ ) is a complete Rie- mannian manifold ( M, g ) together with some function φ 2 C 2 ( M ) where the measure on M is the weighted measure exp( � φ ) dV g . The Bakry–´ Emery Lapla- cian ∆ φ associated with such a manifold is given by ∆ φ = ∆ � r φ · r which is self-adjoint with respect to the inner product associated with the weighted measure. Here ∆ is the standard Laplace-Beltrami operator and r is the gradient operator on the Bakry–´ Emery manifold. Weighted Laplacians were also introduced, in a di ff erent context, by Chavel and Feldman 1991.
The Bakry-´ Emery Laplacian. Typically, in the Bakry-´ Emery Laplacian, the potential φ is smooth, and so is the drift term. However, singular drifts have also been considered in the literature. In fluid mechanics a weighted Laplacian with a singular drift is rather common, but typically the drift is divergence free, in other words, away from the singularities the potential φ is harmonic. Recently (J. Reine und Angew. Math., 2018) A. Grigoryan, S.–X. Ouyang, and M. R¨ ockner considered a drift with a (singular) potential of the form φ ( x ) = | x | − α with α > 0. In that case the singular drift has, generically, a nonzero divergence.
The Brezis–Nirenberg problem for the weighted Laplacian with a Singular Drift In our case we consider the Brezis-Nirenberg problem for the weighted Laplacian in R n ( n ≥ 3), the singular drift derives from a potential of the form φ ( x ) = δ log( | x | ). Notice that in this case the weighted measure, exp( − φ ) dV g becomes | x | − δ dx where dx is the standard Lebesgue measure in R n . Thus the weighted measure can be thought of as the Lebesgue measure on a space of an e ff ective fractional dimension d ≡ n − δ , a fact that we use intensively in the proofs of our theorems.
The Bakry-´ Emery Laplacian. In our case, when we consider the Brezis-Nirenberg problem for the weighted Laplacian in R n ( n � 3), the singular drift derives from a potential of the form � ( x ) = � log( | x | ). Notice that in this case the weighted measure, exp( � � ) dV g becomes | x | − δ dx where dx is the standard Lebesgue measure in R n . Thus the weighted measure can be thought of as the Lebesgue measure on a space of an e ff ective fractional dimension d ⌘ n � � . First consider the problem, x ~ | x | 2 · r u = � u + | u | 4 / ( n − 2 − δ ) u, � ∆ u + � (1) with u 2 H 1 0 ( Ω ), and Ω is the ball in R n , n � 3, centered at the origin. Because of Hardy’s inequality, the operator with the singular drift one considers on the left side of (1) is a positive operator provided � < ( n � 2) / 2. Notice that the critical Sobolev exponent on the right side of (1) depends on the parameter � that characterizes the singular drift. In terms of the “e ff ective dimension” the critical Sobolev exponent is given by the standard form, ( d + 2) / ( d � 2), a remark which is important in the proofs of our theorems. We are interested in the range of values of � and � for which (1) admits positive radial smooth solutions.
Euclidean Case. Theorem [RB, S. Benguria, 2017]. When Ω ⊂ R n , with n ≥ 3, is a unit ball centered at the origin we have, i) If n = 3 and δ ∈ ( − 1 , 1 / 2), (1) has a unique positive radial solution u ∈ H 1 0 ( Ω ) provided j 2 − (1 − δ ) / 2 , 1 < λ < j 2 (1 − δ ) / 2 , 1 , and no positive radial solutions for λ outside that range. If n = 3 and δ ≤ − 1, then (1) has a unique positive radial solution u ∈ H 1 0 ( Ω ) provided 0 < λ < j 2 (1 − δ ) / 2 , 1 , and no positive radial solutions outside that range. ii) If n = 4 and δ ∈ (0 , 1), (1) has a unique positive radial solution u ∈ H 1 0 ( Ω ) provided j 2 − (2 − δ ) / 2 , 1 < λ < j 2 (2 − δ ) / 2 , 1 , and no positive radial solutions for λ outside that range. If n = 4 and δ ≤ 0, then (1) has a unique positive radial solution u ∈ H 1 0 ( Ω ) provided 0 < λ < j 2 (2 − δ ) / 2 , 1 , and no positive radial solutions outside that range.
Euclidean Case. Theorem [RB, S. Benguria, 2017]. (Cont’d): iii) If n = 5 and δ ∈ (1 , 3 / 2), (1) has a unique positive radial solution u ∈ H 1 0 ( Ω ) provided j 2 − (3 − δ ) / 2 , 1 < λ < j 2 (3 − δ ) / 2 , 1 , and no positive radial solutions for λ outside that range. If n = 5 and δ ≤ 1, then (1) has a unique positive radial solution u ∈ H 1 0 ( Ω ) provided 0 < λ < j 2 (3 − δ ) / 2 , 1 , and no positive radial solutions outside that range. Next we consider the Brezis-Nirenberg problem with a singular drift on geodesic balls of S n ( n ≥ 3).
The Brezis–Nirenberg problem on S N We consider the nonlinear eigenvalue problem, − ∆ S n u = λ u + | u | 4 / ( n − 2) u, with u ∈ H 1 0 ( Ω ), where Ω is a geodesic ball in S n . In dimension 3, Bandle and Benguria (JDE, 2002) proved that for λ > − 3 / 4 this problem has a unique positive solution if and only if π 2 − 4 θ 2 < λ < π 2 − θ 2 1 1 4 θ 2 θ 2 1 1 where θ 1 is the geodesic radius of the ball.
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