On isolated singular solutions to Lane-Emden equation Feng ZHOU - PowerPoint PPT Presentation

Outline of the Talk Motivation Main Results Sketch of proofs Further results On isolated singular solutions to Lane-Emden equation Feng ZHOU CPDE and Math. Dept. East China Normal University (based on joint works with H.Y.Chen, X.Huang and Z.M.Guo) Workshop on ”Singular problems associated to quasilinear equations” in honor of 70th birthday of Professors Marie-Fran¸ coise Bidaut-V´ eron and Laurent V´ eron ShanghaiTech University and Masaryk University 2020.06.01-03 UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results 1 Motivation 2 Main Results 3 Sketch of proofs 4 Further results UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular Lane-Emden Equation Consider the following semilinear elliptic equation: � ∆ u = u p in Ω , (2.1) where Ω is a smooth domain in R N with N � 3. x 0 2 Ω , if lim x ! x 0 u ( x ) = + 1 , we call that u is singular at x 0 (singular solution). UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular Lane-Emden Equation Consider the following semilinear elliptic equation: � ∆ u = u p in Ω , (2.1) where Ω is a smooth domain in R N with N � 3. x 0 2 Ω , if lim x ! x 0 u ( x ) = + 1 , we call that u is singular at x 0 (singular solution). Ω = R N \ { 0 } , if p  N N � 2 (Serrin’s exponent), NO positive solution ((M.Bidaut-V´ eron,-S.Pohozaev, 2001, JAM). UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular Lane-Emden Equation Consider the following semilinear elliptic equation: � ∆ u = u p in Ω , (2.1) where Ω is a smooth domain in R N with N � 3. x 0 2 Ω , if lim x ! x 0 u ( x ) = + 1 , we call that u is singular at x 0 (singular solution). Ω = R N \ { 0 } , if p  N N � 2 (Serrin’s exponent), NO positive solution ((M.Bidaut-V´ eron,-S.Pohozaev, 2001, JAM). 2 N � 2 , there exists always a solution w 1 ( x ) ⌘ c p | x | � N If p > p − 1 (slow decay solution). UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Motivation [H.Brezis, L.V´ eron, Arch.Rat.Mech. and Anal., (1980)], N N � 2 , 0 2 Ω ⇢ R N , N � 3 , and If p � in Ω 0 = Ω \ { 0 } , � ∆ u + | u | p � 1 u  C , then lim sup u ( x ) < 1 , x ! 0 and thus if u 2 C 2 ( Ω 0 ) satisfying � ∆ u + | u | p � 1 u = 0 , in Ω 0 , then 9 C 2 function in Ω which coincides with u on Ω 0 . That is the equation � ∆ u + | u | p � 1 u = 0 UPVM has the property that any isolated singularity is ”removable”. Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Motivation [L.V´ eron, Nonl. Anal., (1981)], Classification of isolated singularities of any solution of in Ω 0 = Ω \ { 0 } , � ∆ u + | u | p � 1 u = 0 , If 0 is the singular point, there exists two types of singularities N when 1 < p < N � 2 , and as x ! 0, 2 p − 1 . either u ( x ) ⇠ ± c p | x | � [H.Brezis, L.Oswald, Arch.Rat.Mech. and Anal., (1987)] Some generalization has been made by M-F. Bidaut-V´ eron and L.V´ eron and many others. UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Motivation [L.V´ eron, Nonl. Anal., (1981)], Classification of isolated singularities of any solution of in Ω 0 = Ω \ { 0 } , � ∆ u + | u | p � 1 u = 0 , If 0 is the singular point, there exists two types of singularities N when 1 < p < N � 2 , and as x ! 0, 2 p − 1 . either u ( x ) ⇠ ± c p | x | � or u ( x ) ⇠ c | x | 2 � N , where c is any constant. [H.Brezis, L.Oswald, Arch.Rat.Mech. and Anal., (1987)] Some generalization has been made by M-F. Bidaut-V´ eron and L.V´ eron and many others. UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain Example 1: Consider the Dirichlet boundary value problem: � ∆ u = u p in Ω , u = 0 on @ Ω (2.2) where Ω is a smooth open set in R N and for suitable range for the exponent p . u 2 L p ( Ω ) is called a weak solution of (2.2) if the equality Z Z u p ' dx = 0 u ∆ ' dx + Ω Ω holds for any ' 2 C 2 ( Ω ) and ' = 0 on @ Ω . UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain Example 1: Consider the Dirichlet boundary value problem: � ∆ u = u p in Ω , u = 0 on @ Ω (2.2) where Ω is a smooth open set in R N and for suitable range for the exponent p . u 2 L p ( Ω ) is called a weak solution of (2.2) if the equality Z Z u p ' dx = 0 u ∆ ' dx + Ω Ω holds for any ' 2 C 2 ( Ω ) and ' = 0 on @ Ω . S ✓ Ω is called a singular set for a weak solution u of (2.1) if for any x 2 S , u is not bounded in any neighborhood of x . UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain Example 1: Consider the Dirichlet boundary value problem: � ∆ u = u p in Ω , u = 0 on @ Ω (2.2) where Ω is a smooth open set in R N and for suitable range for the exponent p . u 2 L p ( Ω ) is called a weak solution of (2.2) if the equality Z Z u p ' dx = 0 u ∆ ' dx + Ω Ω holds for any ' 2 C 2 ( Ω ) and ' = 0 on @ Ω . S ✓ Ω is called a singular set for a weak solution u of (2.1) if for any x 2 S , u is not bounded in any neighborhood of x . UPVM S is then a closed subset of Ω . Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain C.C. Chen and C.S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal. 9 (1999), 221-246. UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain C.C. Chen and C.S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal. 9 (1999), 221-246. N � 2 < p < p c ( < N +2 N For N � 2 ), they constructed positive weak solutions with a prescribed singular set. Moreover, as an application to the conformal scalar curvature, they N +2 N − 2 ( S N ) of the problem constructed a weak solution u 2 L N +2 N − 2 = 0 for N � 9 such that S N is the singular set of L 0 u + u u , where L 0 is the conformal Laplacian with respect to the standard metric of S N . UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain C.C. Chen and C.S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal. 9 (1999), 221-246. N � 2 < p < p c ( < N +2 N For N � 2 ), they constructed positive weak solutions with a prescribed singular set. Moreover, as an application to the conformal scalar curvature, they N +2 N − 2 ( S N ) of the problem constructed a weak solution u 2 L N +2 N − 2 = 0 for N � 9 such that S N is the singular set of L 0 u + u u , where L 0 is the conformal Laplacian with respect to the standard metric of S N . By Variational Methods. UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain Many works on the asymptotic behavior of the solutions near a isolated singularity of the equations as (2.2): n � 2 , n +2 n by Gidas and Spruck for p 2 ( n � 2 ) (CPAM, 1981). UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation

Outline of the Talk Motivation Main Results Sketch of proofs Further results Singular solution in bounded domain Many works on the asymptotic behavior of the solutions near a isolated singularity of the equations as (2.2): n � 2 , n +2 n by Gidas and Spruck for p 2 ( n � 2 ) (CPAM, 1981). n by Aviles when p = n � 2 , (Indiana Univ.Math. J. 1983) c 0 + o (1) u ( x ) = as x ! 0 , n − 2 ( � | x | 2 log | x | ) 2 for some constant c 0 . UPVM Feng ZHOU On isolated singular solutions to Lane-Emden equation