Some classes of solutions to quasilinear elliptic equations of p - PowerPoint PPT Presentation

Some classes of solutions to quasilinear elliptic equations of p -Laplace type I. E. Verbitsky University of Missouri, Columbia Singular Problems Associated to Quasilinear Equations Shanghai, China, June 1–3, 2020 In honor of Marie-Fran¸ coise Bidaut-V´ eron and Laurent V´ eron

Abstract This talk is concerned with various classes of solutions, including BMO , Sobolev and Morrey space solutions (along with their local counterparts) to quasilinear elliptic equations of the type − ∆ p u = σ u q + µ, in R n , u ≥ 0 where p > 1 and q > 0 . Here ∆ p is the p -Laplacian, and µ , σ are nonnegative functions (or Radon measures). Solutions u are positive p -superharmonic functions in R n (or local renormalized solutions). More general operators div A ( x , ∇· ) in place of ∆ p will be treated. We will discuss necessary and sufficient conditions for the existence, and pointwise estimates of solutions, along with related weighted norm inequalities. We intend to cover mostly the exponents q above ( q > p − 1 ) and below ( 0 < q < p − 1 ) the natural growth case. Based in part on joint work with Nguyen Cong Phuc (Louisiana State University, USA), Dat Tien Cao (Minnesota State University, USA), and Adisak Seesanea (Hokkaido University, Japan). I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 2 / 38

Publications 1 BMO solutions to quasilinear elliptic equations , in preparation (2020) (with Nguyen Cong Phuc) 2 Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory , Rendiconti Lincei 30 (2020), 733–758 3 Wolff’s inequality for intrinsic nonlinear potentials and quasilinear elliptic equations , Nonlin . Analysis 194 (2020) 4 Solutions in Lebesgue spaces to nonlinear elliptic equations with sub-natural growth terms , St . Petersburg Math . J . 31 (2020), 557-572 (with Adisak Seesanea) 5 Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms , Adv . Calc . Var . 13 (2020), 53–74 (with Adisak Seesanea) 6 Nonlinear elliptic equations and intrinsic potentials of Wolff type , J . Funct . Analysis 272 (2017) 112–165 (with Dat Tien Cao) I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 3 / 38

Quasilinear equations We consider p - superharmonic solutions to the equation − ∆ p u = σ u q + µ, in R n , u ≥ 0 (1) ∆ p u = ∇ · ( |∇ u | p − 2 ∇ u ) is the p -Laplace operator, p > 1 , q > 0 ; u ∈ L q loc ( σ ) ; µ, σ ∈ M + (R n ) (locally finite Radon measures). We actually treat more general quasilinear operators div A ( x , ∇ ) in place of ∆ p . We use local renormalized solutions introduced by Marie-Fran¸ coise [Bidaut-V´ eron ’03]. The equivalence to p -superharmonic solutions was shown by [Kilpel¨ ainen–Kuusi–Tuhola-Kujanp¨ a¨ a ’11]. The case q = p − 1 is called the natural growth case (Schr¨ odinger equation if p = 2 ). We distinguish between the sub-natural-growth case 0 < q < p − 1 and the super-natural-growth case q > p − 1 ( µ � = 0 ). Similar results hold for the fractional Laplace equation ( 0 < α < n ) 2 u = σ u q + µ, α in R n . ( − ∆) u ≥ 0 (2) I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 4 / 38

Classes of measures The p -capacity of a compact set K ⊂ R n is defined by: � � ||∇ u || p u ∈ C ∞ 0 (R n ) cap p ( K ) = inf L p (R n ) : u ≥ 1 on K , . For the existence of a nontrivial solution u to (1) with q > 0 , p > 1 , the measure σ must be absolutely continuous w/r to p -capacity: cap p ( K ) = 0 = ⇒ σ ( K ) = 0 . More precisely, if u is a nontrivial solution to (1), in the case 0 < q ≤ p − 1 we have [Cao-V. ’17] (recall that u ∈ L q loc ( σ ) ) � p − 1 − q �� q p − 1 u q d σ � � σ ( K ) ≤ C cap p ( K ) . p − 1 K In the case q ≥ p − 1 , we have K u ) p − 1 − q . σ ( K ) ≤ C cap p ( K ) (min I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 5 / 38

Classes of measures (continuation) For q = p − 1 , we get the important class of Maz’ya measures σ ( K ) ≤ C cap p ( K ) . Another important class of measures is associated with Riesz capacities � � || f || r f ∈ L r + (R n ) cap α, r ( E ) = inf L r (R n ) : I α f ≥ 1 on E , , for any E ⊂ R n . Here I α = ( − ∆) − α 2 is the Riesz potential of order 0 < α < n and 1 < r < ∞ . Notice that cap 1 , p ( K ) is equivalent to cap p ( K ) . The corresponding class of Maz’ya measures (occurs in the case q q > p − 1 and d σ = dx , with α = p and r = q − p +1 ), σ ( K ) ≤ C cap α, r ( K ) characterizes the weighted norm inequality ∀ f ∈ L r ( dx ) . || I α f || L r ( d σ ) ≤ C || f || L r ( dx ) , (3) I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 6 / 38

Weighted norm inequalities for Riesz potentials More general two-weight inequalities (with measures µ and σ ) ∀ f ∈ L r ( d µ ) , || I α ( fd µ ) || L q ( d σ ) ≤ C || f || L r ( d µ ) , (4) play a role for fractional Laplace equations (2). Here q > 0 and r > 1 . In the end-point case r = 1 , we consider the weighted norm inequality ∀ ν ∈ M + (R n ) , || I α ν || L q ( d σ ) ≤ C || ν || , (5) where || ν || denotes the total variation of the (finite) measure ν . We denote by κ the least constant C in the estimates of type (4), (5). Here the linear Riesz potential of d µ = fd ν is defined by � ∞ f ( y ) d ν µ ( B ( x , r )) dr � I α ( fd ν ) = c | x − y | n − α = c r , r n − α R n 0 where c = c ( α, n ) > 0 . We write I α f if d ν = dx ; I α ν if f ≡ 1 . I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 7 / 38

Nonlinear potentials The Wolff potential (more precisely Havin-Maz’ya-Wolff potential) for µ ∈ M + (R n ) and 1 < p < ∞ , 0 < α < n p , is defined by � ∞ 1 p − 1 dr � µ ( B ( x , r )) � x ∈ R n . W α, p µ ( x ) = r , r n − α p 0 In the special case α = 1 , we use the notation � ∞ 1 p − 1 dr � µ ( B ( x , r )) � x ∈ R n . W p µ ( x ) = r , r n − p 0 Notice that W p µ �≡ ∞ , equivalently W p µ ( x ) < ∞ q.e., iff � ∞ 1 p − 1 dr � µ ( B (0 , r )) � < ∞ . (6) r n − p r 1 I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 8 / 38

Weighted norm inequalities for nonlinear potentials In the case q > p − 1 , the inequality closely related to (1): 1 q p − 1 ( d µ ) . p − 1 ||W p ( fd µ ) || L q ( d σ ) ≤ κ || f || , ∀ f ∈ L q p − 1 ( d µ ) L This is necessary for the existence of a nontrivial supersolution − ∆ p u ≥ u q d σ + µ . A necessary and sufficient condition: W p [( W p µ ) q d σ ] ≤ C W p µ < ∞ . In the case 0 < q < p − 1 , the weighted norm inequality related to (1): 1 p − 1 , ∀ ν ∈ M + (R n ) . ||W p ν || L q ( d σ ) ≤ κ || ν || 1 p − 1 Equivalently, || φ || L q ( d σ ) ≤ κ || ∆ p φ || L 1 (R n ) for all p -superharmonic test functions φ , smooth and vanishing at ∞ in R n . This inequality holds iff there exists a nontrivial supersolution u ∈ L q ( σ ) to − ∆ p u ≥ σ u q . I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 9 / 38

Localized weighted norm inequalities 0 < q < p − 1 For a ball B ⊂ R n , denote by κ ( B ) the least constant in the localized weighted norm inequality for Wolff potentials, 1 p − 1 , ∀ ν ∈ M + (R n ) . ||W p ν || L q ( d σ B ) ≤ κ ( B ) || ν || (7) Here σ B = σ | B is σ restricted to a ball B ; || ν || = ν (R n ) . Equivalently, κ ( B ) can be used in place of κ ( B ) . Here κ ( B ) is the least constant in the localized weighted norm inequality for the p -Laplacian, � 1 �� 1 q | ϕ | q d σ p − 1 ≤ κ ( B ) || ∆ p ϕ || (8) L 1 (R n ) , B for all smooth test functions ϕ such that − ∆ p ϕ ≥ 0 , lim inf x →∞ ϕ ( x ) = 0 . I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 10 / 38

Intrinsic nonlinear potentials 0 < q < p − 1 A new ( intrinsic ) nonlinear potential K p , q σ was introduced in [Cao-Verbitsky ’17] 1 p − 1 dr  q ( p − 1)  � ∞  [ κ ( B ( x , r ))] p − 1 − q K p , q σ ( x ) = (9) r .  r n − p 0 These potentials are closely related to solutions of − ∆ p u ≥ σ u q . A fractional version is defined by 1 p − 1 dr  q ( p − 1)  � ∞  [ κ ( B ( x , r ))] p − 1 − q K p , q ,α σ ( x ) = r ,  r n − α p 0 where κ ( B ) is the constant in the inequality 1 p − 1 . ||W α, p ν || L q ( d σ B ) ≤ κ ( B ) || ν || I. E. Verbitsky (University of Missouri) Classes of solutions to quasilinear equations June 2020 11 / 38