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Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in L 1 ( Ω )

M. F. BettaA. MercaldoF. MuratM. M. Porzio — 2002

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is - div ( a ( x ) ( 1 + | u | 2 ) p - 2 2 u ) + b ( x ) ( 1 + | u | 2 ) λ 2 = f in Ω , u = 0 on Ω , where Ω is a bounded open subset of N , N 2 , 2 - 1 / N < p < N , a belongs to L ( Ω ) , a ( x ) α 0 > 0 , f is a function in L 1 ( Ω ) , b is a function in L r ( Ω ) and 0 λ < λ * ( N , p , r ) , for some r and λ * ( N , p , r ) .

Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in  (Ω)

M. F. BettaA. MercaldoF. MuratM. M. Porzio — 2010

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is
 - div ( a ( x ) ( 1 + | u | 2 ) p - 2 2 u ) + b ( x ) ( 1 + | u | 2 ) λ 2 = f in Ω , u = 0 on Ω , 
where Ω is a bounded open subset of N , N > 2, 2-1/, belongs to  (Ω), a ( x ) α 0 > 0 , is a function in (Ω), is a function in L r ( Ω ) and 0 ≤ λ < λ *(), for some and λ *().

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