Estimates for the green function and existence of positive solutions for higher-order elliptic equations (Abstract and Applied Analysis)
We establish a 3 G -theorem for the iterated Green function of ( − ∆ ) p m , ( p ≥ 1 , m ≥ 1 ), on the unit ball B of ℝ n ( n ≥ 1 ) with boundary conditions ( ∂ / ∂ ν ) j ( − ∆ ) k m u = 0 on ∂ B , for 0 ≤ k ≤ p − 1 and 0 ≤ j ≤ m − 1 . We exploit this result to study a class of potentials 𝒥 m , n ( p ) . Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems ( − ∆ ) p m u = h ( ‧ , u ) , in D (in the sense of distributions), lim | x | → 1 ( ( − ∆ ) k m u ( x ) / ( 1 − | x | ) m − 1 ) = 0 , for 0 ≤ k ≤ p − 1 , where D = B or B { 0 } and h is a Borel measurable function on D × ( 0 , ∞ ) satisfying some appropriate conditions related to 𝒥 m , n ( p ) .