Nonlinear integral equations on bounded domains

Blowup analysis for integral equations on bounded domains

“…We first prove u q (x q ) → ∞ as q → (q α ) + . By contrary, we assume u q (x) ≤ C uniformly, then by the results in [6] it is easy to see that the C 1 norm of u q (x) is also uniformly bounded, thus u q (x) is equicontinuous. Then we conclude that u q (x) → u * (x) ≥ 0 pointwise as q → (q α ) + and u * (x) is a nonnegative solution to (1.1).…”

“…So f * = (u * ) pα−1 is a nonnegative solution to (P qα ). Notice that f q = u p−1 q is the energy maximizing positive solution to (P q ), and ξ α,q (Ω) → ξ α,qα (Ω) > 0 as q → (q α ) + (see [6]). Then…”

“…The domain Ω is strictly convex. By using the method of moving planes to integral equation (1.1), which is omitted here since it is similar to Theorem 3.4 in [6] (see also [3]), we can prove that there exist t 0 > 0, α > 0 depending on the domain only, such that for every x ∈ ∂Ω,…”

“…(2.10) By [6] we know ξ α,qα (Ω) = ξ α,qα (R n ). Then v pα−1 must be an extremal function to ξ α,qα (R n ) since…”

“…For 1 < α < n, the existence of energy maximizing positive solution in subcritical case 2 < q < 2n n+α , and nonexistence of energy maximizing positive solution in critical case q = 2n n+α are proved in [6]. For α > n, the existence of energy minimizing positive solution in subcritical case 0 < q < 2n n+α , and nonexistence of energy minimizing positive solution in critical case q = 2n n+α are also proved in [4]. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as q → ( 2n n+α ) + (in the case of 1 < α < n), and the blowup behaviour of energy minimizing positive solution as q → ( 2n n+α ) − (in the case of α > n) are analyzed.…”