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Additional info for Asymptotic analysis of solutions to elliptic and parabolic problems
P> 7−n n−3 . (iv) Linear combinations with positive coefficients of functions from (i) (iii). 4: The function u is a solution of the problem ∆u + q(u)u = 0 u=0 in C+ on ∂Ω × (0, ∞). 12) u As before, we set G(u) = 0 q(v)v dv. 2, Kozlov , we get 0 = (∆ u + f (u))(x · ∇ u) + uxn xn (x · ∇ u) = div ∇ u(x · ∇ u) − x n−3 |∇ u|2 + x G(u) + |∇ u|2 2 2 − (n − 1)G(u) + uxn xn (x · ∇ u). This implies that 0 = div (∇u(x · ∇ u)) + div − x |∇u|2 + x G(u) + u2xn 2 n−3 + |∇u|2 − (n − 1)G(u). 13) Set CN = Ω × (1, N ) and ΓN = ∂Ω × (1, N ) and observe that ∇u = ∂u ν ∂ν on ΓN .
19 The lemma as well as the corollary also hold with C and ∂C replaced by C+ and ∂Ω × (0, ∞), respectively, but with the additional condition t ≥ 0. References  S. Agmon, A. Douglis, L. Nirenberg, Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions. I. Communications on pure and applied mathematics 12 (1959), 623–727.  C. J. Amick, J. F. Toland, Nonlinear elliptic eigenvalue problems on an infinite strip - global theory of bifurcation and asymptotic bifurcation.
Denote the eigenvalues in increasing order and J be the multiplicity of λ1 . This means that λ1 = . . = λJ < λJ+1 ≤ λJ+2 ≤ . . , λk → ∞ as k → ∞. Let furthermore φ1 , φ2 , . . , φJ be a basis in the eigenspace of λ1 which is (0) orthogonal in L2 -sense. The conditions on Aij , A(0) and Ω imply that N φk ∈ W 1,p (Ω) for some p > 2 and k = 1, 2, . . 2 in Appendix A. 2. 14) where s2 denotes an arbitrary number in (2, n]. 12) we have ex(1) tended the matrices Aij , i, j = 1, . . , n, and A(1) to Ω × R by setting (1) Aij (x, t) = A(1) (x, t) = 0 for t < 0, so κ(t) is defined for every t ∈ R.