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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 [14], 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 [1] 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. [2] 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.