By Benjamin Peirce

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Diff. Int. Equ. 16 (2003), no. 3, 349–384. , Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. 4 (1996), 121-137. , Semi-classcal states for nonlinear schroedinger equations, J. Funct. Anal. 149 (1997), 245-265. M. J. Math. Anal. 31 (1999), 63-79. [13] Floer, A. , and Weinstein, A. , Nonspreading wave packets for the cubic Schrodinger equation with a bounded potential, J. Funct. Anal. 69, (1986), pp. 397-408. [14] Gierer, A. , A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), pp.

Diff. Eq. 163 (2000), no. 2, 429–474. [8] Dancer, E. , Multipeak solutions for a singularly perturbed Neumann problem. Pacific J. Math. 189 (1999), no. 2, 241-262. , On a class of solutions with non-vanishing angular momentum for nonlinear Schrdinger equations. Diff. Int. Equ. 16 (2003), no. 3, 349–384. , Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. 4 (1996), 121-137. , Semi-classcal states for nonlinear schroedinger equations, J. Funct. Anal. 149 (1997), 245-265.

Similarly, if we consider a function u ˜j = ψj (εy1 )vj (y ), see (109), reasoning as before we find det g(−∆g u ˜j + u ˜j − pup−1 ˜j ) = (D1 )j + (D2 )j + (D3 )j + (D4 )j + (D5 )j , 2,ε u (120) where ˜ 1 × ε2 ψ vj + 2ε(2y3 h12 − f ) + G ˜ 2 ) × εψ ∂2 vj ; (D1 )j = − 1 + εy3 (h22 − h11 ) + G 0 j j (D2 )j 2 ˜3 −1 + ε2y3 h22 − ε2 (2y3 h12 − f0 )2 + 2y32 h22 (h22 − h11 ) − y32 (h )12 − 2(y2 − f0 )y3 ∂2 h22 + G = 2 × ψj ∂22 vj ; ˜ 4 ψj ∂ 2 vj + G ˜ 5 εψ vj = − 1 + εy3 (h11 + h22 ) + G 33 j (D3 )j + (D4 )j 1 2 g 11 − y3 ∂2 h11 + ∂2 h22 ε2 2y3 ∂1 h12 − f0 − (y2 − f0 )∂22 2 2 2 2 ˜6 +G × ψj ∂2 vj ; −ε(h11 + h22 ) − ε2 2y3 (h11 + h22 + h12 + h11 h22 ) + (y2 − f0 ) ∂2 h11 + ∂2 h22 = ˜7 +G × ψj ∂3 vj ; (D5 )j 1 ˜8 = −p w0p−1 + ε(p − 1)w0p−2 w1 + ε2 (p − 1)w0p−2 w2 + ε2 (p − 1)(p − 2)w0p−3 w12 + G 2 × ˜ 4 ψj vj .