New PDF release: 1D-grid generation by monotone iteration discretization

By Al-Zanaidi M.

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9] M. del Pino, P. Felmer, M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes. J. Differential Equations 182 (2002), no. 2, 511–540. [10] M. del Pino, P. Felmer, M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries. Bull. London Math. Soc. 35 (2003), no. 4, 513–521. [11] M. del Pino, P. Felmer, J. Wei, On the role of distance function in some singular perturbation problems. Comm. Partial Differential Equations 25 (2000), no.

M }). 16) on B = B(ξj , ε), j = 1, . . 17) to obtain as p → +∞: 1 1 up+1 (∆uξ + upξ )∇uξ = ∂n uξ ∇uξ − |∇uξ |2 n + ξ n 2 p + 1 B ∂B = − 64π 2 e p2 (− ∂B 1 1 x − ξj + ∂n ϕj )(− + ∇ϕj ) 2π 2π |x − ξj |2 1 1 x − ξj 1 |− + ∇ϕj |2 n + o( 2 ) 2 2 2π |x − ξj | p = − 64π 2 e 1 ∇ϕj (ξj ) + o( 2 ), 2 p p 1 since we decompose m l=1 G(x, ξl ) = − 2π ln |x − ξj | + ϕj (x) with ϕj (x) an harmonic function near ξj . 16) ∂B 1 ∂n ϕj ∇ϕj − |∇ϕj |2 n = 2 ∆ϕj ∇ϕj = 0. 15), finally we get: ∂(ξj )i F (ξ) = − 32π 2 e 1 32π 2 1 ∂ ϕ (ξ) + o( ) = − ∂(ξj )i ϕm (ξ) + o( 2 ) (ξj )i m 2 2 2 p p γ p since ∇ϕj (ξj ) = 12 ∇ξj ϕm (ξ).

Y. Li, L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations. Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1445–1490. [22] M. Musso, A. Pistoia, Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent. Indiana Univ. Math. J. 51 (2002), no. 3, 541–579. N. Ni, J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48 (1995), no. 7, 731–768. ∗ [24] D. Passaseo, Multiplicity of positive solutions for the equation ∆u+λu+u2 −1 = 0 in noncontractible domains.

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