By Antonio Ambrosetti, Andrea Malchiodi

This e-book has been provided the Ferran Sunyer i Balaguer 2005 prize. the purpose of this monograph is to debate a number of elliptic difficulties on Rn with major features: they are variational and perturbative in nature, and traditional instruments of nonlinear research in line with compactness arguments can't be utilized in common. For those difficulties, a extra particular technique that takes good thing about any such perturbative surroundings seems the main acceptable. the 1st a part of the booklet is dedicated to those summary instruments, which offer a unified body for numerous purposes, frequently thought of varied in nature. Such purposes are mentioned within the moment half, and comprise semilinear elliptic difficulties on Rn, bifurcation from the fundamental spectrum, the prescribed scalar curvature challenge, nonlinear Schrödinger equations, and singularly perturbed elliptic difficulties in domain names. those themes are provided in a scientific and unified manner.

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**Perturbation Methods and Semilinear Elliptic Problems on R^n **

This ebook has been presented the Ferran Sunyer i Balaguer 2005 prize. the purpose of this monograph is to debate numerous elliptic difficulties on Rn with major features: they are variational and perturbative in nature, and conventional instruments of nonlinear research according to compactness arguments can't be utilized in basic.

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**Additional resources for Perturbation Methods and Semilinear Elliptic Problems on R^n **

**Example text**

21. It follows that uε = zε + wε (zε ) is a critical point of Iε , as required. 3. 6 Morse index of the critical points of Iε Under some further regularity assumptions, it is possible to evaluate the Morse index of the critical points of Iε found above. As before, we will suppose that Z = {zξ : ξ ∈ Rd } is a non-degenerate critical manifold of I0 , with tangent space spanned by qi = ∂ξi zξ / ∂ξi zξ . Moreover, we will assume that (Dk qi | qj ) = 0, ∀ i, j, k = 1, . . , d. 30) Let ξε be a sequence of critical points of Γ = G|Z and suppose that ξε → ξ ∗ as ε → 0.

15 we get Φε (z) = I0 (z + wε ) + G(ε, z + wε ) = c0 + 12 (I0 (z)[wε ] | wε ) + G(ε, z) 2 + (Du G(ε, z) | wε ) + 12 (Duu G(ε, z)[wε ] | wε ) + o( wε 2 ). 26 with β = 12 α we ﬁnd (I0 (z)[wε ] | wε ) = O( wε 2 ) = o(εα ), as ε → 0. 38) One also has 2 (Duu G(ε, z)[wε ] | wε ) = o( wε ) 2 ) = o(εα ), as ε → 0. 39) Moreover, since Du G(ε, z) = o(εα/2 ) and wε = o(εα/2 ), we get (Du G(ε, z) | wε ) = o(εα ), as ε → 0. 40) as ε → 0. 37) we ﬁnd that Φε (z) = c0 + εα G(z) + o(εα ). 16. 28. Suppose that I0 ∈ C 2 (H, R) has a smooth critical manifold Z which is non-degenerate.

1 The abstract setting We will consider the elliptic problem −∆u + u = (1 + εh(x))up , u ∈ W 1,2 (Rn ), u > 0, (Pε ) where n ≥ 3 and p is a subcritical exponent, namely 1

0).