viii INTRODUCTION the late 1970’s and early 1980’s, there was a great deal of interest in the study of semilinear elliptic equations, to a great degree motivated by geometric applications. For instance, recall the Yamabe problem: Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Is there a conformal metric g = cg, so that the scalar curvature of (M, g) is constant? In this context, the following equation was studied extensively: Δu+|u| 4 n−2 u = 0,x ∈ Rn where u ∈ ˙ 1 (Rn) = u : ∇u ∈ L2(Rn) . Using this information, Trudinger, Aubin and Schoen solved the Yamabe problem in the aﬃrmative (see [89] and references therein). We will concentrate in the case n = 3, so that the equation becomes Δu + u5 = 0,x ∈ R3,u ∈ ˙ 1 (R3). This equation is “critical” because the linear part (Δ) and the nonlinear part (u5) have the same “strength”, since if u is a solution, so is 1 λ 1 2 u ( x λ ) and both the linear part and the nonlinear part transform “in the same way” under this change. The equation is “focusing” because the linear part (Δ) and the nonlinear part (u5) have opposite signs and hence they “fight each other”. Note that for the much easier “defocusing” problem Δu−u5 = 0,u ∈ ˙ 1 (R3), it is easy to see that there are no non-zero solutions. The diﬃculty in the study of Δu + u5 = 0 in R3 comes from the “lack of compactness” in the Sobolev embedding u L6(R3) ≤ C3∇u L2(R3) , where C3 is the best con- stant. C3 = π− 1 2 3− 1 2 Γ(3) Γ( 3 2 ) 1 3 (See [100]). Modulo translation and scaling, the only non-negative solution to Δu + u5 = 0,u ∈ ˙ 1 (R3) is W (x) = 1 + |x|2 3 − 1 2 (Gidas-Ni-Nirenberg [42], Kwong [72]). Also W is the unique minimizer to the Sobolev inequality above (Talenti [100]). W is called the “ground state”. W is also the unique radial solution in ˙ 1 ( R3 ) , (without imposing a sign condition). (Pohozaev [88], Kwong [72]). On the other hand, Ding [22] constructed infinitely many variable sign solutions, which are non-radial. Pohozaev [88] also showed that the only solution to the boundary value problem Δu + u5 = 0 in B1ßR3 u|∂B 1 ≡ 0 is u ≡ 0. If instead, we consider the problem Δuε + uε 5 = 0 in B1\Bε uε|∂B 1 ∪∂Bε = 0 then there are non-zero solutions. If we normalize them and let ε → 0, we have uε ∼ ∑ J j=1 (−1)j W x λ j λ 1 2 j , where 0 ≤ λ1(ε) λ2(ε) · · · λJ(ε) (Musso-Pistoia [84], “towers of bubbles”). Through the study of these and related problems, in works of Talenti, Trudinger, Aubin, Schoen, Taubes, Schoen-Uhlenbeck, Sachs-Unlenbeck, Bahri-Coron, Struwe, Br´ ezis-Coron, etc., many important techniques were developed. In particular, through these and others works, the study of the “defect of compactness” and the “bubble decomposition” were systematized through the work of P-L. Lions on concentration-compactness. For nonlinear dispersive equations there are also critical problems, which are related to Δu + u5 = 0.

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