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We seek minimizers of I in M. e. lim I(uk ) = inf I(u) := m. k →∞ u∈M It follows that Ω |Duk |2 dx is bounded, hence {uk } is bounded in H. By the Weak Compactness Theorem, there exist a subsequence (for simplicity, we still denote it by {uk }), which converges weakly to uo in H. The weak lower semi-continuity of the functional I yields I(uo ) ≤ lim inf I(uk ) = m. 4 Variational Methods 57 we find that uk converges strongly to uo in Lp+1 (Ω), and hence |uo |p+1 dx = 1. Ω That is uo ∈ M , and therefore I(uo ) ≥ m.

Suppose u ∈ W01,p (Ω) for some 1 ≤ p ≤ ∞. Then u Lp (Ω) ≤ C Du Lp (Ω) . 1 i) Now based on this inequality, one can take an equivalent norm of W01,p (Ω) as u W 1,p (Ω) = Du Lp (Ω) . 0 ii) Just for this theorem, one can easily prove it by using the Sobolev innq equality u Lp ≤ u Lq with p = n−q and the H¨ older inequality. However, the one we present in the following is a unified proof that works for both this and the next theorem. Proof. For convenience, we abbreviate u Lp (Ω) as u p . 53) does not hold, then there exists a sequence {uk } ⊂ W01,p (Ω), such that Duk p = 1 , while uk p →∞ , as k→∞.

We will use variational methods to obtain the existence. Roughly speaking, we will associate the equation with the functional J(u) = 1 2 (Lu · u)dx − Ω F (x, u)dx Ω where u F (x, u) = f (x, s)ds. 0 We show that the critical point of J(u) is a weak solution of our problem. Then we will focus on how to seek critical points in various situations. 2 Calculus of Variations For a continuously differentiable function g defined on Rn , a critical point is a point where Dg vanishes. The simplest sort of critical points are global or local maxima or minima.