AN OVERVIEW xiii piiiq p-superlinear if lim tÑ˘8 f px, tq |t|p´2 t 8 @x P Ω. Consider the asymptotically p-linear case where lim tÑ˘8 f px, tq |t|p´2 t λ, uniformly in x P Ω with λk ă λ ă λk`1, and assume λ R σp´Δpq to ensure that Φ satisfies the pPSq condition. In the semilinear case p 2, let A v P H ´ : }v} R ( , B H ` with H ˘ as in (4) and R ą 0. Then (8) max ΦpAq ă inf ΦpBq if R is sufficiently large, and A cohomologically links B in dimension k ´ 1 in the sense that the homomorphism r k´1 pH 1 0 pΩqzBq Ñ r k´1 pAq induced by the inclusion is nontrivial. So it follows that problem (7) has a solution u with CkpΦ,uq 0 (see Proposition 3.25). We may ask whether this well-known argument can be modified to obtain the same result in the quasilinear case p 2 where we no longer have the splitting given in (4). We will give an affirmative answer as follows. Let A Ru : u P Ψλk ( , B tu : u P Ψλ k`1 , t ě 0 ( with R ą 0. Then (8) still holds if R is sufficiently large, and A cohomolog- ically links B in dimension k ´ 1 by (5) and the following theorem proved in Section 3.7, so problem (7) again has a solution u with CkpΦ,uq 0. Theorem 1. Let A0 and B0 be disjoint nonempty closed symmetric subsets of the unit sphere S in a Banach space such that ipA0q ipSzB0q k where i denotes the cohomological index, and let A Ru : u P A0 ( , B tu : u P B0, t ě 0 ( with R ą 0. Then A cohomologically links B in dimension k ´ 1. Now suppose f px, 0q 0, so that problem (7) has the trivial solution upxq 0. Assume that (9) lim tÑ0 f px, tq |t|p´2 t λ, uniformly in x P Ω, λk ă λ ă λk`1, and the sign condition (10) p F px, tq ě λk`1 |t|p @px, tq P Ω ˆ R
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