xiv AN OVERVIEW holds. In the p-superlinear case it is customary to also assume the following Ambrosetti-Rabinowitz type condition to ensure that Φ satisfies the pPSq condition: (11) 0 ă μ F px, tq ď tf px, tq @x P Ω, |t| large for some μ ą p. In the semilinear case p “ 2, we can then obtain a nontrivial solution of problem (7) using the well-known saddle point theorem of Rabinowitz as follows. Fix a w0 P H `z t0u and let X “ u “ v ` s w0 : v P H ´ , s ě 0, }u} ď R ( , A “ v P H ´ : }v} ď R ( Y u P X : }u} “ R ( , B “ w P H ` : }w} “ r ( with H ˘ as in (4) and R ą r ą 0. Then (12) max ΦpAq ď 0 ă inf ΦpBq if R is suﬃciently large and r is suﬃciently small, and A homotopically links B with respect to X in the sense that γpXq X B ‰ H @γ P Γ where Γ “ γ P CpX, H1pΩqq 0 : γ| A “ id A ( . So it follows that c :“ inf γPΓ sup uPγpX q Φpuq is a positive critical level of Φ (see Proposition 3.21). Again we may ask whether linking sets that would enable us to use this argument in the quasilinear case p ‰ 2 can be constructed. In Perera and Szulkin [105] the following such construction based on the piercing property of the index (see Proposition 2.12) was given. Recall that the cone CA0 on a topological space A0 is the quotient space of A0 ˆ r0, 1s obtained by collapsing A0 ˆ t1u to a point. We identify A0 ˆ t0u with A0 itself. Fix an h P CpCΨλk,Sq such that hpCΨλk q is closed and h| Ψλk “ id Ψλk , and let X “ tu : u P hpCΨλk q, 0 ď t ď R ( , A “ tu : u P Ψλk, 0 ď t ď R ( Y u P X : }u} “ R ( , B “ ru : u P Ψλ k`1 ( with R ą r ą 0. Then (12) still holds if R is suﬃciently large and r is suﬃciently small, and A homotopically links B with respect to X by (5) and the following theorem proved in Section 3.6, so Φ again has a positive critical level.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.