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 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.
Previous Page Next Page