xii AN OVERVIEW (see Proposition 2.14) and hence (6) CkpΦλ, 0q 0 by (2). The structure provided by this new sequence of eigenvalues is sufficient to adapt many of the standard variational methods for solving semilinear problems to the quasilinear case. In particular, we will construct new linking sets and local linkings that are readily applicable to quasilinear problems. Of course, such constructions cannot be based on linear subspaces since we no longer have eigenspaces to work with. They will instead use nonlinear splittings generated by the sub- and superlevel sets of Ψ that appear in (5), and the indices given there will play a key role in these new topological constructions as we will see next. Consider the boundary value problem (7) $ & % ´Δp u f px, uq in Ω u 0 on where the nonlinearity f is a Carath´ eodory function on Ω ˆ R satisfying the subcritical growth condition |f px, tq| ď C ` |t|r´1 ` 1 ˘ @px, tq P Ω ˆ R for some r P p1, p˚q. Here $ & % np n ´ p , p ă n 8, p ě n is the critical exponent for the Sobolev imbedding W 1, p 0 pΩq ã LrpΩq. Weak solutions of this problem coincide with the critical points of the C1- functional Φpuq ż Ω |∇u|p ´ p F px, uq, u P W 1, p 0 pΩq where F px, tq ż t 0 f px, sq ds is the primitive of f. It is customary to roughly classify problem (7) according to the growth of f as piq p-sublinear if lim tÑ˘8 f px, tq |t|p´2 t 0 @x P Ω, piiq asymptotically p-linear if 0 ă lim inf tÑ˘8 f px, tq |t|p´2 t ď lim sup tÑ˘8 f px, tq |t|p´2 t ă 8 @x P Ω,
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