Frail and strong solutions for a p-Laplace boundary problem with infinitely many discontinuities

Resumen

We consider the problem (P) −∆_p u(x) = h(x)  f (u(x)) + q(x), x ∈ Ω, with u(x) = 0, x ∈ ∂Ω, where p > 1, Ω ⊆ R^N is a bounded domain with smooth boundary, q ∈ L ^p (Ω), 1/p + 1/p′= 1, h ∈ L∞ (Ω) \ {0}. We assume that f has a countable set of upward and downward discontinuities, D ⊆ R, and verifies | f (s)| ≤ C_1 + C_2 |s|^α, s ∈ R, where α, C 1, C2 > 0 and 1 +α ∈ [p, p ∗], p ∗ = pN/(N − p). Since the standard functional, I, associated to (P) is not Fréchet differentiable but locally Lipschitz continuous on W ^{1, p}_0 (Ω), we apply the variational tools developed by Chang and Clarke. We characterize a frail solution of (P), one that verifies a.e. a condition involving an appropriate multivalued function, as a generalized critical point of I. Given u, a frail solution of (P), we find sufficient conditions for u^ −1 (D) to have zero measure; this is enough for u to become a strong solution of (P): it satisfies (P) a.e. We show conditions for the existence of local-extremum strong solutions of (P). Finally we prove that if f verifies a growing condition involving the first eigenvalue of −∆ , then (P) has a ground state, i.e., a strong solution which globaly minimizes I.