2 9 N ov 2 00 5 Ground state alternative for p - Laplacian with potential term

Let Ω be a domain in Rd, d ≥ 2, and 1 < p <∞. Fix V ∈ Lloc(Ω). Consider the functional Q and its Gâteaux derivative Q′ given by Q(u) := ∫ Ω (|∇u|+V |u|)dx, 1 p Q(u) := −∇·(|∇u|∇u)+V |u|u. If Q ≥ 0 on C∞ 0 (Ω), then either there is a positive continuous function W such that ∫ W |u|p dx ≤ Q(u) for all u ∈ C∞ 0 (Ω), or there is a sequence uk ∈ C ∞ 0 (Ω) and a function v > 0 satisfying Q ′(v) = 0, such that Q(uk) → 0, and uk → v in L p loc(Ω). In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′(u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous functionW such that for every ψ ∈ C∞ 0 (Ω) satisfying ∫ ψv dx 6= 0 there exists a constant C > 0 such that C−1 ∫ W |u|p dx ≤ Q(u) + C ∣