Ground state alternative for p-Laplacian with potential term

Abstract

Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given byIn 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 function W such that for every ψ ∈ C ∞ 0 (Ω) satisfying ψv dx = 0 there exists a constant C > 0 such that C −1 W |u| p dx ≤ Q(u) + C uψ dx p for all u ∈ C ∞ 0 (Ω). As a consequence, we prove positivity properties for the quasilinear operator Q ′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

A Liouville-type Theorem for Schrödinger Operators

“…The following theorem was recently proved by K. Tintarev and the author [19] (see also [20]). (Ω) to cϕ, where ϕ is a ground state of the operator P and c is a nonzero constant.…”

“…Fix V ∈ L ∞ loc (Ω). Recently the criticality theory for linear equations was extended in [20] In particular, Theorem 2.4 was proved also for such equations. Therefore, it is natural to pose the following problem.…”

On positive solutions of the (p,A)-Laplacian with potential in Morrey space

“…We refer to [24] and references therein for an up to date account. A generalization of the AP theorem to certain quasilinear equations with A being the identity matrix and V ∈ L ∞ loc (Ω) has been carried out in [38]. This was recently extended in [36] to include Agmon's assumptions on the matrix A.…”

A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic space $${{\mathbb {H}}}^N$$ H N

“…[22], Theorem 1.5 (see also [23], Theorem 1.6), existence of the virtual bound state implies that there is no nonzero nonnegative measurable function W such that Q(u) ≥ W u 2 , i.e., the Hardy potential is optimal. A general result in [14] states that, under general conditions on the elliptic operator, the square root of the positive minimal Green function is always a generalized ground state.…”