On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval.

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation - Δ p u = λ | u | p - 2 u + a (x) | u | q - 2 u in a bounded domain Ω ⊂ R N , where 1 < q < p , λ ∈ R , and a is a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in { x ∈ Ω : a (x) > 0 } , when the parameter λ lies in a neighborhood of the critical value λ ∗ : = inf ∫ Ω | ∇ u | p d x / ∫ Ω | u | p d x : u ∈ W 0 1 , p (Ω) \ { 0 } , ∫ Ω a | u | q d x ≥ 0 . Among main results, we show that if p > 2 q and either ∫ Ω a φ p q d x = 0 or ∫ Ω a φ p q d x > 0 is sufficiently small, then such solutions do exist in a right neighborhood of λ ∗ . Here φ p is the first eigenfunction of the Dirichlet p-Laplacian in Ω . This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case q > p or in the sublinear case q < p = 2 the nonexistence takes place for any λ ≥ λ ∗ . Moreover, we prove that if p > 2 q and ∫ Ω a φ p q d x > 0 is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of λ ∗ , two of which are strictly positive in { x ∈ Ω : a (x) > 0 } . [ABSTRACT FROM AUTHOR]