On subhomogeneous indefinite $p$-Laplace equations in supercritical spectral interval

Abstract

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 continuous 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 criticalmain results, we show that if 𝑝 > 2𝑞 and either ∫︀ Ω 𝑎𝜙 𝑞 𝑝 𝑑𝑥 = 0 or ∫︀ Ω 𝑎𝜙 𝑞 𝑝 𝑑𝑥 > 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 > 2𝑞 and ∫︀ Ω 𝑎𝜙 𝑞 𝑝 𝑑𝑥 > 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}.

On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations

“…Suppose, by contradiction, that ‖∇𝑢 𝑛 ‖ 𝑝 → ∞ along a subsequence. Then, arguing in the same way as in [12,Lemma 2.24], we see that the sequence consisted of normalized functions 𝑣 𝑛 = 𝑢 𝑛 /‖∇𝑢 𝑛 ‖ 𝑝 converges strongly in 𝑊 1,𝑝 0 (Ω) to an eigenfunction 𝑣 0 ∈ 𝐸𝑆(𝜆; 𝑚) ∖ {0}, up to a subsequence. Hence, we obtain a contradiction whenever 𝜆 ̸ ∈ 𝜎(−Δ 𝑝 ; 𝑚).…”

“…Qualitative properties of sign-constant solutions, such as the strict sign vs. dead core formation, continuity with respect to parameters, weights, and exponents, uniqueness issues, etc., are of considerable interest and have been studied, for instance, in [6,18,35,37]. The existence and multiplicity of solutions of (1.5) with respect to parameters 𝜆 and 𝜂 have been investigated, e.g., in [12,13,37,44,50].…”