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

Vladimir Bobkov () and Mieko Tanaka ()
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Vladimir Bobkov: Ufa Federal Research Centre, RAS
Mieko Tanaka: Tokyo University of Science

Partial Differential Equations and Applications, 2023, vol. 4, issue 3, 1-38

Abstract: Abstract We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $$-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$$ - Δ p u = λ m ( x ) | u | p - 2 u + η a ( x ) | u | q - 2 u + f ( x ) in a bounded domain $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N , where $$q 1$$ p > 1 solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the p-Laplacian provided $$m,f \in L^\gamma (\Omega )$$ m , f ∈ L γ ( Ω ) with $$\gamma >N$$ γ > N . For completeness, we also investigate the existence of solutions.

Keywords: p-Laplacian; Sublinear perturbation; Indefinite weight; Antimaximum principle; Maximum principle; Harnack inequality; Picone inequality; Existence; Linking method; 35J92; 35B50; 35B65; 35B09; 35B30; 35A01; 35B38 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00235-1

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