Abstract
We consider the forced problem −Δpu − V(x)∣u∣p−2u = f(x), where Δp is the p-Laplacian (1 < p < ∞) in a domain Ω ⊂ ℝN, V ≥ 0 and QV(u)≔ ∫Ω ∣∇u∣pdx − ∫ΩV∣u∣pdx satisfies the condition (A) below. We show that this problem has a solution for all f in a suitable space of distributions. Then we apply this result to some classes of functions V which in particular include the Hardy potential (1.5) and the potential V(x)= λ1,p(Ω), where λ1,p(Ω) is the Poincaré constant on an infinite strip.
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We would like to thank the referee for useful suggestions.
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Szulkin, A., Willem, M. On some weakly coercive quasilinear problems with forcing. JAMA 140, 267–281 (2020). https://doi.org/10.1007/s11854-020-0088-5
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DOI: https://doi.org/10.1007/s11854-020-0088-5