Abstract
Let 1 < p < N and Ω ⊂ ℝN be an open bounded domain. We study the existence of solutions to equation \((E) - {\Delta _p}u + g(u)\sigma = \mu \) in Ω, where g ∈ C(ℝ) is a nondecreasing function, μ is a bounded Radon measure on Ω and σ is a nonnegative Radon measure on ℝN. We show that if σ belongs to some Morrey space of signed measures, then we may investigate the existence of solutions to equation (E) in the framework of renormalized solutions. Furthermore, imposing a subcritical integral condition on g, we prove that equation (E) admits a renormalized solution for any bounded Radon measure μ. When \(g(t) = |t{|^{q - 1}}t\) with q > p − 1, we give various sufficient conditions for the existence of renormalized solutions to (E). These sufficient conditions are expressed in terms of Bessel capacities.
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Acknowledgement
The author thanks Professor Laurent Véron for proposing the problem and for useful discussions. The research project was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the ���2nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers” (Project Number: 59).
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Gkikas, K.T. Quasilinear elliptic equations involving measure valued absorption terms and measure data. JAMA 153, 555–594 (2024). https://doi.org/10.1007/s11854-023-0321-0
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DOI: https://doi.org/10.1007/s11854-023-0321-0