Abstract
We study the equation \(-\text {div}(A(x,\nabla u))=|u|^{q_1-1}u|\nabla u|^{q_2}+\mu \) where \(A(x,\nabla u)\sim |\nabla u|^{p-2}\nabla u\) in some suitable sense, \(\mu \) is a measure and \(q_1\), \(q_2\) are nonnegative real numbers and satisfy \(q_1+q_2>p-1\). We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.
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References
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory, Grundlehren der Mathematischen Wisenschaften, vol. 31. Springer, Berlin (1999)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)
Bidaut-Véron, M.F.: Removable singularities and existence for a quasilinear equation with absorption or source term and measure data. Adv. Nonlinear Stud. 3, 25–63 (2003)
Bidaut-Véron, M.F., Nguyen, Q.H., Véron, L.: Quasilinear Lane–Emden equations with absorption and measure data. J. Math. Pures Appl. 102, 315–337 (2014)
Bidaut-Véron, M.F., Nguyen, Q.H., Véron, L.: Quasilinear and Hessian Lane–Emden systems with reaction and measure data. Potential Anal. https://doi.org/10.1007/s11118-018-9753-z
Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Sup. Pisa 28, 741–808 (1999)
Duzaar, F., Mingione, G.: Gradient estimates via linear and nonlinear potential. J. Funct. Anal. 259, 2961–2998 (2010)
Fefferman, R.: Strong differentiation with respect to measure. Am. J. Math. 103, 33–40 (1981)
Honzik, P., Jaye, B.: On the good-\(\lambda \) inequality for nonlinear potentials. Proc. Am. Math. Soc. 140, 4167–4180 (2012)
Kuusi, T., Mingione, G.: Linear potential in nonlinear potential theory. Arch. Ration. Mech. Anal. 207, 207–246 (2013)
Maz’ya, V.G., Verbitsky, I.E.: Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Arkiv. Mat. 33, 81–115 (1995)
Nguyen, Q.H.: Potential estimates and quasilinear parabolic equations with measure data, pp. 1–120. arXiv:1405.2587, submitted
Nguyen, Q.-H., Phuc, N.C.: Good-\(\lambda \) and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications. Math. Ann. 374, 67–98 (2019)
Nguyen, Q.-H., Phuc, N.C.: Pointwise gradient estimates for a class of singular quasilinear equation with measure data. J. Funct. Anal. 278(5), 108391 (2020)
Nguyen, Q.-H., Phuc, N.C.: Existence and regularity estimates for quasilinear equations with measure data: the case \(1<p\le \frac{3n-2}{2n-1}\). Submitted for publication
Nguyen, Q.-H., Phuc, N.C.: Quasilinear Riccati type equations with oscillatory and singular data. To appear in Advanced Nonlinear Studies. arXiv:2003.03724
Nguyen, Q.H., Véron, L.: Quasilinear and Hessian type equations with exponential reaction and measure data. Arch. Rat. Mech. Anal. 214, 235–267 (2014)
Phuc, N.C., Verbitsky, I.E.: Quasilinear and Hessian equations of Lane–Emden type. Ann. Math. 168(2008), 859–914 (2008)
Phuc, N.C.: Quasilinear Riccati type equations with super-critical exponents. Commun. Partial Differ. Equ. 35, 1958–1981 (2010)
Phuc, N.C.: Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with application to quasilinear Riccati type equations. Adv. Math. 250, 378–419 (2014)
Trudinger, N.S., Wang, X.J.: Quasilinear elliptic equations with signed measure data. Discrete Contin. Dyn. Syst. 23, 477–494 (2009)
Véron, L.: Local and Global Aspects of Quasilinear Degenerate Elliptic Equations. World Scientific Publishing Co. Pte. Ltd., Hackensack (2017)
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Bidaut-Véron, MF., Nguyen, QH. & Véron, L. Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data. Calc. Var. 59, 148 (2020). https://doi.org/10.1007/s00526-020-01808-3
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DOI: https://doi.org/10.1007/s00526-020-01808-3