Abstract
We investigate Liouville-type theorems for elliptic equations with a drift and with a potential posed in bounded domains. We provide sufficient conditions on the potential and on the drift term in order to the equation does not admit nontrivial bounded solutions. We also show that such conditions are optimal. Indeed, when they fail, the elliptic equation possesses infinitely many bounded solutions.
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References
Fichera, G.: Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine. Mem. Accad. Naz. Lincei I(8), 1–30 (1956)
Friedman, A.: Stochastic Differential Equations and Applications, vol. I, II. Academic Press (1976)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer (1983)
Grigor’yan, A.: Bounded solution for the Schrödinger equation on noncompact Riemannian manifolds. J. Soviet Math. 51, 2340–2349 (1990)
Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)
Khasminskii, R.Z.: Diffusion processes and elliptic differential equations degenerating at the boundary of the domain. Theory Prob. Appl. 4, 400–419 (1958)
Oleinik, O.A., Radkevic, E.V.: Second Order Equations with Nonnegative Characteristic Form. Plenum Press (1973)
Pozio, M.A., Punzo, F., Tesei, A.: Criteria for well-posedness of degenerate elliptic and parabolic problems. J. Math. Pures Appl. 90, 353–386 (2008)
Punzo, F., Tesei, A.: Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients. Ann. Inst. H. Poincaré Anal. Non Lin. 26, 2001–2024 (2009)
Weyl, H.: On the volumes of Tubes. Am. J. Math. 61(2), 461–472 (1939)
Acknowledgements
The authors would like to thank Prof. Dario D. Monticelli and Prof. Matteo Muratori for helpful discussions. The second author is partially supported by the PRIN Project Direct and Inverse Problems for Partial Differential Equations: Theoretical Aspects and Applications (grant no. 201758MTR2, MIUR, Italy).
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Communicated by L. Szekelyhidi.
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Biagi, S., Punzo, F. A Liouville-type theorem for elliptic equations with singular coefficients in bounded domains. Calc. Var. 62, 53 (2023). https://doi.org/10.1007/s00526-022-02389-z
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DOI: https://doi.org/10.1007/s00526-022-02389-z