Abstract
We consider positive solutions to the weighted elliptic problem
where B is the standard unit ball of \({\mathbb {R}}^N\). We give a complete answer for the existence question for \(N':=N+\theta >2\) and \(p > 0\). In particular, for \(N' > 2\) and \(\tau :=\ell -\theta >-2\), it is shown that for \(0< p \le p_s:=\frac{N'+2+2 \tau }{N'-2}\), the only nonnegative solution to the problem is \(u \equiv 0\). This nonexistence result is new, even for the classical case \(\theta = \ell = 0\) and \(\frac{N}{N-2} < p \le \frac{N+2}{N-2}\), \(N \ge 3\). The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption.
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References
Alarcón, S., García-Melián, J., Quaas, A.: Optimal Liouville theorem for supersolutions of elliptic equations with the laplacian. Ann. Sc. Norm. Super. Pisa Cl. Sci. 16, 129–158 (2016)
Armstrong, S.C., Sirakov, B.: Nonexistence of positive supersolutions of elliptic equations via the maximum principle. Commun. PDEs 36, 2011–2047 (2011)
Bandle, C., Marcus, M.: The positive radial solutions of a class of semilinear elliptic equations. J. Reine Angew. Math. 401, 25–59 (1989)
Dávila, J., del Pino, M., Musso, M., Wei, J.: Fast and slow decay solutions for supercritical elliptic problems in exterior domains. Calc. Var. PDEs 32(4), 453–480 (2008)
Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology, vol. 1. Springer, Berlin, Physical origins and classical methods (1990)
Du, Y.H., Guo, Z.M.: Finite Morse index solutions of weighted elliptic equations and the critical exponents. Calc. Var. PDEs 54, 3161–3181 (2015)
Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34(4), 525–598 (1981)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)
Guo, Z.M., Wan, F.S.: Further study of a weighted elliptic equation. Sci. China Math. 60, 2391–2406 (2017)
Liskevich, V., Skrypnik, I.I., Skrypnik, I.V.: Positive supersolutions to general nonlinear elliptic equations in exterior domains. Manuscripta Math. 115, 521–538 (2004)
Ni, W.M.: On the elliptic equation in \(\Delta u+K(x)u^{\frac{N+2}{N-2}}=0\), its generalizations, and applications in geometry. Indiana Univ. Math. J. 31, 493–529 (1982)
Przeradzki, B., Stańczy, R.: Positive solutions for sublinear elliptic equations. Colloquium Mathematicum 92, 141–151 (2002)
Santanilla, J.: Existence and nonexistence of positive radial solutions of an elliptic Dirichlet problem in an exterior domain. Nonlinear Anal. TMA 25, 1391–1399 (1995)
Stańczy, R.: Positive solutions for superlinear elliptic equations. J. Math. Anal. Appl. 283, 159–166 (2003)
Zhang, Q.: A general blow-up result on nonlinear boundary value problems on exterior domains. Proc. R. Soc. Edinburgh 131A, 451–475 (2001)
Acknowledgements
Z.G is supported by NSFC (No. 11571093 and 11171092). X.H is supported by NSFC (No. 11701181). D.Y is partially supported by Science and Technology Commission of Shanghai Municipality (STCSM) (No. 18dz2271000). The authors thank the anonymous referees for their careful reading.
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Guo, Z., Huang, X. & Ye, D. Existence and nonexistence results for a weighted elliptic equation in exterior domains. Z. Angew. Math. Phys. 71, 116 (2020). https://doi.org/10.1007/s00033-020-01338-0
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DOI: https://doi.org/10.1007/s00033-020-01338-0