A constructive approach to positive solutions of \(\delta_pu+f(u,u)\) on Riemannian manifolds. (English) Zbl 1353.58009
Summary: A. Grigor’yan and Y. Sun [Commun. Pure Appl. Math. 67, No. 8, 1336–1352 (2014; Zbl 1296.58011)] (with \(p=2\)) and Y. Sun [J. Math. Anal. Appl. 419, No. 1, 643–661 (2014; Zbl 1297.35297)] (with \(p>1\)) proved that if
\[
\sup\limits_{r\gg 1}\mathrm{vol}(B(x_0,r)) r^{\frac{p\sigma}{p-\sigma -1}}(\ln r)^{\frac{p-1}{p-\sigma -1}}<\infty
\]
then the only non-negative weak solution of \(\Delta_p u+u^\sigma \leq 0\) on a complete Riemannian manifold is identically 0; moreover, the powers of \(r\) and \(\ln r\) are sharp. In this note, we present a constructive approach to the sharpness, which is flexible enough to treat the sharpness for \(\Delta_p u + f(u, \nabla u) \leq 0\). Our construction is based on a perturbation of the fundamental solution to the \(p\)-Laplace equation, and we believe that the ideas introduced here are applicable to other nonlinear differential inequalities on manifolds.
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35J70 | Degenerate elliptic equations |
| 35R01 | PDEs on manifolds |
References:
| [1] | D’Ambrosio, L.; Mitidieri, E., A priori estimates and reduction principles for quasilinear elliptic problems and applications, Adv. Differ. Equ., 17, 935-1000 (2012) · Zbl 1273.35138 |
| [2] | Caristi, G.; D’Ambrosio, L.; Mitidieri, E., Liouville theorems for some nonlinear inequalities, Proc. Steklov Inst. Math., 260, 90-111 (2008) · Zbl 1233.35207 |
| [3] | Caristi, G.; Mitidieri, E.; Pohozaev, S. I., Some Liouville theorems for quasilinear elliptic inequalities, Dokl. Math., 79, 118-124 (2009) · Zbl 1387.35207 |
| [4] | Cheng, S. Y.; Yau, S.-T., Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., 28, 333-354 (1975) · Zbl 0312.53031 |
| [5] | Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34, 525-598 (1981) · Zbl 0465.35003 |
| [6] | Grigor’yan, A.; Sun, Y., On non-negative solutions of the inequality \(\Delta u + u^\sigma \leq 0\) on Riemannian manifolds, Commun. Pure Appl. Math., 67, 1336-1352 (2014) · Zbl 1296.58011 |
| [7] | Mitidieri, E.; Pokhozhaev, S. I., Nonexistence of positive solutions for quasilinear elliptic problems on \(R^N\), Proc. Steklov Inst. Math., 227, 186-216 (1999) · Zbl 1056.35507 |
| [8] | Serrin, J.; Zou, H., Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189, 79-142 (2002) · Zbl 1059.35040 |
| [9] | Sun, Y., Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds, J. Math. Anal. Appl., 419, 643-661 (2014) · Zbl 1297.35297 |
| [10] | Sun, Y., Uniqueness results for non-negative solutions of quasi-linear inequalities on Riemannian manifolds |
| [11] | Sun, Y., On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds · Zbl 1318.58010 |
| [12] | Tersenov, Ar. S., On sufficient conditions for the existence of radially symmetric solutions of the \(p\)-Laplace equation, Nonlinear Anal., 95, 362-371 (2014) · Zbl 1285.35016 |
| [13] | Vèron, L., On the equation \(- \Delta u + e^u - 1 = 0\) with measure as boundary data, Math. Z., 273, 1-17 (2013) · Zbl 1276.35082 |
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