A priori estimates for semistable solutions of semilinear elliptic equations
We consider positive semistable solutions u of Lu + f(u) = 0 with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is an element of C-2 is a positive, nondecreasing, and convex nonlinearity which is super-linear at infinity. Under these assumptions, the boundedness o...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2015 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/80798 |
| Online Access: | https://hdl.handle.net/2117/80798 https://dx.doi.org/10.3934/dcds.2016.36.601 |
| Access Level: | Open access |
| Keyword: | Differential equations, Elliptic Semi-stable solutions extremal solutions regularity boundedness semilinear elliptic equations dimension 4 minimizers Equacions diferencials Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Summary: | We consider positive semistable solutions u of Lu + f(u) = 0 with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is an element of C-2 is a positive, nondecreasing, and convex nonlinearity which is super-linear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension n <= 9, but only established for n <= 4. In this paper we prove the L-infinity bound up to dimension n = 5 under the following further assumption on f: for every epsilon > 0, there exist T = T(epsilon) and C = C(epsilon) such that f '(t) < C f(t)(1+epsilon) for all t > T. This bound will follow from a L-p-estimate for f ' (u) for every p < 3 (and for all n >= 2). Under a similar but more restrictive assumption on f, we also prove the L-infinity estimate when n = 6. We remark that our results do not assume any lower bound on f '. |
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