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Regularity of the extremal solutions associated with elliptic systems

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  27 December 2018

A. Aghajani  and
C. Cowan
A. Aghajani
Affiliation:
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran (aghajani@iust.ac.ir.)
C. Cowan
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada (craig.cowan@umanitoba.ca.)
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Abstract

We examine the elliptic system given by

$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill & {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$
where λ, γ are positive parameters, Ω is a smooth bounded domain in ℝN and f is a C2 positive, nondecreasing and convex function in [0, ∞) such that f(t)/t → ∞ as t → ∞. Assuming
$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$
 we show that the extremal solution (u*, v*) associated with the above system is smooth provided that N < (2α*(2 − τ+) + 2τ+)/(τ+)max{1, τ+}, where α* > 1 denotes the largest root of the second-order polynomial
$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$
 As a consequence, u*, v* ∈ L∞(Ω) for N < 5. Moreover, if τ− = τ+, then u*, v* ∈ L∞(Ω) for N < 10.

Keywords

extremal solutionstable solutionregularity of solutionselliptic systems

MSC classification

Primary: 35J47: Second-order elliptic systems
Secondary: 35J61: Semilinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 1037 - 1046
Copyright
Copyright © Royal Society of Edinburgh 2018 

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