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Regularity of the extremal solutions associated with elliptic systems

Published online by Cambridge University Press:  27 December 2018

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.)

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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