Regularity of the extremal solutions associated with elliptic systems

Abstract

AbstractWe examine the elliptic system given by$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill &amp; {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$where λ, γ are positive parameters, Ω is a smooth bounded domain in ℝNandfis aC2positive, nondecreasing and convex function in [0, ∞) such thatf(t)/t→ ∞ ast→ ∞. 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 thatN&lt; (2α*(2 − τ+) + 2τ+)/(τ+)max{1, τ+}, where α*&gt; 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) forN&lt; 5. Moreover, if τ−= τ+, thenu*,v* ∈L∞(Ω) forN&lt; 10.