Regularity of the extremal solutions associated with elliptic systems

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 ℝ N and f is a C 2 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 < (2 α * (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.

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