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 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.