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In this paper, we analyse the semilinear fourth-order problem ( − Δ)2u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and
for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu > 0 in ℝN, and they do not exist when the previous condition fails.
$$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$
We examine the elliptic system given by
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
$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill & {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$ 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
$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$ As a consequence, u*, v* ∈ L∞(Ω) for N < 5. Moreover, if τ− = τ+, then u*, v* ∈ L∞(Ω) for N < 10.
$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$
We prove a new energy or Caccioppoli type estimate for minimisers of the model functional ∫Ω|Du|2 + (det Du)2, where Ω ⊂ 
2 and u: Ω → 2. We apply this to establish C∞ regularity for minimisers except on a closed set of measure zero. We also prove a maximum principle and use this to establish everywhere continuity of minimisers.
