2 results
Sharp estimates of semistable radial solutions of k-Hessian equations
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 4 / August 2020
- Published online by Cambridge University Press:
- 15 March 2019, pp. 2083-2115
- Print publication:
- August 2020
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We consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data
$u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.
Regularity of the extremal solutions associated with elliptic systems
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 4 / August 2019
- Published online by Cambridge University Press:
- 27 December 2018, pp. 1037-1046
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- August 2019
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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.
$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$
