[Paper Review] Regularity of the extremal solutions associated to elliptic systems

A. Aghajani, Craig Cowan|arXiv (Cornell University)|Jul 21, 2017

Nonlinear Partial Differential Equations18 references3 citations

TL;DR

This paper establishes the boundedness (L∞ regularity) of extremal solutions for a class of fully coupled elliptic systems with superlinear nonlinearities. By analyzing the asymptotic behavior of the nonlinearity via the ratio $ \tau_{\pm} = \lim_{t\to\infty} \frac{f(t)f''(t)}{f'(t)^2} $, it proves that extremal solutions are bounded in dimensions $ N < \frac{2\alpha_*(2-\tau_+)+2\tau_+}{\tau_+}\max\{1,\tau_+\} $, where $ \alpha_* $ is the largest root of a quadratic polynomial derived from $ \tau_- $ and $ \tau_+ $. This yields $ L^\infty $ regularity for $ N < 5 $, and for $ \tau_- = \tau_+ = \tau $, the bound extends to $ N < 10 $.

ABSTRACT

We examine the elliptic system given by \begin{eqnarray*} \qquad \left\{ \begin{array}{lcl} -Δu =λf(v) \quad \mbox{ in } Ω -Δv =γf(u) \quad \mbox{ in } Ω, u=v =0, \quad \mbox{ on } \pOm \end{array} ight. \end{eqnarray*} where $λ,γ$ are positive parameters, $Ω$ is a smooth bounded domain in $\IR^N$ and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t} ightarrow\infty$ as $t ightarrow\infty$. Assuming $$01$ denotes the largest root of the $2^{nd}$ order polynomial $$P_{f}(α,τ_{-},τ_{+}):=(2-τ_{-})^{2} α^{2}- 4(2-τ_{+})α+4(1-τ_{+}).$$ As a consequences, $u^*, v^*\in L^\infty(Ω)$ for $N&lt;5$. Moreover, if $τ_{-}=τ_{+}$, then $u^*, v^*\in L^\infty(Ω)$ for $N&lt;10$.

Motivation & Objective

• To resolve the open problem of regularity for extremal solutions in fully coupled elliptic systems with convex, superlinear nonlinearities.
• To determine the optimal dimension threshold $ N(f) $ beyond which extremal solutions may become singular.
• To extend known results for $ f(t) = e^t $ and $ f(t) = (1+t)^p $ to a broader class of nonlinearities characterized by $ \tau_{-} \leq \tau_{+} < 2 $.
• To derive a sharp bound on the dimension $ N $ for which extremal solutions remain bounded in $ L^\infty(\Omega) $, depending on the asymptotic convexity of $ f $.

Proposed method

• Uses the semistability inequality $ \sqrt{\lambda\gamma}\int \sqrt{f'(u)f'(v)}\phi^2 \leq \int |\nabla\phi|^2 $ for all $ \phi \in H^1_0(\Omega) $, derived from the stability of minimal solutions.
• Introduces a weighted test function technique with $ \tilde{f}(t) = \max\{f(t), T\} $ to localize analysis to large values of $ u $ and $ v $, enabling $ L^1 $ estimates on $ f'(t)^{1-\alpha}\tilde{f}(t)^{2\alpha} $.
• Applies Hölder’s inequality and symmetry arguments to couple estimates on $ I = \int f'(u)^{1/2 - \alpha}f'(v)^{1/2}\tilde{f}(u)^{2\alpha} $ and $ J = \int f'(v)^{1/2 - \alpha}f'(u)^{1/2}\tilde{f}(v)^{2\alpha} $, leading to a contradiction if both are unbounded.
• Derives a quadratic polynomial $ P_f(\alpha, \tau_-, \tau_+) = (2-\tau_-)^2\alpha^2 - 4(2-\tau_+)\alpha + 4(1-\tau_+) $, and selects $ \alpha_* > 1 $ as its largest root to define the critical dimension.
• Employs standard elliptic regularity theory after proving $ f'(v)^{(2-\tau_+)α_* + \tau_+} \in L^1(\Omega) $ and $ f'(v)^{\frac{(2-\tau_+)α_* + \tau_+}{\tau_+}} \in L^1(\Omega) $, ensuring boundedness of $ v^* $, and by symmetry $ u^* $.
• Uses the fact that $ f'(t)^{1/2 - \alpha}\tilde{f}(t)^{2\alpha} $ is increasing for large $ t $ when $ \alpha > 1 $ and $ \tau_+ < 2 $, to deduce integrability and control growth.

Experimental results

Research questions

• RQ1 Under what conditions on the nonlinearity $ f $ is the extremal solution $ (u^*, v^*) $ of the system $ -\Delta u = \lambda f(v), -\Delta v = \gamma f(u) $ in $ \Omega $ bounded in $ L^\infty(\Omega) $?
• RQ2 What is the optimal dimension $ N(f) $ such that $ u^*, v^* \in L^\infty(\Omega) $ for all $ N < N(f) $, depending on the asymptotic behavior of $ f $?
• RQ3 How does the regularity threshold depend on the ratio $ \tau_{\pm} = \lim_{t\to\infty} \frac{f(t)f''(t)}{f'(t)^2} $, particularly when $ \tau_- = \tau_+ $?
• RQ4 Can the known $ L^\infty $ regularity result for $ f(t) = e^t $ (valid up to $ N < 10 $) be generalized to a broader class of nonlinearities with the same $ \tau $-value?
• RQ5 Is there a uniform $ L^1 $ bound on $ f'(t)^{1-\alpha}\tilde{f}(t)^{2\alpha} $ over all semi-stable solutions, independent of $ \lambda, \gamma $, that implies $ L^\infty $ regularity?

Key findings

• The extremal solution $ (u^*, v^*) $ is bounded in $ L^\infty(\Omega) $ for all dimensions $ N < \frac{2\alpha_*(2-\tau_+)+2\tau_+}{\tau_+}\max\{1,\tau_+\} $, where $ \alpha_* > 1 $ is the largest root of the quadratic $ P_f(\alpha, \tau_-, \tau_+) = (2-\tau_-)^2\alpha^2 - 4(2-\tau_+)\alpha + 4(1-\tau_+) $.
• For $ N < 5 $, the extremal solution is always bounded in $ L^\infty(\Omega) $, regardless of the specific $ f $, as long as $ \tau_+ < 2 $.
• If $ \tau_- = \tau_+ = \tau $, then $ u^*, v^* \in L^\infty(\Omega) $ for $ N < 10 $, with equality in the bound $ N(f) = 2 + 4\frac{1+\sqrt{\tau}}{\tau} \geq 10 $.
• For $ f(t) = e^t $ or $ f(t) = e^{t^\alpha} $, the condition $ \tau_- = \tau_+ = 1 $ yields $ N(f) = 10 $, recovering the known result for the exponential nonlinearity.
• For $ f(t) = (1+t)^p $ with $ p > 1 $, the bound becomes $ N(f) = 2 + \frac{4}{p-1}(p + \sqrt{p^2 - p}) $, matching prior results in [2] and [6].
• The proof shows that if $ I $ and $ J $ are both unbounded, then $ P_f(\alpha, \tau_-, \tau_+) \geq 0 $ must hold, which contradicts the choice of $ \alpha $ near $ \alpha_* $, implying uniform $ L^1 $ bounds on the key weight functions.