[Paper Review] Nonexistence of solutions for Dirichlet problems with supercritical growth in tubular domains

Riccardo Molle, Donato Passaseo|arXiv (Cornell University)|May 21, 2019

Advanced Mathematical Modeling in Engineering25 references4 citations

TL;DR

This paper establishes nonexistence results for nontrivial solutions to semilinear elliptic Dirichlet problems with supercritical nonlinearities in tubular domains around compact submanifolds. Using Pohozaev-type integral identities with carefully constructed vector fields, it proves that for tubular domains $ T_\varepsilon(\Gamma_k) $ with $ k=1 $, $ \Gamma_1 $ contractible, and $ \varepsilon $ sufficiently small, no nontrivial solutions exist when the nonlinearity exceeds the critical Sobolev exponent $ p > \frac{2n}{n-2} $.

ABSTRACT

We deal with Dirichlet problems of the form $$ \\Delta u+f(u)=0 \\mbox{ in }\\Omega,\\qquad u=0\\ \\mbox{ on }\\partial \\Omega $$ where $\\Omega$ is a bounded domain of $\\mathbb{R}^n$, $n\\ge 3$, and $f$ has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where $\\Omega$ is a tubular domain $T_\\varepsilon(\\Gamma_k)$ with thickness $\\varepsilon>0$ and centre $\\Gamma_k$, a $k$-dimensional, smooth, compact submanifold of $\\mathbb{R}^n$. Our main result concerns the case where $k=1$ and $\\Gamma_k$ is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for $\\varepsilon>0$ small enough. When $k\\ge 2$ or $\\Gamma_k$ is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on $k$ and $f$.

Motivation & Objective

• To investigate the existence or nonexistence of nontrivial solutions for semilinear elliptic equations with supercritical growth in tubular domains.
• To determine whether the classical Pohozaev nonexistence result for star-shaped domains can be extended to non-star-shaped, contractible tubular domains.
• To analyze how the topology and dimension of the central submanifold $ \Gamma_k $ influence the solvability of the Dirichlet problem with supercritical nonlinearities.
• To establish sharp nonexistence results by identifying critical thresholds for the growth exponent $ p $ and domain thickness $ \varepsilon $.

Proposed method

• Derives a Pohozaev-type integral identity using a vector field $ v_k $ adapted to the geometry of tubular domains $ T_\varepsilon(\Gamma_k) $, where $ \Gamma_k $ is a $ k $-dimensional compact submanifold.
• Constructs vector fields $ v_k $ such that their divergence and directional derivatives are controlled on $ \Gamma_k $, with asymptotic behavior as $ \varepsilon \to 0 $.
• Applies the integral identity to the equation $ \Delta u + f(u) = 0 $ with $ f(u) = |u|^{p-2}u $, leading to an inequality involving $ \int |Du|^2 dx $.
• Uses the asymptotic convergence of $ \mathop{\rm div}\nolimits v_k $ to $ n-k $ and $ dv_k[\eta]\cdot\eta \to 1 $ as $ \varepsilon \to 0 $ to derive a coercive inequality.
• Analyzes the sign of the coefficient $ 1 - \frac{n-k}{2} + \frac{n-k}{p} + \mu_k(\varepsilon) $, showing it becomes negative for $ p > \frac{2(n-k)}{n-k-2} $ and $ n > k+2 $, forcing $ u \equiv 0 $.
• Employs capacity and homology group arguments to explain the sharpness of results, particularly for $ k \geq 2 $ and noncontractible $ \Gamma_k $.

Experimental results

Research questions

• RQ1 Can the Pohozaev nonexistence result for supercritical nonlinearities be extended to non-star-shaped, contractible tubular domains?
• RQ2 What role does the topology (e.g., contractibility) and dimension $ k $ of the central submanifold $ \Gamma_k $ play in the solvability of the Dirichlet problem?
• RQ3 Are the nonexistence results sharp in terms of the growth exponent $ p $ and the thickness $ \varepsilon $ of the tubular domain?
• RQ4 Why do nonexistence results fail for $ k=1 $ when $ \Gamma_1 $ is noncontractible, or for $ k>1 $, even with contractible $ \Gamma_k $, under the same supercritical exponent?
• RQ5 Can similar nonexistence results be obtained for $ q $-Laplacian equations in $ \mathbb{R}^2 $, where supercritical behavior arises even for $ n=2 $?

Key findings

• For $ k=1 $, $ \Gamma_1 $ a compact, smooth, contractible curve in $ \mathbb{R}^n $, $ n \geq 3 $, and $ p > \frac{2n}{n-2} $, there exists $ \bar{\varepsilon} > 0 $ such that the Dirichlet problem has only the trivial solution for all $ \varepsilon \in (0, \bar{\varepsilon}) $.
• When $ k \geq 2 $ or $ \Gamma_k $ is noncontractible, the method fails for $ p > \frac{2n}{n-2} $, but a weaker nonexistence result holds for $ n \geq 4 $ and $ p > \frac{2(n-1)}{n-3} $.
• For $ k \geq 2 $, a nonexistence result holds under $ n > k+2 $ and $ p > \frac{2(n-k)}{n-k-2} $, and this threshold is sharp as shown by existence examples.
• The nonexistence result for $ k=1 $, $ \Gamma_1 $ contractible, is sharp: counterexamples exist for $ k=1 $, $ \Gamma_1 $ noncontractible (e.g., a circle), where solutions exist for $ p \in (1, \frac{2(n-1)}{n-3}) $ when $ n \geq 4 $.
• For $ k \geq 2 $, existence and multiplicity of solutions are demonstrated for $ p \geq \frac{2n}{n-2} $, showing that the nonexistence result for $ k>1 $ cannot be extended under the same $ p $-threshold.
• In $ \mathbb{R}^2 $, replacing the Laplacian with the $ q $-Laplacian ($ 1<q<2 $) allows nonexistence for all contractible domains when $ p \geq \frac{2q}{2-q} $, suggesting a broader extension than in the $ p $-Laplacian case for $ n \geq 3 $.