Liouville results for semilinear integral equations with conical diffusion

TL;DR

This work establishes Liouville-type nonexistence results for supersolutions of semilinear nonlocal inequalities in cone-diffusion settings, where the diffusion operator L is a symmetric stable Lévy-type generator with kernel shaped by a( heta) and active only on a cone. The authors adapt a rescaled test-function method to the anisotropic, weak-diffusion regime and derive cone-aware barrier constructions to handle boundary effects in the half-space. They prove that in the half-space, nonexistence holds for the subcritical range $1 \le p \le \frac{N+s}{N-s}$, and they show the exponent is sharp by constructing positive supersolutions for $p > \frac{N+s}{N-s}$ (in particular for the fractional Laplacian). They also obtain a Liouville-type result in the whole space for $1 \le p \le \frac{N}{N-2s}$, highlighting the different critical thresholds in bounded- versus unbounded-domain settings. Overall, the paper extends nonlocal Liouville theorems to anisotropic diffusion with cone geometry and clarifies the roles of domain, diffusion cone, and nonlinearity in subcritical versus supercritical regimes.

Abstract

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $\mathbb R^N$. The results are sharp: $u\equiv 0$ is the only nonnegative supersolution in the subcritical regime $1\leq p\leq\frac{N+s}{N-s}\,$, while nontrivial supersolutions exist, at least for some specific $-L$, as soon as $p>\frac{N+s}{N-s}$. \\ The arguments used rely on a rescaled test function's method, suitably adapted to such nonlocal setting with weak diffusion; they are quite general and also employed to obtain Liouville type results in the whole space.

Figures (6)

• Figure 1: The graph of $\widetilde{u}$.
• Figure 2: The blue area $\left|\Sigma_{\nu,\tau}\left(x\right)\cap B_{1}\right|>0$.
• Figure 3: $\left|\Sigma_{\nu,\tau}\left(x\right)\cap B_{1}\cap\mathbb{R}^{N}_{+}\right|=0$.
• Figure 4: The blue area $\left|\Sigma_{\nu,\tau}\left(x\right)\cap B_{1}\left(\left(1-\gamma\right)e_{N}\right)\cap\mathbb{R}^{N}_{+}\right|>0$.
• Figure 5: The graph of $\varphi$.

Theorems & Definitions (25)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof