Fully nonlinear logistic equations with sanctuary

Abstract: For the fully nonlinear stationary logistic equation ${\mathcal F}(x,D<sup>2u)+μu=k(x)u<sup>p$ with $p&gt;1$ and $k(x)\geq 0$, in a bounded domain with Dirichlet boundary condition, we determine, in terms of $μ$, the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when $μ$ approaches the boundary points of the existence range.

• The paper establishes sharp existence and uniqueness criteria for fully nonlinear logistic equations by linking solution behavior to principal eigenvalues of the operator.
• It employs viscosity solution techniques, comparison principles, and barrier constructions to handle spatial heterogeneity, including sanctuary regions where diffusion is degenerate.
• The asymptotic analysis reveals that solutions vanish as μ approaches the ambient domain eigenvalue and blow up in sanctuary regions when μ nears the sanctuary eigenvalue.

Fully Nonlinear Logistic Equations with Sanctuary: Existence, Uniqueness, and Asymptotics

Problem Formulation and Context

The paper "Fully nonlinear logistic equations with sanctuary" (2603.29885) investigates stationary fully nonlinear elliptic equations of logistic type, set in a bounded domain $\Omega \subset \mathbb{R}^N$ under homogeneous Dirichlet boundary conditions. The equation studied is

$$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$

where $p > 1$, $\mu \in \mathbb{R}$, and $k \in C(\Omega)$ is nonnegative, possibly vanishing on a nontrivial subset ("sanctuary" $\Omega_0$) while being strictly positive elsewhere. $F$ denotes a general, uniformly elliptic, positively homogeneous of degree 1 operator, which may be fully nonlinear, encompassing but not limited to the Laplacian. Both classical and viscosity solution frameworks are considered, with all results established for viscosity solutions.

The model generalizes standard (semi)linear logistic problems, central in population dynamics, to encompass nonlinear and possibly degenerate diffusion with spatially heterogeneous environments including sanctuaries where the limiting, density-dependent inhibitory term vanishes ($k(x) = 0$). The novelty lies in allowing both nonlinear diffusion and sanctuary, and addressing the impact of these features on existence, uniqueness, and qualitative behavior of positive solutions.

Main Results: Sharp Existence and Uniqueness Criteria

A core contribution is the precise characterization of existence and uniqueness of positive solutions in terms of the principal eigenvalues of $F(x, D^2\cdot)$ on $\Omega$ and the sanctuary region $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$0. The principal eigenvalue is defined in the Berestycki-Nirenberg-Varadhan viscosity sense:

$$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$1

with a similar definition for subdomains such as $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$2.

The main theorem states:

• Case (K1) (No sanctuary, $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$3 in $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$4): Positive solutions exist if and only if $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$5; in this case the solution is unique.
• Case (K2) (Sanctuary present, $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$6 in $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$7 open, nonempty, and $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$8 away from $$(F(x, D^2u) + \mu u = k(x)u^p,~~u > 0~\text{in}~\Omega,~~u=0~\text{on}~\partial\Omega)$$9): Positive solutions exist if and only if $p > 1$0; uniqueness holds.

This pinpoints the principal eigenvalue of the ambient domain as the lower threshold and that of the sanctuary as the upper threshold for existence. The proof is based on comparison principles, strong maximum principles for viscosity solutions, and careful barrier constructions for fully nonlinear operators. Notably, the existence proofs yield precise monotonicity of the solution with respect to $p > 1$1.

Asymptotic Analysis: Extinction and Blow-up

The paper provides a detailed asymptotic analysis as $p > 1$2 approaches the endpoints of the existence interval.

1. Limit $p > 1$3: The unique positive solution vanishes uniformly in $p > 1$4. Upon normalization, solutions converge, locally uniformly, to the positive principal eigenfunction, that is,

$p > 1$5

2. Limit $p > 1$6 (in the sanctuary case): The solution blows up uniformly on $p > 1$7, i.e.,

$p > 1$8

After normalization, on $p > 1$9,

$\mu \in \mathbb{R}$0

On the complement $\mu \in \mathbb{R}$1, the solution converges locally uniformly to the minimal positive blow-up solution of the problem with boundary blow-up on $\mu \in \mathbb{R}$2.

These asymptotics are treated by monotone sequence arguments, compactness in Hölder spaces, and careful local analysis at the interface of the sanctuary.

Analytical Techniques and Structural Insights

The analysis relies on the viscosity solution approach for fully nonlinear operators, using maximum and comparison principles tailored to such settings, including for possibly non-convex operators. The authors make extensive use of the extremal Pucci operators as benchmarks for ellipticity, regularity estimates, and barrier constructions. Principal eigenvalues and associated eigenfunctions for nonlinear operators play a central role, both as bifurcation thresholds and as attractors for normalized solutions.

The construction of global supersolutions via patching of local barriers and the use of blow-up techniques to construct solutions with singular data on the sanctuary boundary are noteworthy contributions. The uniqueness results are obtained via strong comparison principles, taking advantage of the strict monotonicity in $\mu \in \mathbb{R}$3 of the logistic term.

Implications and Outlook

From a theoretical perspective, these results generalize classical logistic-type population dynamics to broad classes of nonlinear systematic diffusions, elucidating the sharp role of principal eigenvalues and spatial refuges. They demonstrate that sanctuaries (regions of "free growth") enlarge the set of parameters supporting persistence; this is sharply quantified by the eigenvalue gap $\mu \in \mathbb{R}$4.

Practically, these findings can inform models of biological populations when spatial heterogeneity and nonlinear dispersal mechanisms are relevant, with implications for persistence/eradication thresholds in invasion ecology, epidemiology, or spatial resource management. The analytical framework also provides a template for nonlinear eigenvalue problems on domains with substructure, which might be relevant for other PDEs in physics and finance.

Future work might extend the analysis to the fully time-dependent (parabolic) problem, consider degenerate/singular nonlinearities, or investigate the optimal design of sanctuaries, possibly within optimal control or game-theoretic frameworks.

Conclusion

This paper establishes a rigorous, comprehensive theory for the existence, uniqueness, and asymptotic behavior of positive stationary solutions to fully nonlinear logistic equations with sanctuaries. The authors provide sharp thresholds for persistence and extinction in terms of principal eigenvalues of general nonlinear elliptic operators, giving new characterizations even in classical linear cases. The techniques, based on viscosity solutions, comparison principles, and nonlinear spectral analysis, should find broader application in nonlinear elliptic and parabolic theory.