Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent
Abstract
We establish a Brézis-Oswald-type spectral principle for semilinear equations −Lu=f(x,u), where L is the generator of a positive C0-semigroup on Lp with compact positivity-improving resolvent. Neither symmetry, variational structure, nor regularizing properties such as ultracontractivity or smoothing are assumed. Let a0 and a∞ denote the asymptotic slopes of the nonlinearity at zero and at infinity, and let λ1(a) denote the generalized principal eigenvalue associated with the perturbed operator −L−a. We prove that λ1(a0)<0<λ1(a∞) implies that the equation admits a strictly positive solution. The proof develops a potential-theoretic sub- and supersolution framework based on the order induced by supermedian functions and combines a Deny-type compactness theorem, a Kato-type inequality, and a Doob transform. The construction also yields an order-preserving selection of solutions and allows the lower control on the nonlinearity to be relaxed. If y↦ f(x,y)/y is strictly decreasing and a0 is bounded, the spectral condition is necessary as well, and the strictly positive solution is unique.
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BibTeX
@misc{klimsiak2026positive,
title = {{Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent}},
author = {Tomasz Klimsiak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.10570},
eprint = {2605.10570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10570},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent
AU - Tomasz Klimsiak
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2605.10570
UR - https://arxiv.org/abs/2605.10570
AB - We establish a Brézis-Oswald-type spectral principle for semilinear equations −Lu=f(x,u), where L is the generator of a positive C_0-semigroup on L^p with compact positivity-improving resolvent. Neither symmetry, variational structure, nor regularizing properties such as ultracontractivity or smoothing are assumed. Let a_0 and a_∞ denote the asymptotic slopes of the nonlinearity at zero and at infinity, and let λ_1(a) denote the generalized principal eigenvalue associated with the perturbed operator −L−a. We prove that λ_1(a_0)<0<λ_1(a_∞) implies that the equation admits a strictly positive solution. The proof develops a potential-theoretic sub- and supersolution framework based on the order induced by supermedian functions and combines a Deny-type compactness theorem, a Kato-type inequality, and a Doob transform. The construction also yields an order-preserving selection of solutions and allows the lower control on the nonlinearity to be relaxed. If y↦ f(x,y)/y is strictly decreasing and a_0 is bounded, the spectral condition is necessary as well, and the strictly positive solution is unique.
ER -