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September/October 2021 Constrained semilinear elliptic systems on $\mathbb R^N$
Wojciech Kryszewski, Jakub Siemianowski
Adv. Differential Equations 26(9/10): 459-504 (September/October 2021).

Abstract

The existence of solutions $u$ in $H^1(\mathbb R^N,\mathbb R^M)\cap H^2_{loc}(\mathbb R^N,\mathbb R^M)$ of a coupled semilinear system of second order elliptic partial differential equations on $\mathbb R^N$ of the form \[ \mathcal{P}[u] = f(x,u,\partial u), \quad x\in \mathbb R^N, \] under pointwise constraints is considered. The problem is studied via the constructed suitable topological invariant, the so-called constrained topological degree, which allows to get the existence of solutions of abstract problems considered as $L^2$-realizations of the approximating sequence of systems obtained by the truncation of the initial system to bounded subdomains. The key step of the proof consists in showing the relative $H^1$-compactness of the sequence of solutions to the truncated systems by the use of the so-called tail estimates. The constructions rely on the semigroup approach combined with topological methods, as well as invariance/viability techniques.

Citation

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Wojciech Kryszewski. Jakub Siemianowski. "Constrained semilinear elliptic systems on $\mathbb R^N$." Adv. Differential Equations 26 (9/10) 459 - 504, September/October 2021.

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Published: September/October 2021
First available in Project Euclid: 12 August 2021

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Rights: Copyright © 2021 Khayyam Publishing, Inc.

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Vol.26 • No. 9/10 • September/October 2021
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