Non-negative solutions of a sublinear elliptic problem

被引:0
作者
Lopez-Gomez, Julian [1 ]
Rabinowitz, Paul H. [2 ]
Zanolin, Fabio [3 ]
机构
[1] Univ Complutense Madrid, Inst Interdisciplinar Matemat, Madrid, Spain
[2] Univ Wisconsin Madison, Dept Math, Madison, WI 53706 USA
[3] Univ Udine, Dipartimento Sci Matemat Informat & Fis, Via Delle Sci 2016, I-33100 Udine, Italy
关键词
Non-negative solutions; sublinear elliptic problems; bifurcation from infinity; singular perturbations; non-negative multi-bump solutions; global structure; BIFURCATION;
D O I
10.1007/s11784-024-01120-z
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this paper, the existence of solutions, (lambda, u), of the problem {-Delta u = lambda u - a(x)|u|(p-1)u in Omega, u = 0 on partial derivative Omega is explored for 0<p<1. When p > 1, it is known that there is an unbounded component of such solutions bifurcating from (sigma(1),0), where sigma 1is the smallest eigenvalue of-Delta in Omega under Dirichlet boundary conditions on partial derivative Omega. These solutions have u is an element of P, the interior of the positive cone. The continuation argument used when p > 1 to keep u is an element of P fails if 0 < p < 1. Nevertheless when 0 < p < 1, we are still able to show that there is a component of solutions bifurcating from (sigma 1,infinity), unbounded outside of a neighborhood of (sigma 1,infinity), and havingu0. This non-negativity for u cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary)number of bumps. Finally, the structure of these components is fully described
引用
收藏
页数:32
相关论文
共 50 条
[41]   Symmetry breaking of solutions of non-cooperative elliptic systems [J].
Stefaniak, Piotr .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2013, 408 (02) :681-693
[42]   Exact multiplicity of solutions to superlinear and sublinear problems [J].
Shi, JP .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2002, 50 (05) :665-687
[43]   A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations [J].
Nakshatrala, K. B. ;
Nagarajan, H. ;
Shabouei, M. .
COMMUNICATIONS IN COMPUTATIONAL PHYSICS, 2016, 19 (01) :53-93
[44]   Variational inequality approach to enforcing the non-negative constraint for advection-diffusion equations [J].
Chang, J. ;
Nakshatrala, K. B. .
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2017, 320 :287-334
[45]   Multiple nonnegative solutions for an elliptic boundary value problem involving combined nonlinearities [J].
Anello, Giovanni .
MATHEMATICAL AND COMPUTER MODELLING, 2010, 52 (1-2) :400-408
[46]   Unbounded sets of solutions of non-cooperative elliptic systems on spheres [J].
Rybicki, Slawomir ;
Stefaniak, Piotr .
JOURNAL OF DIFFERENTIAL EQUATIONS, 2015, 259 (07) :2833-2849
[47]   Quadratic Lifespan for the Sublinear α-SQG Sharp Front Problem [J].
Montalto, Riccardo ;
Murgante, Federico ;
Scrobogna, Stefano .
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 2024,
[48]   Unsupervised machine learning based on non-negative tensor factorization for analyzing reactive-mixing [J].
Vesselinov, V. V. ;
Mudunuru, M. K. ;
Karra, S. ;
O'Malley, D. ;
Alexandrov, B. S. .
JOURNAL OF COMPUTATIONAL PHYSICS, 2019, 395 :85-104
[49]   Non Negative Solutions Conditions of Nonlinear Urban River Model [J].
Song, Huabing ;
Zhang, Xinzheng .
INFORMATION TECHNOLOGY APPLICATIONS IN INDUSTRY, PTS 1-4, 2013, 263-266 :292-+
[50]   Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces [J].
Damek, Ewa ;
Ghardallou, Zeineb .
FORUM MATHEMATICUM, 2019, 31 (04) :1007-1026