Abstract
The paper presents an abstract theory regarding the problems with semilinear operator equations involving iterates of a strongly monotone symmetric linear operator. We obtain existence and localization results of positive solutions for such problems using Krasnosel’skiĭ’s technique and abstract Harnack inequality. In particular, we obtain results for problems with semilinear poly-Laplace operators.
Authors
Radu Precup
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Nataliia Kolun
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Keywords
Krasnosel’skiĭ’s technique; fixed point; Harnack inequality; iterates of a symmetric linear operator. poly-Laplace type operator
Paper coordinates
N. Kolun, R. Precup, Localization of solutions for semilinear problems with poly-Laplace type operators, Applicable Analysis, published online 2023, https://doi.org/10.1080/00036811.2023.2218869
About this paper
Journal
Applicable Analysis\Taylor & Francis Online
Publisher Name
Print ISSN
Online ISSN
google scholar link
[1] Precup R. Semilinear problems with poly-Laplace type operators. Proc Rom Acad Ser A.2022;23:319–328.
[2] Nicolescu M. Opera matematică: funcţii poliarmonice. Bucureşti: Ed. Academiei;1980.
[3] Bernis F, Garcia-Azorebo J, Peral I. Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv Differ Equ.1996;1:210–240.[4] Bhakta M. Solutions to semilinear elliptic PDE’s with biharmonic operator and singular potential. Electron J DifferEqu.2016;261:1–17.
[5] Cheng X, Feng Z, Wei L. Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equationwith weight functions. Discrete Cont Dyn Syst Ser S.2021;14:3067–3083.
[6] Pérez-Llanos M, Primo A. Semilinear biharmonic problems with a singular term. J Differ Equ.2014;257:3200–3225.
[7] GazzolaF,GrunauHC,SweersG.Polyharmonicboundaryvalueproblems.Berlin:Springer;2009.
[8] Zhang Y, Lü Y, Wang N. Existence of positive solutions of semilinear biharmonic equations. Abstr Appl Anal.2014;2014:11 p. Article ID 624328.
[9] Jungy T, Choi QH. Applications of topological methods to the semilinear biharmonic problem with differentpowers. Korean J Math.2017;25:455–468.
[10] Wei G, Zeng L. Estimates for eigenvalues of Poly-Harmonic operators. Adv Nonlinear Stud.2016;16:31–44.
[11] Yolcu SY, Yolcu T. Eigenvalue bounds for the poly-harmonic operators. Illinois J Math.2014;58:847–865.
[12] Cheng QM, Qi X, Wei G. A lower bound for eigenvalues of the poly-Laplace with arbitrary order. Pacific J Math.2013;262:35–47.
[13] Precup R. Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations. J Fixed Point TheoryAppl.2012;12:193–206.
[14] Precup R. Compression-expansion fixed point theorems in two norms. Ann Tiberiu Popoviciu Semin Funct EquApprox Convexity.2005;3:157–163.
[15] O’Regan D, Precup R. Compression-expansion fixed point theorem in two norms and applications. J Math AnalAppl.2005;309:383–391.
[16] Precup R. Moser-Harnack inequality, Krasnoselskiı type fixed point theorems in cones and elliptic problems.Topol Methods Nonlinear Anal.2012;40:301–313.
[17] Precup R. Critical point theorems in cones and multiple positive solutions of elliptic problems. Nonlinear Anal.2012;75:834–851.
[18] Jost J. Partial differential equations. New York: Springer;2007.
[19] Azizieh C, Clement P. A priori estimates and continuation methods for positive solutions of p-Laplace equations.J Differ Equ.2002;179:213–245.
[20] Brezis H. Functional analysis, Sobolev spaces and partial differential equations. New York: Springer;2011.
[21] Precup R. Linear and semilinear partial differential equations. Berlin: De Gruyter;2013.
[22] Herlea DR. Positive solutions for second-order boundary-value problems with phi-Laplacian. Addendum Elec-tron J Differ Equ.2016;51:1–12.