Semilinear problems with poly-Laplace type operators


The paper deals with semilinear operator equations involving iterates of a strongly monotone symmetric linear operator. In particular there are consider semilinear polyharmonic equations subject to the Navier boundary conditions. A careful analysis is made on the energetic spaces associated to such problems and a number of existence results are obtained by using a fixed point approach.


Radu Precup
Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


polyharmonic equation, iterates of symmetric linear operators, energetic space

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R. Precup, Semilinear problems with poly-Laplace type operators, Proceedings of the Romanian Academy Series A, 23 (2022) no. 4, pp. 319-328.


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[1] F. BERNIS, J. GARCIA-AZOREBO, I. PERAL, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1, pp. 210–240, 1996.
[2] M. BHAKTA, Solutions to semilinear elliptic PDE’s with biharmonic operator and singular potential, Electronic J. Differential Equations, 2016, 261, pp. 1–17, 2016.
[3] H. BREZIS, Functional analysis, Sobolev spaces and partial differential equations, Springer, 2011.
[4] X. CHENG, Z. FENG, L. WEI, Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with weight functions, Discrete Cont. Dyn. Syst. Ser. S, 14, pp. 3067–3083, 2021.
[5] F. GAZZOLA, H.-C. GRUNAU, G. SWEERS, Polyharmonic boundary value problems, Springer, 2009.
[6] L.V. KANTOROVICH, G.P. AKILOV, Functional analysis, Pergamon Press, 1982.
[7] G. LAURICELLA, Integrazione dell’equazione ∆2(∆2u) = 0 in un campo di forma circolare, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 31, pp. 1010, 1895-96.
[8] E. MATHIEU, Memoire sur l’ equation aux diff erences partielles du quatrieme ordre  ∆∆u = 0 et sur l’equilibre d’ elasticite d’un  corps solide, J. Math. Pures Appl. 2e serie, 14, pp. 378–421, 1869.
[9] M. NICOLESCU, Opera matematica. Functii poliarmonice , Ed. Academiei, Bucuresti, 1980.
[10] M. NICOLESCU, Les fonctions polyharmoniques, Actualite Sci.  331, Herman, 1936.
[11] M. PEREZ-LLANOS, A. PRIMO, Semilinear biharmonic problems with a singular term, J. Differential Equations, 257, pp. 3200–3225, 2014.
[12] R. PRECUP, Linear and semilinear partial differential equations, De Gruyter, 2013.
[13] CH. RIQUIER, Sur quelques problemes relatifs a l’ equation aux derivees partielles  ∆nu = 0, J. Math. Pures Appl., 5, 9, pp. 297–394, 1926.
[14] E. ZEIDLER, Applied functional analysis: applications to mathematical physics, Springer, 1995

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