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June  2018, 11(3): 357-377. doi: 10.3934/dcdss.2018020

Entire solutions of nonlocal elasticity models for composite materials

Giuseppina Autuori and  Patrizia Pucci ,

Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli, 1, 06123, Perugia, Italy

* Corresponding author: Patrizia Pucci

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: This research has been developped within a scientific project Nonlocal Elasticity Models for Composite Materials with Professors F. Cluni and V. Gusella of the Dipartimento di Ingegneria Civile ed Ambientale of the Università degli Studi di Perugia.

Many structural materials, which are preferred for the developing of advanced constructions, are inhomogeneous ones. Composite materials have complex internal structure and properties, which make them to be more effectual in the solution of special problems required for civil and environmental engineering. As a consequence of this internal heterogeneity, they exhibit complex mechanical properties. In this work, the analysis of some features of the behavior of composite materials under different loading conditions is carried out. The dependence of nonlinear elastic response of composite materials on loading conditions is studied. Several approaches to model elastic nonlinearity such as different stiffness for particular type of loadings and nonlinear shear stress–strain relations are considered. Instead of a set of constant anisotropy coefficients, the anisotropy functions are introduced. Eventually, the combined constitutive relations are proposed to describe simultaneously two types of physical nonlinearities. The first characterizes the nonlinearity of shear stress–strain dependency and the latter determines the stress state susceptibility of material properties. Quite satisfactory correlation between the theoretical dependencies and the results of experimental studies is demonstrated, as described in [2,3] as well as in the references therein.

Keywords: Existence theorems, entire solutions, fractional elliptic operators, nonlocal elasticity models, composite materials.
Mathematics Subject Classification: Primary: 35R11, 35A15, 35J60; Secondary: 74B20, 74E30.
Citation: Giuseppina Autuori, Patrizia Pucci. Entire solutions of nonlocal elasticity models for composite materials. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 357-377. doi: 10.3934/dcdss.2018020
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

G. Autuori, F. Cluni, V. Gusella and P. Pucci, Mathematical models for nonlocal elastic composite materials Adv. Nonlinear Anal. (2017), pages 39. doi: 10.1515/anona-2016-0186.  Google Scholar

[3]

G. Autuori, F. Cluni, V. Gusella and P. Pucci, Effects of the fractional Laplacian order on the nonlocal elastic rod response, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 3 (2017), 030902, 5pp. doi: 10.1115/1.4036806.  Google Scholar

[4]

G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar

[5]

G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, Nonlinear Differ. Equ. Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.  Google Scholar

[6]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.  Google Scholar

[7]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Universitext Springer, New York, 2011, xiv+599 pp.  Google Scholar

[8]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[9]

L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, 37-52, Abel Symp. 7, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[10]

M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x.  Google Scholar

[11]

X. Chang, Ground states of some fractional Schrödinger equations on $\mathbb R^N$, Proc. Edinb. Math. Soc., 58 (2015), 305-321.  doi: 10.1017/S0013091514000200.  Google Scholar

[12]

D. C. de Morais Filho, M. A. S. Souto and J. M. do Ó., A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones, 19 (2000), 1-17.  doi: 10.4067/S0716-09172000000100001.  Google Scholar

[13]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[14]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb R^n$ Lecture Notes, Scuola Normale Superiore di Pisa (New Series), 15, Edizioni della Normale, Pisa, 2017, viii+152 pp. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[15]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.  Google Scholar

[16]

A. Fiscella and P. Pucci, p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.  doi: 10.1016/j.nonrwa.2016.11.004.  Google Scholar

[17]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[18]

Y. Lei, Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in $\mathbb R^n$, J. Differential Equations, 258 (2015), 4033-4061.  doi: 10.1016/j.jde.2015.01.043.  Google Scholar

[19]

P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6.  Google Scholar

[20]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations 2nd edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[21]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[22]

P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., 62 (2018), pages 34.   Google Scholar

[23]

P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré A.N.L., 25 (2008), 505-537.  doi: 10.1016/j.anihpc.2007.02.004.  Google Scholar

[24]

P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $R^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.  Google Scholar

[25]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128.  doi: 10.1016/j.jde.2014.10.012.  Google Scholar

[26]

C. E. Torres Ledesma, Multiplicity result for non-homogeneous fractional Schrödinger-Kirchhoff type equations in $\mathbb R^n$, Adv. Nonlinear Anal., 5 (2016), 133-146.  doi: 10.1515/anona-2015-0076.  Google Scholar

[27]

Z. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508.  doi: 10.3934/dcds.2016.36.499.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

G. Autuori, F. Cluni, V. Gusella and P. Pucci, Mathematical models for nonlocal elastic composite materials Adv. Nonlinear Anal. (2017), pages 39. doi: 10.1515/anona-2016-0186.  Google Scholar

[3]

G. Autuori, F. Cluni, V. Gusella and P. Pucci, Effects of the fractional Laplacian order on the nonlocal elastic rod response, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 3 (2017), 030902, 5pp. doi: 10.1115/1.4036806.  Google Scholar

[4]

G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar

[5]

G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, Nonlinear Differ. Equ. Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.  Google Scholar

[6]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.  Google Scholar

[7]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Universitext Springer, New York, 2011, xiv+599 pp.  Google Scholar

[8]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[9]

L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, 37-52, Abel Symp. 7, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[10]

M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x.  Google Scholar

[11]

X. Chang, Ground states of some fractional Schrödinger equations on $\mathbb R^N$, Proc. Edinb. Math. Soc., 58 (2015), 305-321.  doi: 10.1017/S0013091514000200.  Google Scholar

[12]

D. C. de Morais Filho, M. A. S. Souto and J. M. do Ó., A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones, 19 (2000), 1-17.  doi: 10.4067/S0716-09172000000100001.  Google Scholar

[13]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[14]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb R^n$ Lecture Notes, Scuola Normale Superiore di Pisa (New Series), 15, Edizioni della Normale, Pisa, 2017, viii+152 pp. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[15]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.  Google Scholar

[16]

A. Fiscella and P. Pucci, p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.  doi: 10.1016/j.nonrwa.2016.11.004.  Google Scholar

[17]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[18]

Y. Lei, Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in $\mathbb R^n$, J. Differential Equations, 258 (2015), 4033-4061.  doi: 10.1016/j.jde.2015.01.043.  Google Scholar

[19]

P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6.  Google Scholar

[20]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations 2nd edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[21]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[22]

P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., 62 (2018), pages 34.   Google Scholar

[23]

P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré A.N.L., 25 (2008), 505-537.  doi: 10.1016/j.anihpc.2007.02.004.  Google Scholar

[24]

P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $R^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.  Google Scholar

[25]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128.  doi: 10.1016/j.jde.2014.10.012.  Google Scholar

[26]

C. E. Torres Ledesma, Multiplicity result for non-homogeneous fractional Schrödinger-Kirchhoff type equations in $\mathbb R^n$, Adv. Nonlinear Anal., 5 (2016), 133-146.  doi: 10.1515/anona-2015-0076.  Google Scholar

[27]

Z. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508.  doi: 10.3934/dcds.2016.36.499.  Google Scholar

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