# American Institute of Mathematical Sciences

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

## Entire solutions of nonlocal elasticity models for composite materials

 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.

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
 [1] Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure & Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361 [2] Yajing Zhang, Jianghao Hao. Existence of positive entire solutions for semilinear elliptic systems in the whole space. Communications on Pure & Applied Analysis, 2009, 8 (2) : 719-724. doi: 10.3934/cpaa.2009.8.719 [3] Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-25. doi: 10.3934/dcdss.2020077 [4] Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039 [5] Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531 [6] Yu-Juan Sun, Li Zhang, Wan-Tong Li, Zhi-Cheng Wang. Entire solutions in nonlocal monostable equations: Asymmetric case. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1049-1072. doi: 10.3934/cpaa.2019051 [7] C. T. Cremins. Existence theorems for weakly inward semilinear operators. Conference Publications, 2003, 2003 (Special) : 200-205. doi: 10.3934/proc.2003.2003.200 [8] Jun Wang, Qiuping Geng, Maochun Zhu. Existence of the normalized solutions to the nonlocal elliptic system with partial confinement. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2187-2201. doi: 10.3934/dcds.2019092 [9] Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311 [10] Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335 [11] Patrizia Pucci, Marco Rigoli. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 115-137. doi: 10.3934/dcds.2008.20.115 [12] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248 [13] Fang-Di Dong, Wan-Tong Li, Li Zhang. Entire solutions in a two-dimensional nonlocal lattice dynamical system. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2517-2545. doi: 10.3934/cpaa.2018120 [14] Vincenzo Ambrosio, Giovanni Molica Bisci. Periodic solutions for nonlocal fractional equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 331-344. doi: 10.3934/cpaa.2017016 [15] Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707 [16] Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50 [17] Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607 [18] Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505 [19] Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110 [20] Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113

2018 Impact Factor: 0.545