September  2019, 14(3): 567-587. doi: 10.3934/nhm.2019022

Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations

1. 

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy

2. 

Dipartimento di Ingegneria Civile Edile e Architettura, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy

3. 

Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via Valerio 12/1, I-34127, Trieste, Italy

* Corresponding author: Piero Montecchiari

Received  July 2018 Revised  February 2019 Published  May 2019

We study systems of elliptic equations $ -\Delta u(x)+F_{u}(x, u) = 0 $ with potentials $ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $ which are periodic and even in all their variables. We show that if $ F(x, u) $ has flip symmetry with respect to two of the components of $ x $ and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on $ {\mathbb{R}}^{n} $.

Citation: Francesca Alessio, Piero Montecchiari, Andrea Sfecci. Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations. Networks & Heterogeneous Media, 2019, 14 (3) : 567-587. doi: 10.3934/nhm.2019022
References:
[1]

S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in ${\mathbb{R}}^{2}$ for an Allen-Cahn system with multiple well potential, Calc. Var. Partial Differential Equations, 5 (1997), 359–390. doi: 10.1007/s005260050071.  Google Scholar

[2]

F. AlessioG. Autuori and P. Montecchiari, Saddle type solutions for a class of reversible elliptic equations, Adv. Differential Equations, 21 (2016), 1-30.   Google Scholar

[3]

F. AlessioM. L. Bertotti and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys., 50 (1999), 860-891.  doi: 10.1007/s000330050184.  Google Scholar

[4]

F. AlessioA. Calamai and P. Montecchiari, Saddle type solutions to a class of semilinear elliptic equations, Adv. Differential Equations, 12 (2007), 361-380.   Google Scholar

[5]

F. AlessioC. Gui and P. Montecchiari, Saddle solutions to Allen-Cahn equations in doubly periodic media, Indiana Univ. Math. J., 65 (2016), 199-221.  doi: 10.1512/iumj.2016.65.5772.  Google Scholar

[6]

F. Alessio and P. Montecchiari, Layered solutions with multiple asymptotes for non autonomous Allen-Cahn equations in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 46 (2013), 591-622.  doi: 10.1007/s00526-012-0495-2.  Google Scholar

[7]

F. Alessio and P. Montecchiari, Saddle solutions for bistable symmetric semilinear elliptic equations, NoDEA Nonlinear Differential Equation Appl., 20 (2013), 1317-1346.  doi: 10.1007/s00030-012-0210-1.  Google Scholar

[8]

F. Alessio and P. Montecchiari, Multiplicity of layered solutions for Allen-Cahn systems with symmetric double well potential, J. Differential Equations, 257 (2014), 4572-4599.  doi: 10.1016/j.jde.2014.09.001.  Google Scholar

[9]

S. Aubry and P. Y. LeDaeron, The discrete Frenkel–Kantorova model and its extensions I–Exact results for the ground states, Physica, 8D (1983), 381-422.  doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

[10]

U. Bessi, Many solutions of elliptic problems on ${\mathbb{R}}^{n}$ of irrational slope, Comm. Partial Differential Equations, 30 (2005), 1773-1804.  doi: 10.1080/03605300500299992.  Google Scholar

[11]

U. Bessi, Slope-changing solutions of elliptic problems on ${\mathbb{R}}^n$, Nonlinear Anal., 68 (2008), 3923-3947.  doi: 10.1016/j.na.2007.04.031.  Google Scholar

[12]

S. Bolotin and P. H. Rabinowitz, Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 103-128.  doi: 10.1016/j.anihpc.2013.02.003.  Google Scholar

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V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138.  doi: 10.1016/S0294-1449(16)30328-6.  Google Scholar

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X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of ${\mathbb{R}}^{2m}$, J. Eur. Math. Soc. (JEMS), 11 (2009), 819-943.  doi: 10.4171/JEMS/168.  Google Scholar

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X. Cabré and J. Terra, Qualitative properties of saddle-shaped solutions to bistable diffusion equations, Comm. Partial Differential Equations, 35 (2010), 1923-1957.  doi: 10.1080/03605302.2010.484039.  Google Scholar

[16]

H. DangP. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys, 43 (1992), 984-998.  doi: 10.1007/BF00916424.  Google Scholar

[17]

R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344.  doi: 10.1016/j.anihpc.2008.11.002.  Google Scholar

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M. del PinoM. KowalczykF. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in R2, J. Funct. Anal., 258 (2010), 458-503.  doi: 10.1016/j.jfa.2009.04.020.  Google Scholar

[19]

C. Gui, Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions, J. Differential Equations, 252 (2012), 5853-5874.  doi: 10.1016/j.jde.2012.03.004.  Google Scholar

[20]

C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Indiana Univ. Math. J., 57 (2008), 781-836.  doi: 10.1512/iumj.2008.57.3089.  Google Scholar

[21]

M. Kowalczyk and Y. Liu, Nondegeneracy of the saddle solution of the Allen-Cahn equation, Proc. Amer. Math. Soc., 139 (2011), 4319-4329.  doi: 10.1090/S0002-9939-2011-11217-6.  Google Scholar

[22]

M. KowalczykY. Liu and F. Pacard, The space of 4-ended solutions to the Allen-Cahn equation on the plane, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 761-781.  doi: 10.1016/j.anihpc.2012.04.003.  Google Scholar

[23]

J. N. Mather, Existence of quasi–periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467.  doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[24]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.  Google Scholar

[25]

J. Moser, Minimal solutions of a variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  doi: 10.1016/S0294-1449(16)30387-0.  Google Scholar

[26]

P. Montecchiari and P. H. Rabinowitz, On the existence of multi-transition solutions for a class of elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 199-219.  doi: 10.1016/j.anihpc.2014.10.001.  Google Scholar

[27]

F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension $8$ and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[28]

P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784.  doi: 10.1017/S0143385700000985.  Google Scholar

[29]

P. H. Rabinowitz, On a class of reversible elliptic systems, Netw. Heterog. Media, 7 (2012), 927-939.  doi: 10.3934/nhm.2012.7.927.  Google Scholar

[30]

P. H. Rabinowitz, A note on a class of reversible elliptic systems, Adv. Nonlinear Stud., 12 (2012), 851-875.  doi: 10.1515/ans-2012-0411.  Google Scholar

[31]

P. H. Rabinowitz and E. Stredulinsky, Extensions of Moser–Bangert Theory: Locally Minimal Solutions, Progr. Nonlinear Differential Equations Appl., 81, Birkhauser, Boston, 2011. Google Scholar

[32]

M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1241-1275.  doi: 10.1017/S0308210500030493.  Google Scholar

[33]

E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185.  doi: 10.1515/crll.2004.068.  Google Scholar

show all references

References:
[1]

S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in ${\mathbb{R}}^{2}$ for an Allen-Cahn system with multiple well potential, Calc. Var. Partial Differential Equations, 5 (1997), 359–390. doi: 10.1007/s005260050071.  Google Scholar

[2]

F. AlessioG. Autuori and P. Montecchiari, Saddle type solutions for a class of reversible elliptic equations, Adv. Differential Equations, 21 (2016), 1-30.   Google Scholar

[3]

F. AlessioM. L. Bertotti and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys., 50 (1999), 860-891.  doi: 10.1007/s000330050184.  Google Scholar

[4]

F. AlessioA. Calamai and P. Montecchiari, Saddle type solutions to a class of semilinear elliptic equations, Adv. Differential Equations, 12 (2007), 361-380.   Google Scholar

[5]

F. AlessioC. Gui and P. Montecchiari, Saddle solutions to Allen-Cahn equations in doubly periodic media, Indiana Univ. Math. J., 65 (2016), 199-221.  doi: 10.1512/iumj.2016.65.5772.  Google Scholar

[6]

F. Alessio and P. Montecchiari, Layered solutions with multiple asymptotes for non autonomous Allen-Cahn equations in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 46 (2013), 591-622.  doi: 10.1007/s00526-012-0495-2.  Google Scholar

[7]

F. Alessio and P. Montecchiari, Saddle solutions for bistable symmetric semilinear elliptic equations, NoDEA Nonlinear Differential Equation Appl., 20 (2013), 1317-1346.  doi: 10.1007/s00030-012-0210-1.  Google Scholar

[8]

F. Alessio and P. Montecchiari, Multiplicity of layered solutions for Allen-Cahn systems with symmetric double well potential, J. Differential Equations, 257 (2014), 4572-4599.  doi: 10.1016/j.jde.2014.09.001.  Google Scholar

[9]

S. Aubry and P. Y. LeDaeron, The discrete Frenkel–Kantorova model and its extensions I–Exact results for the ground states, Physica, 8D (1983), 381-422.  doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

[10]

U. Bessi, Many solutions of elliptic problems on ${\mathbb{R}}^{n}$ of irrational slope, Comm. Partial Differential Equations, 30 (2005), 1773-1804.  doi: 10.1080/03605300500299992.  Google Scholar

[11]

U. Bessi, Slope-changing solutions of elliptic problems on ${\mathbb{R}}^n$, Nonlinear Anal., 68 (2008), 3923-3947.  doi: 10.1016/j.na.2007.04.031.  Google Scholar

[12]

S. Bolotin and P. H. Rabinowitz, Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 103-128.  doi: 10.1016/j.anihpc.2013.02.003.  Google Scholar

[13]

V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138.  doi: 10.1016/S0294-1449(16)30328-6.  Google Scholar

[14]

X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of ${\mathbb{R}}^{2m}$, J. Eur. Math. Soc. (JEMS), 11 (2009), 819-943.  doi: 10.4171/JEMS/168.  Google Scholar

[15]

X. Cabré and J. Terra, Qualitative properties of saddle-shaped solutions to bistable diffusion equations, Comm. Partial Differential Equations, 35 (2010), 1923-1957.  doi: 10.1080/03605302.2010.484039.  Google Scholar

[16]

H. DangP. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys, 43 (1992), 984-998.  doi: 10.1007/BF00916424.  Google Scholar

[17]

R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344.  doi: 10.1016/j.anihpc.2008.11.002.  Google Scholar

[18]

M. del PinoM. KowalczykF. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in R2, J. Funct. Anal., 258 (2010), 458-503.  doi: 10.1016/j.jfa.2009.04.020.  Google Scholar

[19]

C. Gui, Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions, J. Differential Equations, 252 (2012), 5853-5874.  doi: 10.1016/j.jde.2012.03.004.  Google Scholar

[20]

C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Indiana Univ. Math. J., 57 (2008), 781-836.  doi: 10.1512/iumj.2008.57.3089.  Google Scholar

[21]

M. Kowalczyk and Y. Liu, Nondegeneracy of the saddle solution of the Allen-Cahn equation, Proc. Amer. Math. Soc., 139 (2011), 4319-4329.  doi: 10.1090/S0002-9939-2011-11217-6.  Google Scholar

[22]

M. KowalczykY. Liu and F. Pacard, The space of 4-ended solutions to the Allen-Cahn equation on the plane, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 761-781.  doi: 10.1016/j.anihpc.2012.04.003.  Google Scholar

[23]

J. N. Mather, Existence of quasi–periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467.  doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[24]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.  Google Scholar

[25]

J. Moser, Minimal solutions of a variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  doi: 10.1016/S0294-1449(16)30387-0.  Google Scholar

[26]

P. Montecchiari and P. H. Rabinowitz, On the existence of multi-transition solutions for a class of elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 199-219.  doi: 10.1016/j.anihpc.2014.10.001.  Google Scholar

[27]

F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension $8$ and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[28]

P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784.  doi: 10.1017/S0143385700000985.  Google Scholar

[29]

P. H. Rabinowitz, On a class of reversible elliptic systems, Netw. Heterog. Media, 7 (2012), 927-939.  doi: 10.3934/nhm.2012.7.927.  Google Scholar

[30]

P. H. Rabinowitz, A note on a class of reversible elliptic systems, Adv. Nonlinear Stud., 12 (2012), 851-875.  doi: 10.1515/ans-2012-0411.  Google Scholar

[31]

P. H. Rabinowitz and E. Stredulinsky, Extensions of Moser–Bangert Theory: Locally Minimal Solutions, Progr. Nonlinear Differential Equations Appl., 81, Birkhauser, Boston, 2011. Google Scholar

[32]

M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1241-1275.  doi: 10.1017/S0308210500030493.  Google Scholar

[33]

E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185.  doi: 10.1515/crll.2004.068.  Google Scholar

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