March  2022, 42(3): 1039-1065. doi: 10.3934/dcds.2021146

Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

1. 

Universidade Federal de Goiás, IME, Goiánia-GO, Brazil

2. 

Universidade Federal de Jataí, Jataí-GO, Brazil

*Corresponding author: Edcarlos D. Silva, email: edcarlos@ufg.br

Received  March 2021 Revised  August 2021 Published  March 2022 Early access  October 2021

Fund Project: The first author was partially supported by CNPq and FAPDF with grants 309026/2020-2 and 16809.78.45403.25042017, respectively

It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space $ \mathbb{R}^N $. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by $ \alpha \Delta^2 u + \beta \Delta u + V(x)u $ where $ \alpha > 0, \beta \in \mathbb{R} $ and $ V: \mathbb{R}^N \rightarrow \mathbb{R} $ is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where $ \beta $ is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.

Citation: Edcarlos D. Silva, Marcos L. M. Carvalho, Claudiney Goulart. Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth. Discrete & Continuous Dynamical Systems, 2022, 42 (3) : 1039-1065. doi: 10.3934/dcds.2021146
References:
[1]

C. O. AlvesJ. M. do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponent, Nonlinear Anal., 46 (2001), 121-133.  doi: 10.1016/S0362-546X(99)00449-6.  Google Scholar

[2]

C. O. Alves and A. B. Nóbrega, Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator, Monatsh. Math., 183 (2017), 35-60.  doi: 10.1007/s00605-016-0967-0.  Google Scholar

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D. BonheureJ. B. CastérasT. Gou and L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime, Trans. Amer. Math. Soc., 372 (2019), 2167-2212.  doi: 10.1090/tran/7769.  Google Scholar

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D. Bonheure, J. B. Castéras, T. Gou and L. Jeanjean, Strong instability of ground states to a fourth order Schrödinger equation, Int. Math. Res. Not. IMRN, (2019), 5299–5315. doi: 10.1093/imrn/rnx273.  Google Scholar

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F. BernisJ. Garcia-Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.   Google Scholar

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H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

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E. BerchioF. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differ. Equ., 229 (2006), 1-23.  doi: 10.1016/j.jde.2006.04.003.  Google Scholar

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E. BerchioF. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov type boundary conditions, Adv. Differ. Equ., 12 (2007), 381-406.   Google Scholar

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J. Chabrowski and J. M. do Ó, On some fourth-order semilinear elliptic problems in $\mathbb{R}^N$, Nonlinear Anal., 49 (2002), 861-884.  doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

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D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.  Google Scholar

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G. M. Figueiredo and M. T. O. Pimenta, Multiplicity of solutions for a biharmonic equation with subcritical or critical growth, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), 519-534.   Google Scholar

[13]

G. M. FigueiredoM. F. Furtado and J. P. P. da Silva, Existence and multiplicity of positive solutions for a fourth-order elliptic equation, Manuscripta Math., 160 (2019), 199-215.   Google Scholar

[14]

J. Garcia Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problem with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.1090/S0002-9947-1991-1083144-2.  Google Scholar

[15]

F. Gazzola, On the moments of solutions to linear parabolic equations involving the biharmonic operator, Discrete Contin. Dyn. Syst., 33 (2013), 3583-3597.  doi: 10.3934/dcds.2013.33.3583.  Google Scholar

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F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 251-263.  doi: 10.1017/S0308210500012774.  Google Scholar

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F. Gazzola and H. C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936.  doi: 10.1007/s00208-005-0748-x.  Google Scholar

[18]

F. Gazzola and E. Berchio, Best constants and minimizers for embeddings of second order Sobolev spaces, J. Math. Anal. Appl., 320 (2006), 718-735.  doi: 10.1016/j.jmaa.2005.07.052.  Google Scholar

[19]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger type equations, Phys. Rev. E, 53 (1996), 1336-1339.   Google Scholar

[20]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.  Google Scholar

[21]

H. F. Lins, Asymptotically periodic quasilinear elliptic equations with critical growth, PhD Thesis, UnB, 2004. Google Scholar

[22]

H. F. Lins and E. A. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.  Google Scholar

[23]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[24]

O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N$ with critical growth, Nonlinear Anal., 29 (1997), 773-781.  doi: 10.1016/S0362-546X(96)00087-9.  Google Scholar

[25]

M. T. O. Pimenta and S. H. M. Soares, Existence and concentration of solutions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274-289.  doi: 10.1016/j.jmaa.2012.01.039.  Google Scholar

[26]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[27]

E. D. Silva and T. R. Cavalcante, Multiplicity of solutions to fourth-order superlinear elliptic problems under Navier condictions, Electron. J. Differential Equations, (2017), 1–16.  Google Scholar

[28]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[29]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.  Google Scholar

[30]

J. SunJ. Chu and T. F. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian, J. Differential Equations, 262 (2017), 945-977.  doi: 10.1016/j.jde.2016.10.001.  Google Scholar

[31]

C. A. Swanson, The best Sobolev constant, Appl. Anal, 47 (1992), 227-239.  doi: 10.1080/00036819208840142.  Google Scholar

[32]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

C. O. AlvesJ. M. do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponent, Nonlinear Anal., 46 (2001), 121-133.  doi: 10.1016/S0362-546X(99)00449-6.  Google Scholar

[2]

C. O. Alves and A. B. Nóbrega, Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator, Monatsh. Math., 183 (2017), 35-60.  doi: 10.1007/s00605-016-0967-0.  Google Scholar

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Part. Diff. Eq., 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[4]

D. BonheureJ. B. CastérasT. Gou and L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime, Trans. Amer. Math. Soc., 372 (2019), 2167-2212.  doi: 10.1090/tran/7769.  Google Scholar

[5]

D. Bonheure, J. B. Castéras, T. Gou and L. Jeanjean, Strong instability of ground states to a fourth order Schrödinger equation, Int. Math. Res. Not. IMRN, (2019), 5299–5315. doi: 10.1093/imrn/rnx273.  Google Scholar

[6]

F. BernisJ. Garcia-Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.   Google Scholar

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[8]

E. BerchioF. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differ. Equ., 229 (2006), 1-23.  doi: 10.1016/j.jde.2006.04.003.  Google Scholar

[9]

E. BerchioF. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov type boundary conditions, Adv. Differ. Equ., 12 (2007), 381-406.   Google Scholar

[10]

J. Chabrowski and J. M. do Ó, On some fourth-order semilinear elliptic problems in $\mathbb{R}^N$, Nonlinear Anal., 49 (2002), 861-884.  doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

[11]

D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.  Google Scholar

[12]

G. M. Figueiredo and M. T. O. Pimenta, Multiplicity of solutions for a biharmonic equation with subcritical or critical growth, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), 519-534.   Google Scholar

[13]

G. M. FigueiredoM. F. Furtado and J. P. P. da Silva, Existence and multiplicity of positive solutions for a fourth-order elliptic equation, Manuscripta Math., 160 (2019), 199-215.   Google Scholar

[14]

J. Garcia Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problem with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.1090/S0002-9947-1991-1083144-2.  Google Scholar

[15]

F. Gazzola, On the moments of solutions to linear parabolic equations involving the biharmonic operator, Discrete Contin. Dyn. Syst., 33 (2013), 3583-3597.  doi: 10.3934/dcds.2013.33.3583.  Google Scholar

[16]

F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 251-263.  doi: 10.1017/S0308210500012774.  Google Scholar

[17]

F. Gazzola and H. C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936.  doi: 10.1007/s00208-005-0748-x.  Google Scholar

[18]

F. Gazzola and E. Berchio, Best constants and minimizers for embeddings of second order Sobolev spaces, J. Math. Anal. Appl., 320 (2006), 718-735.  doi: 10.1016/j.jmaa.2005.07.052.  Google Scholar

[19]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger type equations, Phys. Rev. E, 53 (1996), 1336-1339.   Google Scholar

[20]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.  Google Scholar

[21]

H. F. Lins, Asymptotically periodic quasilinear elliptic equations with critical growth, PhD Thesis, UnB, 2004. Google Scholar

[22]

H. F. Lins and E. A. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.  Google Scholar

[23]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[24]

O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N$ with critical growth, Nonlinear Anal., 29 (1997), 773-781.  doi: 10.1016/S0362-546X(96)00087-9.  Google Scholar

[25]

M. T. O. Pimenta and S. H. M. Soares, Existence and concentration of solutions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274-289.  doi: 10.1016/j.jmaa.2012.01.039.  Google Scholar

[26]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[27]

E. D. Silva and T. R. Cavalcante, Multiplicity of solutions to fourth-order superlinear elliptic problems under Navier condictions, Electron. J. Differential Equations, (2017), 1–16.  Google Scholar

[28]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[29]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.  Google Scholar

[30]

J. SunJ. Chu and T. F. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian, J. Differential Equations, 262 (2017), 945-977.  doi: 10.1016/j.jde.2016.10.001.  Google Scholar

[31]

C. A. Swanson, The best Sobolev constant, Appl. Anal, 47 (1992), 227-239.  doi: 10.1080/00036819208840142.  Google Scholar

[32]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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