May  2020, 19(5): 2887-2906. doi: 10.3934/cpaa.2020126

Existence of ground state solution and concentration of maxima for a class of indefinite variational problems

Universidade Federal de Campina Grande, Unidade Acdêmica de Matemática, CEP: 58429-900, Campina Grande - Pb, Brazil

* Corresponding author

Received  July 2019 Revised  November 2019 Published  March 2020

Fund Project: Claudianor O. Alves was partially supported by CNPq/Brazil 304804/2017-7

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like
$ \begin{cases} -\Delta u+V(x)u = A(\epsilon x)f(u) \quad \mbox{in} \quad \mathbb{R}^{N}, \\ u\in H^{1}( \mathbb{R}^{N}), \end{cases} \qquad\qquad\qquad{(P)_{\epsilon}} $
where
$ N \geq 1 $
,
$ \epsilon $
is a positive parameter,
$ f: \mathbb{R} \to \mathbb{R} $
is a continuous function with subcritical growth and
$ V,A: \mathbb{R}^{N} \to \mathbb{R} $
are continuous functions verifying some technical conditions. Here
$ V $
is a
$ \mathbb{Z}^N $
-periodic function,
$ 0 \not\in \sigma(-\Delta + V) $
, the spectrum of
$ -\Delta +V $
, and
$ 0 < \inf\limits_{x \in \mathbb{R}^{N}}A(x)\leq \lim\limits_{|x|\rightarrow+\infty}A(x)<\sup\limits_{x \in \mathbb{R}^{N}}A(x). $
Citation: Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126
References:
[1]

C. O. Alves and G. F. Germano, Ground state solution for a class of indefinite variational problems with critical growth, J. Differ. Equ., 265 (2018), 444-477.  doi: 10.1016/j.jde.2018.02.039.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Concentration phenomena for for NLS: recent results and new perspectives, in Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, Vol. 446, American Mathematical Society, Providence, RI, (2007), 19–30. doi: 10.1090/conm/446/08624.  Google Scholar

[3]

A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

[4]

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

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J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc., 130 (2001), 85-93.  doi: 10.1090/S0002-9939-01-06143-3.  Google Scholar

[6]

M. del Pino and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

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J. M. B. do Ó and B. Ruf, On a Schrödinger equation with periodic potential and critical growth in $\mathbb{R}^2$, Nonlinear Differ. Equ. Appl., 13 (2006), 167-192.  doi: 10.1007/s00030-005-0034-3.  Google Scholar

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A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[9]

M. F. Furtado and R. Marchi, Existence of solutions to asymptotically periodic Schrödinger equations, Electron. J. Differ. Equ., 2017 (2017), 1-7.   Google Scholar

[10]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472.   Google Scholar

[11]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part Ⅱ, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 4 (1984), 223-283.   Google Scholar

[12]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar

[13]

Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_\alpha$, Commun. Partial Differ. Equ., 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.  Google Scholar

[14]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.  Google Scholar

[15]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[16]

A. Pankov and K. Pflüger, On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. Theory Methods Appl., 33 (1998), 593-609.  doi: 10.1016/S0362-546X(97)00689-5.  Google Scholar

[17]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reginal Conference Series in Mathematics, Vol. 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[18]

P. H. Rabinowitz, A note on semilinear elliptic equation on $\mathbb{R}^N$, Nonlinear Analysis: A Tribute in Honour of G. Prodi, Quad. Scu. Norm. Super. Pisa., (1991), 307–318.  Google Scholar

[19]

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

[20]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[21]

A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, International Press, (2010), 597–632.  Google Scholar

[22]

M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1596-7.  Google Scholar

[23]

M. Schechter, Nonlinear Schrödinger operators with zero in the spectrum, Z. Angew. Math. Phys., 66 (2015), 2125-2141.  doi: 10.1007/s00033-015-0511-4.  Google Scholar

[24]

M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), 601–619. (electronic) doi: 10.1051/cocv:2003029.  Google Scholar

[25]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.  Google Scholar

[26]

X. H. Tang, New super-quadratic conditions for asymptotically periodic Schrödinger equation, Canadian Math. Bull., 60 (2017), 422-435.  doi: 10.4153/CMB-2016-090-2.  Google Scholar

[27]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 53 (1993), 229-244.   Google Scholar

[28]

M. Willem, Minimax Theorems, Birkhauser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52 (2003), 109-132.  doi: 10.1512/iumj.2003.52.2273.  Google Scholar

[30]

M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627.  doi: 10.1016/j.na.2009.11.009.  Google Scholar

[31]

H. ZhangJ. Xu and F. Zhang, Ground state solutions asymptotically periodic Schrödinger equations with indefinite linear part, Math. Meth. Appl. Sci., 38 (2015), 113-122.  doi: 10.1002/mma.3054.  Google Scholar

[32]

H. ZhangJ. Xu and F. Zhang, On a class of semilinear Schrödinger equation with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.  doi: 10.1016/j.jmaa.2014.01.001.  Google Scholar

show all references

References:
[1]

C. O. Alves and G. F. Germano, Ground state solution for a class of indefinite variational problems with critical growth, J. Differ. Equ., 265 (2018), 444-477.  doi: 10.1016/j.jde.2018.02.039.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Concentration phenomena for for NLS: recent results and new perspectives, in Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, Vol. 446, American Mathematical Society, Providence, RI, (2007), 19–30. doi: 10.1090/conm/446/08624.  Google Scholar

[3]

A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

[4]

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

[5]

J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc., 130 (2001), 85-93.  doi: 10.1090/S0002-9939-01-06143-3.  Google Scholar

[6]

M. del Pino and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[7]

J. M. B. do Ó and B. Ruf, On a Schrödinger equation with periodic potential and critical growth in $\mathbb{R}^2$, Nonlinear Differ. Equ. Appl., 13 (2006), 167-192.  doi: 10.1007/s00030-005-0034-3.  Google Scholar

[8]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[9]

M. F. Furtado and R. Marchi, Existence of solutions to asymptotically periodic Schrödinger equations, Electron. J. Differ. Equ., 2017 (2017), 1-7.   Google Scholar

[10]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472.   Google Scholar

[11]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part Ⅱ, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 4 (1984), 223-283.   Google Scholar

[12]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar

[13]

Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_\alpha$, Commun. Partial Differ. Equ., 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.  Google Scholar

[14]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.  Google Scholar

[15]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[16]

A. Pankov and K. Pflüger, On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. Theory Methods Appl., 33 (1998), 593-609.  doi: 10.1016/S0362-546X(97)00689-5.  Google Scholar

[17]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reginal Conference Series in Mathematics, Vol. 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[18]

P. H. Rabinowitz, A note on semilinear elliptic equation on $\mathbb{R}^N$, Nonlinear Analysis: A Tribute in Honour of G. Prodi, Quad. Scu. Norm. Super. Pisa., (1991), 307–318.  Google Scholar

[19]

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

[20]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[21]

A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, International Press, (2010), 597–632.  Google Scholar

[22]

M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1596-7.  Google Scholar

[23]

M. Schechter, Nonlinear Schrödinger operators with zero in the spectrum, Z. Angew. Math. Phys., 66 (2015), 2125-2141.  doi: 10.1007/s00033-015-0511-4.  Google Scholar

[24]

M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), 601–619. (electronic) doi: 10.1051/cocv:2003029.  Google Scholar

[25]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.  Google Scholar

[26]

X. H. Tang, New super-quadratic conditions for asymptotically periodic Schrödinger equation, Canadian Math. Bull., 60 (2017), 422-435.  doi: 10.4153/CMB-2016-090-2.  Google Scholar

[27]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 53 (1993), 229-244.   Google Scholar

[28]

M. Willem, Minimax Theorems, Birkhauser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52 (2003), 109-132.  doi: 10.1512/iumj.2003.52.2273.  Google Scholar

[30]

M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627.  doi: 10.1016/j.na.2009.11.009.  Google Scholar

[31]

H. ZhangJ. Xu and F. Zhang, Ground state solutions asymptotically periodic Schrödinger equations with indefinite linear part, Math. Meth. Appl. Sci., 38 (2015), 113-122.  doi: 10.1002/mma.3054.  Google Scholar

[32]

H. ZhangJ. Xu and F. Zhang, On a class of semilinear Schrödinger equation with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.  doi: 10.1016/j.jmaa.2014.01.001.  Google Scholar

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