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A non-periodic indefinite variational problem in ℝN with critical exponent

Part of: Partial differential equations Elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  26 June 2023

Gustavo S. do Amaral Costa ,
Giovany M. Figueiredo Open the ORCID record for Giovany M. Figueiredo [Opens in a new window]  and
José Carlos de O. Junior Open the ORCID record for José Carlos de O. Junior [Opens in a new window]
Gustavo S. do Amaral Costa
Affiliation:
Departamento de Matemática, Universidade Federal do Maranhão, São Luís, Maranhão, Brazil (gsa.costa@ufma.br)
Giovany M. Figueiredo
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, Distrito Federal, Brazil (giovany@unb.br)
José Carlos de O. Junior
Affiliation:
Colegiado de Matemática, Universidade Federal do Norte do Tocantins, Araguaína, Tocantins, Brazil (jc.oliveira@uft.edu.br)
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Abstract

We consider the non-linear Schrödinger equation(Pμ)

\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*}
in $\mathbb{R}^N$, $N\geq3$, where V changes sign and $f(s)/s$, s ≠ 0, is bounded, with V non-periodic in x. The existence of a solution is established employing spectral theory, a general linking theorem due to [12] and interaction between translated solutions of the problem at infinity with some qualitative properties of them.

Keywords

critical exponentabstract linking theoremindefinite problemsspectral theory

MSC classification

Primary: 35A15: Variational methods 35J60: Nonlinear elliptic equations 35D30: Weak solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 2 , May 2023 , pp. 579 - 612
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Alves, C. O. and Figueiredo, G. M., Multiplicity and Concentration o Positive Solutions for a Class of Quasilinear Problems Adv. Nonlinear Stud. 11 (2011), 265–294.10.1515/ans-2011-0203CrossRefGoogle Scholar
Alves, C. O., and Germano, G. F., Ground state of solution for a class of indefinite variational problems with critical growth, J. Differential Equations 265(1) (2018), 444–447.10.1016/j.jde.2018.02.039CrossRefGoogle Scholar
Berestycki, H. and Lions, P. L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–345.10.1007/BF00250555CrossRefGoogle Scholar
Chabrowski, J. and Szulkin, A., On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc. 130 (2001), 85–93.10.1090/S0002-9939-01-06143-3CrossRefGoogle Scholar
Costa, D. G. and Tehrani, H., Existence and multiplicity results for a class of Schrödinger equations with indefinite nonlinearities, Adv. Differential Equations 8 (2003), 1319–1340.10.57262/ade/1355926119CrossRefGoogle Scholar
Egorov, Y. and Kondratiev, V., On Spectral Theorey of Elliptic Operators (Birkhäuser, Basel, 1996).10.1007/978-3-0348-9029-8CrossRefGoogle Scholar
Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.10.1017/S0308210500013147CrossRefGoogle Scholar
Kryszewski, W. and Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations 3 (1998), 441–472.10.57262/ade/1366399849CrossRefGoogle Scholar
Li, G. and Szulkin, A., An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math. 4 (2002), 763–776.10.1142/S0219199702000853CrossRefGoogle Scholar
Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case. Parts I and II, Ann. Inst. H. Poincaré C Anal. Non Linéaire 1 (1984), .Google Scholar
Maia, L. A., Junior, J. C. O. and Ruviaro, R., A non-periodic and asymptotically linear indefinite variational problem in $\mathbb{R}^N$, Indiana Univ. Math. J. 66(1) (2017), 31–54.10.1512/iumj.2017.66.5955CrossRefGoogle Scholar
Maia, L. A. and Soares, M., An indefinite elliptic problem on RN autonomous at infinity: the crossing effect of the spectrum and the nonlinearity, Calc. Var. Partial Differential Equations 41(59) (2020), 1–22.Google Scholar
Pankov, A. A. and Pflüger, K., On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. 33 (1998), 593–690.10.1016/S0362-546X(97)00689-5CrossRefGoogle Scholar
Schechter, M. and Zou, W., Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var. 9 (2003), 601–619.10.1051/cocv:2003029CrossRefGoogle Scholar
Stuart, C. A., An Introduction to Elliptic Equation in RN (River Edge, NJ: World Sci. Publ.), Trieste Notes, (1998).Google Scholar
Stuart, C. A. and Zhou, H. S., Applying the mountain pass theorem to an asymptotically linear elliptic equation on $\mathbb{R}^N$, Comm. Partial Differential Equations 24 (1999), 1731–1758.10.1080/03605309908821481CrossRefGoogle Scholar
Szulkin, A. and Weth, T., Ground state solutions for some indefinite variational problems, J. Func. Anal. 257 (2009), 3802–3822.10.1016/j.jfa.2009.09.013CrossRefGoogle Scholar
Zhang, H., Xu, J. and Zhang, F., On a class of semilinear Schrödinger equation with indefinite linear part, J. Math. Anal. Appl. 414 (2014), 710–724.10.1016/j.jmaa.2014.01.001CrossRefGoogle Scholar