Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation
Tóm tắt
This paper treats the existence of positive solutions of $$-\Delta u + V(x) u = \uplambda f(u)$$ in $${\mathbb {R}}^N$$ . Here $$N \ge 1$$ , $$\uplambda > 0$$ is a parameter and f(u) satisfies conditions only in a neighborhood of $$u=0$$ . We shall show the existence of positive solutions with potential of trapping type or $${\mathcal {G}}$$ -symmetric potential where $${\mathcal {G}} \subset O(N)$$ . Our results extend previous results (Adachi and Watanabe in J Math Anal Appl 507:125765, 2022; Costa and Wang in Proc Am Math Soc 133(3):787–794, 2005; do Ó et al. in J Math Anal Appl 342:432–445, 2008) as well as we also study the asymptotic behavior of a family $$(u_\uplambda )_{\uplambda \ge \uplambda _0}$$ of positive solutions as $$\uplambda \rightarrow \infty $$ .
Tài liệu tham khảo
citation_journal_title=Commun. PDE; citation_title=A positive solution of a nonhomogeneous elliptic equation in
with
-invariant nonlinearity; citation_author=S Adachi; citation_volume=27; citation_publication_date=2002; citation_pages=1-22; citation_doi=10.1081/PDE-120002781; citation_id=CR1
citation_journal_title=Adv. Differ. Eqns.; citation_title=
-invariant positive solutions for a quasilinear Schrödinger equation; citation_author=S Adachi, T Watanabe; citation_volume=16; citation_publication_date=2011; citation_pages=289-324; citation_id=CR2
citation_journal_title=J. Math. Anal. Appl.; citation_title=G-invariant positive solutions for a class of locally superlinear Schrödinger equations; citation_author=S Adachi, T Watanabe; citation_volume=507; citation_publication_date=2022; citation_doi=10.1016/j.jmaa.2021.125765; citation_id=CR3
citation_journal_title=Manuscr. Math.; citation_title=Existence of solution for a class of problem in whole
without the Ambrosetti-Rabinowitz condition; citation_author=C Alves, M Souto; citation_volume=165; citation_publication_date=2021; citation_pages=453-468; citation_doi=10.1007/s00229-020-01231-0; citation_id=CR4
citation_journal_title=Rev. Mat. Iberoamericana; citation_title=On a min-max procedure for the existence of a positive solution for certain scalar field equations in
; citation_author=A Bahri, YY Li; citation_volume=6; citation_issue=1–2; citation_publication_date=1990; citation_pages=1-15; citation_doi=10.4171/RMI/92; citation_id=CR5
Bartsch, T., Wang, Z.-Q., Willem, M.: The Dirichlet problem for superlinear elliptic equations. Stationary partial differential equations. Vol. II, 1–55, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005)
citation_journal_title=Arch. Rational. Mech. Anal.; citation_title=Nonlinear scalar field equations. I. Existence of a ground state; citation_author=H Berestycki, PL Lions; citation_volume=82; citation_issue=4; citation_publication_date=1983; citation_pages=313-345; citation_doi=10.1007/BF00250555; citation_id=CR7
citation_journal_title=C. R. Acad. Sci. Paris Sér. I Math.; citation_title=Éuations de champs scalaires euclidiens non linéaires dans le plan; citation_author=H Berestycki, T Gallouët, O Kavian; citation_volume=297; citation_issue=5; citation_publication_date=1983; citation_pages=307-310; citation_id=CR8
Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture notes in Mathematics, AMS (2003)
citation_journal_title=Istit. Lombardo Accad. Sci. Lett. Rend. A; citation_title=Un criterio di esistenza per i punti critici su varieta’illimitate; citation_author=G Cerami; citation_volume=112; citation_issue=2; citation_publication_date=1978; citation_pages=332-336; citation_id=CR10
citation_journal_title=J. Math. Anal. Appl.; citation_title=On a nonlinear elliptic eigenvalue problem; citation_author=S Chen, S Li; citation_volume=307; citation_issue=2; citation_publication_date=2005; citation_pages=691-698; citation_doi=10.1016/j.jmaa.2005.02.061; citation_id=CR11
citation_journal_title=Proc. Am. Math. Soc.; citation_title=Multiplicity results for a class of superlinear elliptic problems; citation_author=DG Costa, ZQ Wang; citation_volume=133; citation_issue=3; citation_publication_date=2005; citation_pages=787-794; citation_doi=10.1090/S0002-9939-04-07635-X; citation_id=CR12
De Bièvre, S., Genoud, F., Rota Nodari, S.: Orbital stability: analysis meets geometry. Nonlinear optical and atomic systems, 147–273, Lecture Notes in Math., 2146, CEMPI Ser., Springer, Cham (2015)
citation_journal_title=J. Math. Anal. Appl.; citation_title=On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations; citation_author=JM do Ó, E Medeiros, U Severo; citation_volume=342; citation_publication_date=2008; citation_pages=432-445; citation_doi=10.1016/j.jmaa.2007.11.058; citation_id=CR14
Ekeland, I.: Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19. Springer, Berlin (1990)
Enguiça, R., Ricardo, S., Sanchez, L.: A second order non-autonomous problem on the half-line: a variational approach. Mathematical models in engineering, biology and medicine, 119–128, AIP Conf. Proc., 1124, Amer. Inst. Phys., Melville, NY (2009)
citation_journal_title=Nonlinear Anal.; citation_title=Solutions of second-order and fourth-order ODEs on the half-line; citation_author=R Enguiça, A Gavioli, L Sanchez; citation_volume=73; citation_publication_date=2010; citation_pages=2968-2979; citation_doi=10.1016/j.na.2010.06.062; citation_id=CR17
citation_journal_title=Differ. Int. Eqns.; citation_title=Positive homoclinic solutions to some Schrödinger type equations; citation_author=A Gavioli, L Sanchez; citation_volume=29; citation_publication_date=2016; citation_pages=665-682; citation_id=CR18
Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in
$${\mathbf{R}}^n$$
. Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London (1981)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin (2001)
Han, Q., Lin, F.: Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2011)
citation_journal_title=Adv. Differ. Eqns.; citation_title=A positive solution of a nonlinear elliptic equation in
with G-symmetry; citation_author=J Hirata; citation_volume=12; citation_publication_date=2007; citation_pages=173-199; citation_id=CR22
citation_journal_title=Nonlinear Anal.; citation_title=A positive solution of a nonlinear Schrödinger equation with G-symmetry; citation_author=J Hirata; citation_volume=69; citation_issue=9; citation_publication_date=2008; citation_pages=3174-3189; citation_doi=10.1016/j.na.2007.09.010; citation_id=CR23
citation_journal_title=Proc. R. Soc. Edinburgh Sect. A; citation_title=On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on
; citation_author=L Jeanjean; citation_volume=129; citation_issue=4; citation_publication_date=1999; citation_pages=787-809; citation_doi=10.1017/S0308210500013147; citation_id=CR24
citation_journal_title=Proc. Am. Math. Soc.; citation_title=A remark on least energy solutions in
; citation_author=L Jeanjean, K Tanaka; citation_volume=131; citation_issue=8; citation_publication_date=2003; citation_pages=2399-2408; citation_doi=10.1090/S0002-9939-02-06821-1; citation_id=CR25
citation_journal_title=Adv. Nonlinear Stud.; citation_title=A note on a mountain pass characterization of least energy solutions; citation_author=L Jeanjean, K Tanaka; citation_volume=3; citation_issue=4; citation_publication_date=2003; citation_pages=445-455; citation_doi=10.1515/ans-2003-0403; citation_id=CR26
citation_journal_title=Israel J. Math.; citation_title=Schrödinger operators with singular potentials; citation_author=T Kato; citation_volume=13; citation_publication_date=1972; citation_pages=135-148; citation_doi=10.1007/BF02760233; citation_id=CR27
citation_journal_title=Arch. Rational Mech. Anal.; citation_title=Uniqueness of positive solutions of
in
; citation_author=MK Kwong; citation_volume=105; citation_issue=3; citation_publication_date=1989; citation_pages=243-266; citation_doi=10.1007/BF00251502; citation_id=CR28
citation_journal_title=Ann. Inst. H. Poincaré Anal. Non Linéaire; citation_title=The concentration-compactness principle in the calculus of variations. The locally compact case. I.; citation_author=P-L Lions; citation_volume=1; citation_issue=2; citation_publication_date=1984; citation_pages=109-145; citation_doi=10.1016/s0294-1449(16)30428-0; citation_id=CR29
citation_journal_title=Ann. Inst. H. Poincaré Anal. Non Linéaire; citation_title=The concentration-compactness principle in the calculus of variations. The locally compact case. II.; citation_author=P-L Lions; citation_volume=1; citation_issue=4; citation_publication_date=1984; citation_pages=223-283; citation_doi=10.1016/s0294-1449(16)30422-x; citation_id=CR30
citation_journal_title=Commun. Math. Phys.; citation_title=The principle of symmetric criticality; citation_author=RS Palais; citation_volume=69; citation_publication_date=1979; citation_pages=19-30; citation_doi=10.1007/BF01941322; citation_id=CR31
citation_journal_title=Ric. Mat.; citation_title=On the Ekeland-Ghoussoub-Preiss and Stuart criteria for locating Cerami sequences; citation_author=PJ Rabier; citation_volume=61; citation_issue=1; citation_publication_date=2012; citation_pages=19-29; citation_doi=10.1007/s11587-011-0112-2; citation_id=CR32
citation_journal_title=Z. Angew. Math. Phys.; citation_title=On a class of nonlinear Schrödinger equations; citation_author=PH Rabinowitz; citation_volume=43; citation_issue=2; citation_publication_date=1992; citation_pages=270-291; citation_doi=10.1007/BF00946631; citation_id=CR33
citation_journal_title=Nonlinear Anal. RWA.; citation_title=The existence and non-existence of solutions for the nonlinear Schrödinger equation in one dimension; citation_author=Y Sato; citation_volume=43; citation_publication_date=2018; citation_pages=477-494; citation_doi=10.1016/j.nonrwa.2018.03.013; citation_id=CR34
citation_journal_title=Milan J. Math.; citation_title=Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation; citation_author=CA Stuart; citation_volume=76; citation_publication_date=2008; citation_pages=329-399; citation_doi=10.1007/s00032-008-0089-9; citation_id=CR35
citation_journal_title=Commun. Appl. Anal.; citation_title=Locating Cerami sequences in a mountain pass geometry; citation_author=CA Stuart; citation_volume=15; citation_issue=2–4; citation_publication_date=2011; citation_pages=569-588; citation_id=CR36
Willem, M.: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA (1996)