A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains

Alireza Khatib; Liliane A. Maia

doi:10.3934/cpaa.2018132

Article Contents

2018, Volume 17, Issue 6: 2789-2812. Doi: 10.3934/cpaa.2018132

This issue Previous Article Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems Next Article On variational and topological methods in nonlinear difference equations

A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains

Alireza Khatib and
Liliane A. Maia^,

Universidade de Brasília, Departamento de Matemática, 70910-900 Brasília-DF, Brazil

* Corresponding author
* Corresponding author

Received: September 30, 2017

Revised: December 31, 2017

Published: June 2018

Research supported by FAPDF 193.000.939/2015 and 0193.001300/2016, CNPq/PQ 308173/2014-7 and PROEX/CAPES (Brazil).

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We establish the existence of a positive solution for semilinear elliptic equation in exterior domains

$\begin{array}{lc}-Δ u + V(x) u = f(u),\ \ {\rm{in}} \ \ Ω \subseteq \mathbb{R}^N &&&(P_V)\end{array}$

where $N≥2$, $Ω$ is an open subset of $\mathbb{R}^N$ and $ \mathbb{R}^N \setminus Ω $ is bounded and not empty but there is no restriction on its size, nor any symmetry assumption. The nonlinear term $f$ is a non homogeneous, asymptotically linear or superlinear function at infinity. Moreover, the potential Ⅴ is a positive function, not necessarily symmetric. The existence of a solution is established in situations where this problem does not have a ground state.

Keywords:

Mathematics Subject Classification: Primary: 35Q55, 35J61; Secondary: 35J20.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	N. Ackermann, M. Clapp and F. Pacella, Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains, Comm. Partial Differential Equations, 38 (2013), 751-779.
[2]	A. Bahri and Y.Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}^N$, Rev. Mat. Iberoamericana 6, 1/2 (2013), 751-779. doi: 10.4171/RMI/92.
[3]	A. Bahri and P.L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincare Anal. Non Lineaire, 14 (1997), 365-413. doi: 10.1016/S0294-1449(97)80142-4.
[4]	T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005.
[5]	V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048.
[6]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[7]	H. Berestycki, T. Gallouet and O. Kavian, Equations de champs scalaires euclidiens non lineaires dans le plan, C. R. Math. Acad. Sci., 297 (1983), 307-310.
[8]	A. Bonnet, A deformation lemma on $C^1$ manifold, Manuscripta Math, 81 (1993), 339-359. doi: 10.1007/BF02567863.
[9]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science+Business Media, LLC 2011.
[10]	G. Cerami, Un criterio di esistenza per i punti critici su variet`a illimitate, Rend. Accad. Sc. Lett. Inst. Lombardo, 112 (1978), 332-336.
[11]	G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math, 74 (2006), 47-77. doi: 10.1007/s00032-006-0059-z.
[12]	G. Cerami and D. Passaseo, Existence and multiplicity results for semilinear elliptic dirichlet problems in exterior domains, Nonlinear Analysis TMA 24, 11 (1995), 1533-1547. doi: 10.1016/0362-546X(94)00116-Y.
[13]	G. Citti, On the exterior Dirichlet problem for $\Delta u-u+f(x,u) = 0$, Rendiconti del seminario matematico dell'università di Padova, 88 (1992), 83-100.
[14]	M. Clapp and L. A. Maia, A positive bound state for an asymptotically linear or superlinear Schrödinger equation, J. Differential Equation, 260 (2016), 3173-3192. doi: 10.1016/j.jde.2015.09.059.
[15]	C. V. Coffman and M. Marcus, Superlinear elliptic Dirichlet problems in almost spherically symmetric exterior domains, Arch Rational Mech. Anal., 96 (1986), 167-196. doi: 10.1007/BF00251410.
[16]	R. Dautray and J-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1. Physical Origins and Classical Methods, Springer-Verlag, Berlin, 1990. ⅹⅷ+695 pp.
[17]	W-Y. Ding and W-M. Ni, On the existence of positive entire solution of semilinear elliptic equation, Arch Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.
[18]	I. Ekeland, On the variational principle, J. Math. anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[19]	M. Esteban and P. L. Lions, Existence and nonexistence results for semi-linear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 93, 1-2 (1982), 1-14. doi: 10.1017/S0308210500031607.
[20]	G. Évéquoz and T. Weth, Entire solutions to nonlinear scalar field equations with indefinite linear part, Adv. Nonlinear Stud., 12 (2012), 281-314. doi: 10.1515/ans-2012-0206.
[21]	G. P. Galdi and C. R. Grisanti, Existence and regularity of steady flows for shear-thinning liquids in exterior two-dimensional, Arch. Rational Mech. Anal., 200 (2011), 533-559. doi: 10.1007/s00205-010-0364-0.
[22]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, Adv. in Math. Suppl. Stud., 7a (1981), 369-402.
[23]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Parts Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145 and 223-283.
[24]	L. A. Maia, O. H. Miyagaki and S. M. Soares, A sign changing solution for an asymptotically linear Schödinger equation, Proc. Edinb. Math. Soc., (2015), 697-716. doi: 10.1017/S0013091514000339.
[25]	L. A. Maia and B. Pellacci, Positive solutions for asymptotically linear problems in exterior domains, Annali di Matematica Pura ed Applicata, (2016), 196 (2017), 4, 1399-1430. doi: 10.1007/s10231-016-0621-4.
[26]	K. McLeod, Uniqueness of positive radial solutions of $ Δ u+ f(u) = 0$ in $\mathbb{R}^N$ Ⅱ, Trans. Amer. Math. Soc., 339 (1993), 495-505. doi: 10.2307/2154282.
[27]	Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175. doi: 10.1007/BF02559588.
[28]	Z. Nehari, A nonlinear oscillation theorem, Duke Math. J., 42 (1975), 183-189.
[29]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.
[30]	J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893.
[31]	C. A. Stuart, An Introduction to Elliptic Equation in $\mathbb{R}^N$, Trieste Notes, 1998.
[32]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24, Birkhauser Boston, Inc., Boston, MA, doi: 10.1007/978-1-4612-4146-1.