November  2018, 17(6): 2789-2812. doi: 10.3934/cpaa.2018132

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

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

* Corresponding author

Received  October 2017 Revised  January 2018 Published  June 2018

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

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.
Citation: Alireza Khatib, Liliane A. Maia. A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2789-2812. doi: 10.3934/cpaa.2018132
References:
[1]

N. AckermannM. 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. BerestyckiT. 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. GidasW. 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. MaiaO. 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.

show all references

References:
[1]

N. AckermannM. 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. BerestyckiT. 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. GidasW. 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. MaiaO. 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.

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