### Existence and Concentrate Behavior of Schrödinger equations with critical exponential growth in $\mathbb{R}^N$

DOI: http://dx.doi.org/10.12775/TMNA.2016.058

#### Abstract

#### Keywords

#### References

C.O. Alves, M.A.S. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations 43 (2012), 537–554.

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 140 (1997), 285–300.

A. Ambrosetti, M. Badiale and S. Cingolani, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal. 159 (2001), 253–271.

T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3 (2001), 549–569.

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN , Comm. Partial Differential Equations 20(1995), 1725–1741.

T. Bartsch and Z.-Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys. 51 (2000), 366–384.

H. Berestycki, T. Gallouët and O. Kavian, Equations de champs scalaires euclidiens non linéaire dans le plan, C.R. Acad. Sci. Paris Paris Ser. I Math. 297 (1983), 307–310.

H. Berestycki and P.L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–346.

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), 185–200.

G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math. 74 (2006), 47–77.

M. Clapp and Y.H. Ding, Positive solutions of a Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys. 55 (2004), 592–605.

S. Cingolani and A. Pistoia, Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent, Z. Angew. Math. Phys. 55 (2004), 201–215.

D.G. de Figueiredo and Y.H. Ding, Solutions of a nonlinear Schrödinger equation, Discrete Contin. Dynam. Sys. 8(2002), 563–584.

M. del Pino and P. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.

M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrodinger equations, Ann. Inst. H. Poincaré Anal. Nonlinéaire 15 (1998), 127–149.

M. del Pino and P. Felmer, Semiclassical states for nonlinear Schrodinger equations, J. Funct. Anal. 149 (1997), 245–265.

M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrodinger equations: a variational reduction method, Math. Ann. 324 (2002), 1–32.

Y.H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrodinger equation, Manuscripta Math. 112 (2003), 109–135.

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational methods, Comm. Partial Differential Equations 21 (1996), 787–820.

L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on RN , Proc. Roy. Soc. Edinburgh 129 (1999), 787–809.

L. Jeanjean and K. Tanaka, A positive solution for asymptotically linear elliptic problem on RN autonomous at infinity, ESAIM Control Optim. Calc. Var. 7 (2002), 597–614.

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on RN , Indiana Univ. Math. J. 54 (2005), 443–464.

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations 21 (2004), 287–318.

Y.S. Jiang and H.S. Zhou, Schrödinger–Poisson system with steep potential well, J. Differential Equations 251 (2011), 582–608.

Y.Q. Li, Z.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non-linéaire, 23 (2006), 829–837.

Y.Y. Li, On a singularly perturbed elliptic equation, Adv. Differential Equations 2 (1997), 955–980.

Y.-G. Oh, Existence of semi-classical bound states of nonlinear Schrodinger equation with potential on the class (V )a , Comm. Partial Differential Equations 13 (1988), 1499–1519.

, On positive multi-lump bound states of nonlinear Schrodinger equations under multiple well potential, Comm. Math. Phys. 131 (1990), 223–253.

P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.

Y. Sato and K. Tanaka, Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells, Transactions of the American Mathematical Society 361 (2009), 6205–6253.

W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.

C.A. Stuart and H.S. Zhou, Global branch of solutions for non-linear Schrodinger equations with deepening potential well, Proc. London Math. Soc. 92 (2006), 655–681.

F.A. van Heerden, Multiple solutions for a Schrödinger type equation with an asymptotically linear term, Nonlinear Anal. 55 (2003), 739–758.

F.A. van Heerden and Z.-Q. Wang, Schrödinger type equations with asymptotically linear nonlinearities, Differential Integral Equations 16 (2003), 257–280.

M. Willem, Minimax Theorems, Birkhäuser, Boston (1996).

J.J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, submitted.

J. Zhang and W. Zou, The critical case for a Berestycki–Lions theorem, Sci. China Math. 57 (2014), 541–555.

J. Zhang and W. Zou, A Berestycki–Lions theorem revisited, Comm. Contemp. Math. 14 (2012), 1250033.

### Refbacks

- There are currently no refbacks.