Singularly perturbed N-Laplacian problems with a nonlinearity in the critical growth range

Jianjun Zhang, João Marcos do Ó, Olímpio H. Miyagaki

DOI: http://dx.doi.org/10.12775/TMNA.2017.021

Abstract


We consider the following singularly perturbed problem: $$ -\e^N\Delta_N u+V(x)|u|^{N-2}u= f(u),\quad u(x)> 0\quad \mbox{in } \mathbb R^N, $$ \noindent where $N\ge 2$ and $\Delta_N u$ is the $N$-Laplacian operator. In this paper, we construct a solution $u_\e$ which concentrates around any given isolated positive local minimum component of $V$, as $\e\rg 0$, in the Trudinger-Moser type of subcritical or critical case. In the subcritical case, we only impose on $f$ the Berestycki and Lions conditions. In the critical case, a global condition on the nonlinearity $f$ is imposed. However, any {\it monotonicity} of $f(t)/t^{N-1}$ or {\it Ambrosetti-Rabinowitz} type conditions are not required.

Keywords


Nonlinear Schrödinger equations; semiclassical states; potential well; critical growth; penalization

Full Text:

PREVIEW FULL TEXT

References


Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), 393–413.

S. Adachi and K. Tanaka, Trudinger type inequalities in RN and their best exponents, Proc. Amer. Math. Soc. 128 (2000), 2051–2057.

C. Alves and G. Figueiredo, Existence and multiplicity of positive solutions to a p-Laplacian equation in RN , Differential Integral Equations 19 (2006), 143–162.

C. Alves and G. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN , J. Differential Equations 246 (2009), 1288–1311.

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

H. Berestycki, T. Gallouöt and O. Kavian, Equations de champs scalaires euclidens non linéires dans le plan, C.R. Acad. Sci. Paris Sér. I Math. 297 (1983), 307–310; Publications du Laboratoire d’Analyse Numérique, Université de Paris VI (1984).

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, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc. 362 (2010), 1981–2001.

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

J. Byeon, L. Jeanjean and M. Maris, Symmetric and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations 36 (2009), 481–492.

J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimentional case, Comm. Partial Differential Equations 33(2008), 1113–1136.

J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc. 15 (2013), 1859–1899.

J. Byeon and K. Tanaka, Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations, Mem. Amer. Math. Soc., vol. 229, American Mathematical Society, Providence, 2014

J. Byeon and Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differential Equuations 18 (2003), 207–219.

D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2, Comm. Partial Differential Equations 17 (1992), 407–435.

P. D’Avenia, A. Pomponio and D. Ruiz, Semi-classical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal. 262 (2012), 4600–4633.

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

M. del Pino and P.L. Felmer, Multi-peak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non-Linéaire 15 (1998), 127–149.

M. del Pino and P.L. Felmer, Spike-layered solutions of singularlyly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), 883–898.

M. del Pino and P.L. Felmer, Semiclasscial states for nonlinear Schrödinger equations: a variational reduction method, Math. Ann. 324 (2002), 1–32.

J.M. do Ó, N -Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.

J.M. do Ó, On existence and concentration of positive bound states of p-Laplacian equations in RN involving critical growth, Nonlinear Analysis 62 (2005), 777–801.

J.M. do Ó and E.S. Medeiros, Remarks on least energy solutions for quasilinear elliptic problems in RN , Electronic J. Differential Equations 83 (2003), 1–14.

J.M. do Ó, F. Sani and J.J. Zhang, Stationary nonlinear Schrödinger equations in R2 with potentials vanishing at infinity, Annali di Matematica Pura ed Applicata (2016), published online, DOI: 10.1007/s10231-016-0576-5.

G.M. Figueiredo and M.F. Furtado, Positive solutions for a quasilinear Schröinger equation with critical growth, J. Dynamics and Differential Equations, 24 (2012), 13–28.

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

A. Giacomini and M. Squassina, Multi-peak solutions for a class of degenerate elliptic equations, Asymptot. Anal. 36 (2003), no. 2, 115–147.

E. Gloss, Existence and concentration of bound states for a p-Laplacian equation in RN, Adv. Nonlinear Stud. 10 (2010), 273–296.

O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Springer, Heidelberg, 1983.

G.B. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. Math. 14 (1989), 27–36.

Y.G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V )a , Comm. Partial Differential Equations 13(1988), 1499–1519.

P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681–703.

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

J. Serrin, Local behavior of solutions of qusai-linear equations, Acta. Math. 111 (1964), 248–302.

W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1997), 149–162.

P. Tolksdorf, Regularity for a more general class of qusilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.

N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747.

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (1993), 229–244.

G.Q. Zhang and J. Sun, Ground-state solutions for a class of N -Laplacian equation with critical growth, Abstr. Appl. Anal. 2012 (2012), 1–14, DOI: 10.1155/2012/831468.

J.J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, J. London Math. Soc. 90 (2014), 827–844.

J.J. Zhang and J.M. do Ó, Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger–Moser type, Z. Angew. Math. Phys. 66 (2015), 3049–3060.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism