Quasilinear elliptic equations involving the \(N\)-Laplacian with critical exponential growth in \(\mathbb R^N\). (English) Zbl 1180.35262

Wang, Youjun; Yang, Jun; Zhang, Yimin

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, 6157-6169 (2009).

Summary: This paper shows the existence of nontrivial solutions for the quasilinear equations of the form \[ -\Delta_N u+V(x)|u|^{N-2}u-\Delta_N(u^2)u= h(u)\quad \text{in }\mathbb R^N, \]
where \(\Delta_N\) is the \(N\)-Laplacian operator, \(V\) is a continuous function bounded from below away from zero and \(h(u)\) is a continuous function having critical exponential growth.

MSC:

35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35J20	Variational methods for second-order elliptic equations
35B33	Critical exponents in context of PDEs

Keywords:

quasilinear elliptic equations; critical exponential growth; Trudinger-Morser equality

References:

[1]	Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. funct. anal., 69, 397-408, (1986) · Zbl 0613.35076
[2]	Bertone, A.M.; do Ó, J.M., On a class of semilinear Schrödinger equations involving critical growth and discontinuous nonlinearities, Nonlinear anal., 54, 885-906, (2003) · Zbl 1290.35090
[3]	Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. agrew math. phys., 43, 270-291, (1992) · Zbl 0763.35087
[4]	Jaen jean, L.; Tanaka, K., A remark on the least energy solutions in \(\mathbb{R}^N\), Proc. amer. math. soc., 131, 2399-2408, (2003) · Zbl 1094.35049
[5]	Chabrowski, J.; Yang, J.F., Existence theorems for the Schrödinger equation involving a critical exponent, Z. agrew math. phys., 49, 276-293, (1998) · Zbl 0903.35021
[6]	Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[7]	Bartsch, T.; Wang, Z.Q., Existence and multiplicity results for some superlinear elliptic problem on \(\mathbb{R}^N\), Comm. partial differential equations, 20, 1725-1741, (1995) · Zbl 0837.35043
[8]	Liu, J.Q.; Wang, Z.Q., Soliton solutions for quasilinear Schrödinger equations I, Proc. amer. math. soc., 131, 441-448, (2003) · Zbl 1229.35269
[9]	Liu, J.Q.; Wang, Y.Q.; Wang, Z.Q., Soliton solutions to quasilinear Schrödinger equations II, J. differential equations, 187, 473-493, (2003) · Zbl 1229.35268
[10]	Liu, J.Q.; Wang, Y.Q.; Wang, Z.Q., Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. partial differential equations, 29, 5-6, 879-901, (2004) · Zbl 1140.35399
[11]	Colin, M.; Jeanjean, L., Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear anal., 56, 213-226, (2004) · Zbl 1035.35038
[12]	Poppenberg, M.; Schmitt, K.; Wang, Z.Q., On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. var. partial differential equations, 14, 329-344, (2002) · Zbl 1052.35060
[13]	do Ó, J.M.; Miyagaki, O.H.; Soares, S.H.M., Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear anal., 67, 3357-3372, (2007) · Zbl 1151.35016
[14]	Bartsch, T.; Wang, Z.Q., Multiple positive solutions for a nonlinear Schrödinger equation, Z. angew. math. phys., 51, 366-384, (2000) · Zbl 0972.35145
[15]	Moameni, A., On a class of periodic quasilinear Schrödinger equations involving critical growth in \(\mathbb{R}^2\), J. math. anal. appl., 334, 775-786, (2007) · Zbl 1156.35031
[16]	Moameni, A., Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in \(\mathbb{R}^N\), J. differential equations, 229, 570-587, (2006) · Zbl 1131.35080
[17]	Ritchie, B., Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. rev. E, 50, 687-689, (1994)
[18]	Lang, H.; Poppenberg, M.; Teismann, H., Nash-Morse methods for the solution of quasilinear Schrödinger equations, Comm. partial differential equations, 24, 1399-1418, (1999) · Zbl 0935.35153
[19]	Ambrosetti, A.; Wang, Z.Q., Positive solutions to a class quasilinear elliptic equations on \(\mathbb{R}\), Discrete contin. dyn. syst., 9, 56-68, (2003)
[20]	Borovskii, A.V.; Galkin, A.L., Dynamical modulation of an ultrashort high-intensity laser pulse in matter, Jetp, 77, 562-573, (1993)
[21]	Alves, M.J.; Carriã, P.C.; Miyagaki, O.H., Non-autonomous perturbations for a class of quasilinear elliptic equations on \(\mathbb{R}\), J. math. anal. appl., 344, 186-203, (2008) · Zbl 1143.35033
[22]	Severo, U., Existence of weak solutions for quasilinear elliptic equations involving the \(p\)-Laplacian, Electron. J. differential equations, 56, 1-16, (2008) · Zbl 1173.35483
[23]	do Ó, J.M.; Medeiros, E.S., Remark on least energy solutions for quasilinear elliptic problems in \(\mathbb{R}^N\), Electron. J. differential equations, 83, 1-14, (2003)
[24]	Trudinger, N.S., On the imbedding into Orlicz spaces and some applications, J. math. mech., 17, 473-484, (1967) · Zbl 0163.36402
[25]	do Ó, J.M., N-Laplacian equations in \(\mathbb{R}^N\) with critical growth, Abstr. appl. anal., 2, 301-315, (1997) · Zbl 0932.35076
[26]	do Ó, J.M., Semilinear Dirichlet problems for the N-Laplacian in \(\mathbb{R}^N\) with nonlinearities in critical growth range, Differential integral equations, 9, 967-979, (1996) · Zbl 0858.35043
[27]	Lions, P.L., The concentration compactness principle in the calculus of variations. the locally compact case. part I and II, Ann. inst. H. Poincaré anal. non linéaire, 1, 109-145, (1984), and 223-283 · Zbl 0541.49009