Previous |  Up |  Next

Article

MSC: 35D30, 35J20, 35J92
Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
$N$-Laplacian equation; critical exponential growth; Schwarz symmetrization; Nehari manifold
Summary:
In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation $$ -\Delta _Nu+b|u|^{N-2}u-\Delta _N(u^2)u=h(u), \quad x\in \mathbb {R}^N, $$ where $\Delta _N$ is the $N$-Laplacian operator, $h(u)$ is continuous and behaves as $\exp (\alpha |u|^{{N}/{(N-1)}})$ when $|u|\to \infty $. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u(x)\in W^{1,N}(\mathbb {R}^N)$ with $u(x)\to 0$ as $|x|\to \infty $ is established.
References:
[1] Adachi, S., Tanaka, K.: Trudinger type inequalities in $\mathbb R^N$ and their best exponents. Proc. Am. Math. Soc. 128 (2000), 2051-2057. DOI 10.1090/S0002-9939-99-05180-1 | MR 1646323
[2] Badiale, M., Serra, E.: Semilinear Elliptic Equations for Beginners. Existence Results via the Variational Approach. Universitext Springer, London (2011). MR 2722059 | Zbl 1214.35025
[3] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82 (1983), 313-345. MR 0695535 | Zbl 0533.35029
[4] Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88 (1983), 486-490. DOI 10.2307/2044999 | MR 0699419 | Zbl 0526.46037
[5] Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics 10 New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (2003). MR 2002047 | Zbl 1055.35003
[6] Colin, M., Jeanjean, L.: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 56 (2004), 213-226. DOI 10.1016/j.na.2003.09.008 | MR 2029068 | Zbl 1035.35038
[7] Bouard, A. de, Hayashi, N., Saut, J.-C.: Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun. Math. Phys. 189 (1997), 73-105. DOI 10.1007/s002200050191 | MR 1478531 | Zbl 0948.81025
[8] Figueiredo, D. G. de, Miyagaki, O. H., Ruf, B.: Elliptic equations in {$R^2$} with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3 (1995), 139-153. DOI 10.1007/BF01205003 | MR 1386960
[9] 'O, J. M. B. do: Semilinear Dirichlet problems for the {$N$}-Laplacian in {$\mathbb R^N$} with nonlinearities in the critical growth range. Differ. Integral Equ. 9 (1996), 967-979. MR 1392090
[10] 'O, J. M. B. do: $N$-Laplacian equations in $\mathbb R^N$ with critical growth. Abstr. Appl. Anal. 2 (1997), 301-315. DOI 10.1155/S1085337597000419 | MR 1704875
[11] Ó, J. M. B. do, Medeiros, E., Severo, U.: On a quasilinear nonhomogeneous elliptic equation with critical growth in $\Bbb R^N$. J. Differ. Equations 246 (2009), 1363-1386. DOI 10.1016/j.jde.2008.11.020 | MR 2488689
[12] Ó, J. M. B. do, Miyagaki, O. H., Soares, S. H. M.: Soliton solutions for quasilinear Schrödinger equations: the critical exponential case. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67 (2007), 3357-3372. DOI 10.1016/j.na.2006.10.018 | MR 2350892
[13] Ó, J. M. B. do, Severo, U.: Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. Partial Differ. Equ. 38 (2010), 275-315. DOI 10.1007/s00526-009-0286-6 | MR 2647122
[14] Edmunds, D. E., Ilyin, A. A.: Asymptotically sharp multiplicative inequalities. Bull. London Math. Soc. 27 (1995), 71-74. DOI 10.1112/blms/27.1.71 | MR 1331684 | Zbl 0840.46018
[15] Fang, X.-D., Szulkin, A.: Multiple solutions for a quasilinear Schrödinger equation. J. Differ. Equations 254 (2013), 2015-2032. DOI 10.1016/j.jde.2012.11.017 | MR 3003301 | Zbl 1263.35113
[16] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics Springer, Berlin (2001). MR 1814364
[17] Hasse, R. W.: A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z. Phys., B 37 (1980), 83-87. DOI 10.1007/BF01325508 | MR 0563644
[18] Kurihara, S.: Exact soliton solution for superfluid film dynamics. J. Phys. Soc. Japan 50 (1981), 3801-3805. DOI 10.1143/JPSJ.50.3801 | MR 0638806
[19] Laedke, E. W., Spatschek, K. H., Stenflo, L.: Evolution theorem for a class of perturbed envelope soliton solutions. J. Math. Phys. 24 (1983), 2764-2769. DOI 10.1063/1.525675 | MR 0727767 | Zbl 0548.35101
[20] Lieb, E. H., Loss, M.: Analysis. Graduate Studies in Mathematics 14 American Mathematical Society, Providence (2001). MR 1817225 | Zbl 0966.26002
[21] Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984), 223-283. MR 0778974 | Zbl 0704.49004
[22] Liu, J.-Q., Wang, Y.-Q., Wang, Z.-Q.: Soliton solutions for quasilinear Schrödinger equations. II. J. Differ. Equations 187 (2003), 473-493. DOI 10.1016/S0022-0396(02)00064-5 | MR 1949452 | Zbl 1229.35268
[23] Liu, J.-Q., Wang, Y.-Q., Wang, Z.-Q.: Solutions for quasilinear Schrödinger equations via the Nehari method. Commun. Partial Differ. Equations 29 879-901 (2004). DOI 10.1081/PDE-120037335 | MR 2059151 | Zbl 1140.35399
[24] Makhankov, V. G., Fedyanin, V. K.: Nonlinear effects in quasi-one-dimensional models and condensed matter theory. Phys. Rep. 104 1-86 (1984). DOI 10.1016/0370-1573(84)90106-6 | MR 0740342
[25] Quispel, G. R. W., Capel, H. W.: Equation of motion for the Heisenberg spin chain. Physica A 110 (1982), 41-80. DOI 10.1016/0378-4371(82)90104-2 | MR 0647411
[26] Ruiz, D., Siciliano, G.: Existence of ground states for a modified nonlinear Schrödinger equation. Nonlinearity 23 (2010), 1221-1233. DOI 10.1088/0951-7715/23/5/011 | MR 2630099 | Zbl 1189.35316
[27] Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111 (1964), 247-302. DOI 10.1007/BF02391014 | MR 0170096 | Zbl 0128.09101
[28] Severo, U.: Existence of weak solutions for quasilinear elliptic equations involving the {$p$}-Laplacian. Electron. J. Differ. Equ. (electronic only) 2008 (2008), 16 pages. MR 2392960 | Zbl 1173.35483
[29] Wang, Y., Yang, J., Zhang, Y.: Quasilinear elliptic equations involving the {$N$}-Laplacian with critical exponential growth in {$\Bbb R^N$}. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 6157-6169. DOI 10.1016/j.na.2009.06.006 | MR 2566522 | Zbl 1180.35262
Partner of
EuDML logo