On a semilinear Schrödinger equation with critical Sobolev exponent

Chabrowski, Jan; Szulkin, Andrzej

doi:10.1090/S0002-9939-01-06143-3

On a semilinear Schrödinger equation with critical Sobolev exponent
HTML articles powered by AMS MathViewer

by Jan Chabrowski and Andrzej Szulkin PDF

Proc. Amer. Math. Soc. 130 (2002), 85-93 Request permission

Abstract:

We consider the semilinear Schrödinger equation $-\Delta u+V(x)u = K(x)|u|^{2^{*}-2}u+g(x,u)$, $u\in W^{1,2}(\mathbf {R}^{N})$, where $N\ge 4$, $V,K,g$ are periodic in $x_{j}$ for $1\le j\le N$, $K>0$, $g$ is of subcritical growth and 0 is in a gap of the spectrum of $-\Delta +V$. We show that under suitable hypotheses this equation has a solution $u\ne 0$. In particular, such a solution exists if $K\equiv 1$ and $g\equiv 0$.

References

Stanley Alama and Yan Yan Li, On “multibump” bound states for certain semilinear elliptic equations, Indiana Univ. Math. J. 41 (1992), no. 4, 983–1026. MR 1206338, DOI 10.1512/iumj.1992.41.41052
Wolfgang Arendt, Gaussian estimates and interpolation of the spectrum in $L^p$, Differential Integral Equations 7 (1994), no. 5-6, 1153–1168. MR 1269649
Thomas Bartsch and Yanheng Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann. 313 (1999), no. 1, 15–37. MR 1666801, DOI 10.1007/s002080050248
Vieri Benci and Giovanna Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{(N+2)/(N-2)}$ in $\textbf {R}^N$, J. Funct. Anal. 88 (1990), no. 1, 90–117. MR 1033915, DOI 10.1016/0022-1236(90)90120-A
B. Buffoni, L. Jeanjean, and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993), no. 1, 179–186. MR 1145940, DOI 10.1090/S0002-9939-1993-1145940-X
Jan Chabrowski and Jianfu Yang, On Schrödinger equation with periodic potential and critical Sobolev exponent, Topol. Methods Nonlinear Anal. 12 (1998), no. 2, 245–261. MR 1701262, DOI 10.12775/TMNA.1998.040
José F. Escobar, Positive solutions for some semilinear elliptic equations with critical Sobolev exponents, Comm. Pure Appl. Math. 40 (1987), no. 5, 623–657. MR 896771, DOI 10.1002/cpa.3160400507
Rainer Hempel and Jürgen Voigt, The spectrum of a Schrödinger operator in $L_p(\textbf {R}^\nu )$ is $p$-independent, Comm. Math. Phys. 104 (1986), no. 2, 243–250. MR 836002
L. Jeanjean, Solutions in spectral gaps for a nonlinear equation of Schrödinger type, J. Differential Equations 112 (1994), no. 1, 53–80. MR 1287552, DOI 10.1006/jdeq.1994.1095
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Diff. Eq. 3 (1998), 441–472.
Peter Kuchment, Floquet theory for partial differential equations, Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. MR 1232660, DOI 10.1007/978-3-0348-8573-7
A. A. Pankov and K. Pflüger, On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. 33 (1998), no. 6, 593–609. MR 1635911, DOI 10.1016/S0362-546X(97)00689-5
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
C. A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc. Simon Stevin suppl. (1995), 59. MR 1361485
A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, Preprint.
C. Troestler, Bifurcation into spectral gaps for a noncompat semilinear Schrödinger equation with nonconvex potential, Preprint.
C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations 21 (1996), no. 9-10, 1431–1449. MR 1410836, DOI 10.1080/03605309608821233
Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1