×

Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. (English) Zbl 1142.35082

The authors consider the nonlinear Schrödinger problem \(-\varepsilon ^{2}\Delta u+V(x)u=K(x)u^{p}\), \(u>0\), posed in \(\mathbb{R}^{n}\), where \( \varepsilon >0\), \(1<p<(n+2)/(n-2)\). Here \(V\) and \(K\) are smooth bounded and positive potentials. The authors assume that \(A_{0}/(1+| x| ^{\alpha })\leq V(x)\leq A_{1}\) and \(0\leq K(x)\leq k/(1+| x| ^{\beta })\) on \(\mathbb{R}^{n}\) with some hypotheses on \(\alpha \) and \(\beta \). They are looking for generalized solutions of this problem which belong to \(W^{1,2}(\mathbb{R}^{n})\). They introduce the function \(Q(x)=V^{\theta }(x)K^{2/(p-1)}(x)\), with \(\theta =(p+1)/(p-1)-n/2\).
In the main result, the authors prove that if \(x_{0}\) is an isolated stable stationary point of \(Q\), for \(\varepsilon \) small enough, then, the Schrödinger problem has a finite energy solution which concentrates at \(x_{0}\). For the proof, the authors quote the previous result [see A. Ambrosetti, V. Felli and A. Malchiodi, J. Eur. Math. Soc. 7, No. 1, 117–144 (2005; Zbl 1064.35175)], where the existence was proved for the problem, assuming other hypotheses. The authors introduce some change of variables and a truncated nonlinearity which lead to an energy functional \( I_{\varepsilon }\) defined on a Hilbert space \(E\). The authors then use functional analysis arguments and intricate computations.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 1064.35175
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A. Ambrosetti and M. Badiale,Homoclinics: Poincaré-Melnikov type results via a variational approach, Ann. Inst. H. Poincaré Anal. Non Linéaire15 (1998), 233–252. · Zbl 1004.37043
[2] A. Ambrosetti and A. Malchiodi,Perturbation Methods and Semilinear Elliptic Problems on \(\mathbb{R}\) n , Progress in mathematics, Vol. 240, Birkhäuser, Basel, 2005. · Zbl 1115.35004
[3] A. Ambrosetti, V. Felli and A. Malchiodi,Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. European Math. Soc.7 (2005), 117–144. · Zbl 1064.35175
[4] A. Ambrosetti, A. Malchiodi and W.-M. Ni,Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part I, Comm. Math. Phys.235 (2003), 427–466. · Zbl 1072.35019
[5] A. Ambrosetti, A. Malchiodi and S. Secchi,Multiplicity results for some nonlinear singularly perturbed elliptic problems on R n , Arch. Rational Mech. Anal.159 (2001), 253–271. · Zbl 1040.35107
[6] A. Bahri and P. L. Lions,On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire14 (1997), 365–413. · Zbl 0883.35045
[7] J. Chabrowski,Variational Methods for Potential Operator Equations. With Applications to Nonlinear Elliptic Equations, Walter de Gruyter & Co., Berlin, 1997. · Zbl 1157.35338
[8] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[9] N. K. Kwong,Uniqueness of positive solutions of {\(\Delta\)}u+u p =0 in \(\mathbb{R}\) n , Arch. Rational Mech. Anal.105 (1989), 243–266. · Zbl 0676.35032
[10] N. N. Lebedev,Special Functions and their Applications, Prentice Hall, Inc., Englewood Cliffs, N.J., 1965. · Zbl 0131.07002
[11] E. S. Noussair and C. A. Swanson,Decaying solutions of semilinear elliptic equations in \(\mathbb{R}\) n , SIAM J. Math. Anal.20 (1989), 1336–1343. · Zbl 0696.35051
[12] Y-G. Oh,On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials, Comm. Math. Phys.131 (1990), 223–253. · Zbl 0753.35097
[13] P. H. Rabinowitz,On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys.43 (1992), 270–291. · Zbl 0763.35087
[14] M. Schneider,Entire solutions of semilinear elliptic problems with indefinite nonlinearities, PhD Thesis, Universität Mainz, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.