×

Multi-peak bound states for nonlinear Schrödinger equations. (English) Zbl 0901.35023

The authors consider the equation \[ \varepsilon^2 \Delta u-V(x)u +f(u)=0 \] in a smooth, possibly unbounded domain \(\Omega \subset \mathbb{R}^n\). Here \(\varepsilon\) is a small parameter, \(f\) a superlinear function, and the potential \(V\) is positive, locally Hölder continuous, and bounded away from zero. This equation arises from studying standing wave solutions of nonlinear Schrödinger equations. Assume that there are \(K\) disjoint bounded domains \(\Lambda_i \Subset \Omega\) such that \(\inf_{\Lambda_i} V<\inf_{\partial \Lambda_i} V\). If the nonlinearity \(f\) is appropriate, then for all sufficiently small \(\varepsilon>0\) the authors prove the existence of a positive solution \(u_\varepsilon \in H^1_0 (\Omega)\) possessing exactly \(K\) local maxima \(x_{\varepsilon,i} \in \Lambda_i\). Moreover, for certain positive constants \(\alpha,\beta\), \[ u_\varepsilon (x)\leq \alpha \exp \left(-{\beta \over \varepsilon} | x-x_{\varepsilon,i} |\right) \text{ in } \Omega \setminus \bigcup_{j\neq i} \Lambda_j \] and \(V(x_{\varepsilon,i}) \to \inf_{\Lambda_i} V\).

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Chen, C.C.; Lin, C.S., Uniqueness of the ground state solutions of δu + f(u) = 0 in \(R\)N, N ≥ 3, Comm. in P.D.E., Vol. 8-9, 16, 1549-1572, (1991) · Zbl 0753.35034
[2] Coti Zelati, V.; Rabinowitz, P., Homoclinic type solutions for semilinear elliptic PDE on \(R\)^{N}, Comm. pure and applied math, Vol. XLV, 1217-1269, (1992) · Zbl 0785.35029
[3] Del Pino, M.; Felmer, P., Local mountain passes for semilinear elliptic problems in unbounded domains, Calculus of variations and PDE, Vol. 4, 121-137, (1996) · Zbl 0844.35032
[4] Esteban, M.J.; Lions, P.L., Existence and non-existence results for semilinear problems in unbounded domains, (), 1-14 · Zbl 0506.35035
[5] Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, Journal of functional analysis, Vol. 69, 397-408, (1986) · Zbl 0613.35076
[6] Kwong, M.K.; Zhang, L., Uniqueness of positive solutions of δu + f(u) = 0 in an annulus, Differential and integral equations, Vol. 4, 583-599, (1991) · Zbl 0724.34023
[7] Lions, P.L., The concentration-compactness principle in the calculus of variations, The locally compact case. part II analyse nonlin., Vol. 1, 223-283, (1984) · Zbl 0704.49004
[8] Oh, Y.J., Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class (V)a, Comm. partial diff., eq., Vol. 13, 1499-1519, (1988) · Zbl 0702.35228
[9] Oh, Y.J., Corrections to existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class (V)a, Comm. partial diff. eq., Vol. 14, 833-834, (1989) · Zbl 0714.35078
[10] Oh, Y.J., On positive multi-lump bound states nonlinear Schrödinger equations under multiple well potential, Comm. math. phys., Vol. 131, 223-253, (1990) · Zbl 0753.35097
[11] Rabinowitz, P., On a class of nonlinear Schrödinger equations, Z. angew math phys., Vol. 43, 270-291, (1992) · Zbl 0763.35087
[12] Spradlin, G., ()
[13] Thandi, N., ()
[14] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. math. phys., Vol. 153, No 2, 229-244, (1993) · Zbl 0795.35118
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.