Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials

Ba, Na; Deng, Yinbin; Peng, Shuangjie

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1205-1226

In this paper, we will study the existence and qualitative property of standing waves $\psi (x,t)=e^{-\frac{iEt}{ϵ}}u\left(x\right)$ for the nonlinear Schrödinger equation $iϵ\frac{\partial \psi }{\partial t}+\frac{{ϵ}^2}{2m}{\Delta }_x\psi -(V\left(x\right)+E)\psi +K\left(x\right){\left|\psi \right|}^{p-1}\psi =0$, $(t,x)\in {ℝ}_{+}\times {ℝ}^N$. Let $G\left(x\right)={\left[V\right(x\left)\right]}^{\frac{p+1}{p-1}-\frac{N}{2}}{\left[K\right(x\left)\right]}^{-\frac{2}{p-1}}$ and suppose that $G\left(x\right)$ has k local minimum points. Then, for any $l\in \{1,\cdots ,k\}$, we prove the existence of the standing waves in $H^1\left({ℝ}^N\right)$ having exactly l local maximum points which concentrate near l local minimum points of $G\left(x\right)$ respectively as $ϵ\to 0$. The potentials $V\left(x\right)$ and $K\left(x\right)$ are allowed to be either compactly supported or unbounded at infinity. Therefore, we give a positive answer to a problem proposed by Ambrosetti and Malchiodi (2007) [2].

Classification : 35J20, 35J60
Keywords: Bound state, Multi-peak solutions, Nonlinear Schrödinger equation, Potentials compactly supported or unbounded at infinity

[1] A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. 7 (2005), 117-144 | EuDML | Zbl

[2] A. Ambrosetti, A. Malchiodi, Concentration phenomena for NLS: recent results and new perspectives, Perspectives in Nonlinear Partial Differential Equations, Contemp. Math. vol. 446, Amer. Math. Soc., Providence, RI (2007), 19-30 | Zbl

[3] A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006), 317-348 | Zbl

[4] A. Ambrosetti, A. Malchiodi, S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal. 159 (2001), 253-271 | Zbl

[5] A. Ambrosetti, Z.Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Diff. Int. Equats. 18 (2005), 1321-1332 | Zbl

[6] T. Bartsch, S. Peng, Symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres, Z. Angew. Math. Phys. 58 (2007), 778-804 | Zbl

[7] F.A. Berezin, M.A. Shubin, The Schrödinger Equation, Kluwer Acad. Publ. (1991) | Zbl

[8] J. Byeon, Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 165 (2002), 295-316 | Zbl

[9] J. Byeon, Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II, Calc. Var. Partial Differential Equations 18 (2003), 207-219 | Zbl

[10] J. Byeon, Z.Q. Wang, Standing wave for nonlinear Schrödinger equations with singular potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 943-958 | EuDML | Zbl | Numdam

[11] J. Byeon, Z.Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials, J. Eur. Math. Soc. 8 (2006), 217-228 | EuDML | Zbl

[12] D. Cao, H.P. Heinz, Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations, Math. Z. 243 (2003), 599-642 | Zbl

[13] D. Cao, E.S. Noussair, Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, J. Diff. Equats. 203 (2004), 292-312 | Zbl

[14] D. Cao, E.S. Noussair, S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 360 (2008), 3813-3837 | Zbl

[15] D. Cao, S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency, Math. Ann. 336 (2006), 925-948 | Zbl

[16] D. Cao, S. Peng, Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations 34 (2009), 1566-1591 | Zbl

[17] V. Coti Zelati, P. Rabinowitz, Homoclinic type solutions for semilinear elliptic PDE on ${ℝ}^N$, Comm. Pure Appl. Math. 45 (1992), 1217-1269 | Zbl

[18] E.N. Dancer, S. Yan, Singularly perturbed elliptic problems in exterior domains, Diff. Int. Equats. 13 (2000), 747-777 | Zbl

[19] M. Del Pino, P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 127-149 | EuDML | Zbl | Numdam

[20] M. Del Pino, P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal. 149 (1997), 245-265 | Zbl

[21] M. Del Pino, P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121-137 | Zbl

[22] M. Fei, H. Yin, Existence and concentration of bound states of nonlinear Schrödinger equations with compactly supported and competing potentials, Pacific J. Math. 244 (2010), 261-296 | Zbl

[23] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397-408 | Zbl

[24] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${ℝ}^N$, Adv. Math. Suppl. Stud. 7A (1981), 369-402

[25] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. vol. 224, Springer, Berlin, Heidelberg, New York (1998) | Zbl

[26] M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 261-280 | EuDML | Zbl | Numdam

[27] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations 21 (1996), 787-820 | Zbl

[28] M.K. Kwong, Uniqueness of positive solutions of $\Delta u-\lambda u+u^p=0$ in ${ℝ}^N$, Arch. Ration. Mech. Anal. 105 (1989), 243-266 | Zbl

[29] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223-283 | EuDML | Zbl | Numdam

[30] E.S. Noussair, S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc. 62 (2000), 213-227 | Zbl

[31] Y.G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class ${\left(V\right)}_{\alpha }$, Comm. Partial Differential Equations 13 (1988), 1499-1519 | Zbl

[32] Y.G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys. 131 (1990), 223-253 | Zbl

[33] P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270-291 | Zbl

[34] Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency, Calc. Var. Partial Differential Equations 29 (2007), 365-395 | Zbl

[35] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (1993), 229-244 | Zbl

[36] X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997), 633-655 | Zbl

[37] Z.Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, J. Diff. Equats. 159 (1999), 102-137 | Zbl