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Multi-bump type nodal solutions having a prescribed number of nodal domains. II. (English) Zbl 1330.35154

Summary: This paper is a sequel to Part I [ Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, 597–608 (2005; Zbl 1130.35054)] in which we studied nodal property of multi-bump type sign-changing solutions constructed by V. Coti-Zelati and P. H. Rabinowitz [Commun. Pure Appl. Math. 45, 1217–1269 (1992; Zbl 0785.35029)]. In this paper we remove a technical condition that the nonlinearity is odd, which was used in [Coti Zelati te al. (loc. cit.); Liu and Wang, Part I ], for constructing multi-bump type nodal solutions having a prescribed number of nodal domains.

MSC:

35J60 Nonlinear elliptic equations
35B10 Periodic solutions to PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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References:

[1] Bartsch, T.; Liu, Z.L.; Weth, T., Sign changing solutions of superlinear Schrödinger equations, Comm. partial differential equations, 29, 25-42, (2004) · Zbl 1140.35410
[2] Coti Zelati, V.; Rabinowitz, P.H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. amer. math. soc., 4, 623-627, (1991) · Zbl 0744.34045
[3] Coti Zelati, V.; Rabinowitz, P.H., Homoclinic type solutions for a semilinear elliptic PDE on \(\mathbf{R}^n\), Comm. pure appl. math., 45, 1217-1269, (1992) · Zbl 0785.35029
[4] Liu, Z.L.; Sun, J.X., Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. differential equations, 172, 257-299, (2001) · Zbl 0995.58006
[5] Liu, Z.L.; Wang, Z.-Q., Multi-bump type nodal solutions having a prescribed number of nodal domains: I, Ann. I. H. Poincaré - AN, 22, 597-608, (2005) · Zbl 1130.35054
[6] van Heerden, F., Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity, Calc. var. partial differential equations, 20, 431-455, (2004) · Zbl 1142.35422
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