We consider the combined effect of concave-convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When α + β < 4, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais-Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.
We construct the Hirota bilinear form of the nonlocal Boussinesq (nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nlBq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nlBq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form. The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.
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