By means of nonlinear analysis theory and methods, many existence and multiplicity results of solutions or positive solutions have been obtained, see [1 29] and the references therein.
With the aids of the variational methods, the existence and multiplicity of homoclinic orbits for (1.2) have been extensively investigated in many recent papers; see [2 24].
In the past 20 years, with the aid of the variational methods, the existence and multiplicity of homoclinic solutions for system (1.1) have been extensively investigated by many authors; see [3, 4, 6 9, 11 30] and references therein.
With the variational methods, the existence and multiplicity of homoclinic orbits of problem (1) have been obtained in many papers (see [1 16]), mainly in the case that W satisfies some global assumptions for all t and u.
Using the Nehari manifold and variational methods, the existence and multiplicity of positive solutions for the multi-singular semilinear elliptic system with critical growth terms in bounded domains are investigated.
(1.2) With the aid of variational methods, the existence and multiplicity of homoclinic orbits for system (1.2) or its special form has been extensively investigated in many recent papers; see [1 16].
By using the variational method, the existence and multiplicity of positive solutions are obtained.
Furthermore, in [8], by using the modification function technique and the Leray-Schauder degree method, some existence and multiplicity results for sign-changing solutions of certain three-point boundary value problems were obtained.
By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
For example, Liu et al. in [35], employing variational methods, studied the existence and multiplicity of nontrivial solutions for fourth-order elliptic equations.
