Sentence examples for use the variational methods from inspiring English sources

We will use the variational methods to solve the problem of (1.1).

In order to treat the singularity properly, we use the variational method proposed in Greenleaf et al. (2007) [1] and prove the possibility of the perfect cloaking.

The competing effect of the non-local term with the critical nonlinearity and the lack of compactness of the embedding of H 1 into the space L p prevent us from using the variational methods in a standard way.

In this section, we shall state and prove our main results by using the variational methods.

By using the variational methods to study problem (1), the growth conditions of (W t,x)) are needed.

Second, in Section 3, we shall state and prove our main results by using the variational methods.

The existence and multiplicity results of nontrivial solutions for the system (1.3) were obtained by using the variational methods and the Nehari manifold.

In this paper, by using the variational methods, the existence of nontrivial solutions and the concentration phenomena of the solutions to equation (1.1) were established.

In this paper, by using the variational methods and some weaker conditions, the existence of Nehari-type solutions to equation (1.1) is established.

There have been many papers concerning this topic by using the variational methods since the remarkable results by Ambrosetti and Rabinowitz [1].

Zhang [36] obtained infinitely many solutions for a class of general second-order damped vibration systems by using the variational methods.