Sentence examples for using the variational method from inspiring English sources

As we will see later, the competing effect of (lambda W x) vert u vert ^{{p - 2}}u + vert u vert ^{4}u) ((2 < p le 4)) and the lack of compactness of the embedding prevent us from using the variational method in a standard way.

By using the variational method, some existence results are obtained.

They obtained the existence of solutions for problems by using the variational method.

By using the variational method, the existence and multiplicity of positive solutions are obtained.

We obtain this result by using the variational method, critical point theory for indefinite functional.

We obtain this result by using the variational method and critical point theory for indefinite functional.

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.