Sentence examples for by using variational methods from inspiring English sources

We denote and we study (1.1) by using variational methods.

By using variational methods, we study the multiplicity of solutions for Kirchhoff type problems, in ;, on.

The positive solutions of such a problem are obtained by using variational methods [3 5].

By using variational methods, the existence of at least two positive solutions was obtained.

Moreover, the ground state solution and infinitely many nontrivial solutions are obtained by using variational methods.

By using variational methods, the authors obtained that system (1.5) has at least three weak solutions.

has been studied by using variational method.

We prove the existence of periodic solutions under a generalized Lazer-Leach type condition by using variational method.

They proved that IBVP (1.4) has at least one solution and infinitely many solutions by using variational method and critical point theorem.

Motivated by the above facts, in this paper, our aim is to study the variational structure of IBVP (1.1) in an appropriate space of functions and obtain the existence and multiplicity of solutions for IBVP (1.1) by using variational method.

In this paper, by using variational method, we have dealt with a kind of ordinary p-Laplacian system and obtained the existence of fast homoclinic solutions to the above system.