Sentence examples for by the variational methods from inspiring English sources

The existence of solutions of impulsive problems was also treated by the variational methods and critical point theorems (see [20 22]).

However, up to now, there are few results on the solutions to fractional boundary value problems that are established by the variational methods; see, for example, [7 18].

For the case (s=1) and (lambda =1), Ding and Tang in [17] obtained the existence of positive solutions for problem (1) by the variational methods and some analysis techniques with f satisfying the (AR) condition.

Since the functionals E ( u ) = ∫ − ∞ + ∞ [ 1 2 u L ( u ) − F ( u ) ] d x. and Q ( u ) = 1 2 ∫ − ∞ + ∞ u 2 d x. are two conserved quantities with (1.1), for studying the existence of solitary wave solutions to (1.1), by the variational methods, the solitary wave solutions to equation (1.1) will be founded as minimizers of.

Recently, by the variational methods, Ma and Wang etc. studied (1.4) and the following fourth-order equation of Kirchhoff type: { △ 2 u − M ( ∫ Ω | ∇ u | 2 d x ) ∇ u = f ( x, u ) in  Ω, u = ∇ u = 0 on  ∂ Ω. and obtained the existence and multiplicity of solutions; see [4] [6].

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

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.

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.