Sentence examples for used variational methods and from inspiring English sources

In 2011, Zhang and Wang [15] used variational methods and Morse theory to study the multiplicity of periodic solutions for (P) with double resonance between two consecutive eigenvalues at infinity.

By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.

By using variational methods and elementary analysis, the conditions for the existence of positive and negative solutions to a nonlinear system are given.

By using variational methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation.

In [6], Sun and Wu investigated the existence and the non-existence of nontrivial solution of problem (1.1) by using variational methods and explored the concentration of solution.

By using variational methods and critical point theory, we obtain that such a system possesses at least one, two periodic solutions generated by impulses under different conditions, respectively.

We study the existence of distinct pairs of nontrivial solutions for impulsive differential equations with Dirichlet boundary conditions by using variational methods and critical point theory.

By using variational methods and Morse theory, we study the multiplicity of the periodic solutions for a class of difference equations with double resonance at infinity.

In this article, under the guidance of [8], we consider multiple solutions of problem (1) with the asymmetric nonlinearity by using variational methods and Morse theory.

In [42] using variational methods and critical point theory, the multiplicity results of solutions for a class of impulsive fractional differential systems was established.

Recently, many researchers [11 16] have studied the existence and multiplicity of solutions of impulsive problems by using variational methods and critical point theory.