Sentence examples for by using variational arguments from inspiring English sources

By using variational arguments, many authors investigated nonlinear biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions and got interesting results (see [4 10]).

They obtained infinitely many nontrivial solutions of (1.3) under the assumptions ( H ˜ 1 ) to ( H ˜ 4 ) by using variational arguments and a fountain theorem.

Using variational arguments, Carvalho-Goncalves-Silva [23] established the existence of multiple solutions for quasilinear elliptic problems driven by the Φ-Laplacian operator.

We use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem.

has been studied by using variational method.

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.