Extended duality methods for nonlinear Helmholtz equations

We provide extensions of the dual variational method for the nonlinear Helmholtz equation from Evéquoz and Weth. In particular we prove the existence of dual ground state solutions in the Sobolev critical case, extend the dual method beyond the standard Stein Tomas and Kenig Ruiz Sogge range and generalize the method for sign changing nonlinearities.

Periodic Navier Solutions for the Plate Equation with Non-Monotone Nonlinearities: the Multidimensional Case

We discuss the solvability of the periodic Navier problem for the plate equation with forced vibrationsxtt(t, y)+Δ2x(t, y)+l(t, y, x(t, y)) = 0 in higher dimensions with side lengths being irrational numbers and the nonlinearity being superlinear. We also derive a new dual variational method.

On a fourth order Dirichlet Problem

We consider by a dual variational method the existence of solutions to certain fourth order Dirichlet problems with nonlinearities corresponding to the derivatives of a sum of a convex and a concave function. The growth conditions are imposed only on the convex part.

On the Dirichlet problem with nonconvex nonlinearity

We provide existence results for 2 m order Dirichlet problems with nonconvex nonlinearity which satisfies general local growth conditions. In doing so we construct a dual variational method. Problem considered relates to the problem of nonlinear eigenvalue.

A note on a dirichlet problem with concave-convex nonlinearity

We prove an existence principle that would apply for elliptic problems with nonlinearity separating into a difference of derivatives of two convex functions in the case when the growth conditions are imposed only on the minuend term. We present abstract result and its application. We modify the so called dual variational method.

Dual Variational Method for a Fourth Order Dirichlet Problem

We obtain the existence and stability results for a fourth order Dirichlet problems with nonlinearity being convex in a certain interval. A dual variational method is introduced, which relies on investigating the primal and dual action functionals on certain subsets of their domains. The dependence on a functional parameter for a fourth order Dirichlet problem is considered as a consequence of stability.

The Existence of Solutions for Nonlinear Operator Equations

We provide the existence results for a nonlinear operator equation Λ*Φ′ (Λ𝑥) = 𝐹′(𝑥), in case 𝐹 – Φ is not necessarily convex. We introduce the dual variational method which is based on finding global minima of primal and dual action functionals on certain nonlinear subsets of their domains and on investigating relations between the minima obtained. The solution is a limit of a minimizng sequence whose existence and convergence are proved. The application for the non-convex Dirichlet problem with P.D.E. is given.

On the Nonlinear Dirichlet Problem with P(x)-Laplacian

Using a dual variational method which we develop, we show the existence and stability of solutions for a family of Dirichlet problems k = 0, 1,… in a bounded domain in ℝN and with the nonlinearity satisfying some general growth conditions. The assumptions put on v are satisfied by p(x)-Laplacian operators.

A new variational method for thep(x)-Laplacian equation

Using a dual variational method we shall show the existence of solutions to the Dirichlet problemwithout assuming Palais-Smale condition.