Multiplicity results for superquadratic Dirichlet boundary value problems in \(\mathbb R^2\)
The authors consider with the boundary value problem of the form \[ z'=J\nabla F(t, z), \quad z=(x,y)\in \mathbb{R}^2,\;t\in [0,\pi], \qquad x(0)=0=x(\pi), \] where \(J\) is the standard symplectic matrix and \(F\in C^2([0,\pi]\times \mathbb{R}^2; R^+)\) satisfies \(\nabla F(t, 0)=0\) for \(t\in [0,\pi]\). They prove the existence of infinitely many solutions in the case when \(F\) has a superquadratic growth. The main result is based on a global bifurcation theorem obtained by \textit{A. Capietto} and \textit{W. Walter} [J. Differ. Equations 227, No. 2, 692--713 (2006; Zbl 1106.34006)].
scientific article; zbMATH DE number 5310040