A multiplicity result for a class of superquadratic Hamiltonian systems. (English) Zbl 1034.35046

Let \(\Omega\subset\mathbb{R}^n\) be a bounded smooth domain. For a small real parameter \(\lambda>0\) the authors study the system \[ -\Delta v=\lambda f(u),\;-\Delta u=g(v)\quad \text{in }\Omega,\quad u= v=0\text{ on }\partial\Omega. \] Since the function \(g\) is assumed to be continuous and strictly increasing and to satisfy \(g(\mathbb{R})=\mathbb{R}\), the inverse function \(g^{-1}:\mathbb{R} \to\mathbb{R}\) exists; and the system may be rewritten as quasilinear fourth order equation \[ \Delta\bigl(g^{-1} (\Delta u)\bigr)=\lambda f(u) \] under Navier boundary conditions: \[ u=\Delta u=0\text{ on }\partial\Omega. \] For the latter problem, the existence of two nontrivial solutions is shown under superlinearity and subcriticality assumptions on the nonlinearities \(f\) and \(g\). One solution is constructed by means of the classical mountain pass lemma, and the second by local minimization.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J35 Variational methods for higher-order elliptic equations
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