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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.

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