Multiplicity results for systems of superlinear second order equations
The authors study the existence of solutions to the Dirichlet problem \[ x''+\nabla F(t,x)=q(t,x,x'),\quad x(0)=0,\quad x(\pi)=0, \] where \(F:[0,\pi]\times \mathbb R^m\to\mathbb R\) is a \({\mathcal C}^1\)-function and \(q:[0,\pi]\times \mathbb R^{2m}\to\mathbb R^m\) is continuous. The main result proves that for any \((n_1,\ldots,n_m)\in{\mathbb N}^m\) there exist \(2^m\) solutions so that the \(i\)th component of each of them has exactly \(n_i\) zeros. These are large solutions which are obtained assuming a superlinear behavior of the nonlinearity at infinity \[ \lim_{|x_i|\to\infty}\frac{1}{x_i} \frac{\partial F}{\partial x_i}(t,x)= +\infty,\quad |q_i(t,x,y)|\leq A_1|x_i|+ A_2|y_i|+ A_3, \] together with \[ x_i\frac{\partial F}{\partial x_i}(t,x)\geq K_0\quad\text{and}\quad\left |\frac{\partial F}{\partial t}(t,x)\right |\leq \alpha(t)F(t,x)+C. \] As corollaries, the authors obtain results on the Liénard equation and on weakly coupled systems. The proof uses a topological degree approach. The system is homotoped to uncoupled equations to which a time-map technique can be applied. An important step is based on the ``elastic property'', i.e. a control of the maximum of the norm of a solution in phase planes from an estimate on its minimum.
scientific article; zbMATH DE number 1523554
scientific article