On existence and stability of solutions for higher order semilinear Dirichlet problems (Q732844)

The author considers the fourth order ordinary differential equation \[ \beta_kx^{iv}+\gamma_kx''+\delta_kx = F^k_x(t,x) \] with zero Dirichlet boundary data, where \(\beta_k>0\), \(F^k_x(t,0)\neq 0\) and \(x\mapsto F^k(t,x)\) is convex for \(x\) in a certain interval \([-d_0,d_0]\). It is shown that, under appropriate conditions, this equation has a solution \(x_k\). The proof is achieved by minimization of a dual functional obtained by means of the Fenchel conjugate of \(F^k\). The following stability result is also shown. Suppose \(k=1,2,\ldots\) and \(\beta_k,\gamma_k,\delta_k,F^k\) converge as \(k\to\infty\) (\(\beta_k>0\) decreasing). Then, passing to a subsequence, \(x_k\to\bar x\) and \(\bar x\) is a solution of a limiting problem. If \(\beta_k\to \beta_0>0\), the limiting problem is of fourth order, and if \(\beta_k\to 0\), it is of second order.