Periodic problems with double resonance
The authors study the second-order periodic problem \[ -u''=f(t,u(t)),\quad u(0)=u(b),\;u'(0)=u'(b),\tag{1} \] where \(f\) is \(C^1\) with respect to the second variable; \(f(t,0)\equiv0\); and, at \(0\) and at infinity, the ratio \(\frac{f(t,x)}{x}\) tends to distinct non-zero eigenvalues of the linear \(b\)-periodic problem for \(u''+\lambda u=0\). They show that, under certain supplementary growth assumptions of technical character, the problem has at least six nontrivial solutions. This is done by applying critical point theory to the functional \(\varphi\) whose Euler equation is the equation in (1). Main ingredients of the proof are minimization, the mountain pass theorem and Morse inequalities. First, two constant sign solutions are found by minimization. The mountain pass theorem of a truncated functional yields two further solutions, respectively, above the positive solution and below the negative solution already found. Comparing the critical groups of these critical points with the critical groups at infinity another critical point appears different from the so far found ones and nontrivial. Finally, applying the Morse inequalities, it is shown that one more nontrivial solution must exist. Some mistakes in the paper easy to correct do not harm the reading.
scientific article; zbMATH DE number 6062889
scientific article