Multiple solutions for asymptotically linear Duffing equations with Neumann boundary value conditions. II
The authors consider multiple solutions of the Duffing equation \[ \quad x''(t)+ f(t,x(t)) = 0 \tag{1.1} \] subject to \[ \quad x'(1)=0=x'(0),\tag{1.2} \] where \(f:[0,1]\times \mathbb{R}\to \mathbb{R}\) is a continuously differentiable function. They define \[ f_1(t,x)=f(t,x)-a_0(t) x, \] where \(a_0\in H_0(1,(0,1),\frac{\pi}{2},\frac{\pi}{2})\), i.e., \(x''(t)+ a_0(t) x = 0\) and (1.2) has a nontrivial solution \(x=x_0(t)>0\;\forall t\in [0,1]\). They assume that \(f\) satisfies the conditions \((A_1)\) \(f\) is odd with respect to \(x\), i.e., \(-f(t,x)=f(t,-x)\); \((A_2)\) \(f(t,x)=b(t)x+o(|x|)\) as \(|x|\to 0,\) where \(b\in L^\infty[0,1]\), \(i(b)\geq 2, \nu(b)=0\). Here, for any \(b\in L^\infty[0,1],\) \(i(b)\) and \(\nu(b)\) denote its index and the nullity of the associated linear Duffing equations. The main results of this paper are the following theorems. Theorem 1.1. Let \((A_1)\) and \((A_2)\) hold. Assume \((H_1)\) that there exists a constant \(r>0\) and \(\lambda\in L^\infty[0,1]\) such that \(a_0(t)\leq \frac{f(t,x)}{x}<\lambda(t)\) as \(|x|\geq r\), with \(i(\lambda)=1\), \(\nu(\lambda)=0\); and \((H_2)\) \[ \int_0^1 f_1(t,+\infty) x_0\, dt>0>\int_0^1 f_1(t,-\infty) x_0\, dt, \] where \(f_1(t,+\infty)=\liminf_{x\to+\infty}f_1(t,x)\), \(f_1(t,-\infty)=\limsup_{x\to-\infty}f_1(t,x)\). Then system (1.1), (1.2) has at least \(i(b)-1\) pairs of solutions. Theorem 1.2. Assume \((A_1)\) and \((A_2)\) hold. Assume \((H_1')\) that there exists a constant \(r>0\) such that \( \frac{f(t,x)}{x}\leq a_0(t)\) as \(|x|\geq r\); and \((H_2')\) \[ \int_0^1 f_1(t,+\infty) x_0\, dt<0<\int_0^1 f_1(t,-\infty) x_0\, dt, \] where \(f_1(t,+\infty)=\limsup_{x\to+\infty}f_1(t,x)\), \(f_1(t,-\infty)=\liminf_{x\to-\infty}f_1(t,x)\). Then system (1.1), (1.2) has at least \(i(b)-1\) pairs of solutions. The proofs are based on index theory and Lusternik-Schnirelmann theory.
asymptotically linear Duffing equations with resonance
scientific article; zbMATH DE number 6156264