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Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition. (English) Zbl 1265.34058

The paper investigates the periodic boundary value problem \[ Ju'=F(t,u),\qquad u(0)=u(T),\tag{1} \] where \(F:[0,T]\times \mathbb R^2\to \mathbb R^2\) has the form \[ F(t,u)=\gamma (t,u)\nabla H_0(u)+(1-\gamma (t,u))\nabla H_1(u)+r(t,u). \] Here, \(J\) is the standard symplectic matrix, \(\gamma (t,u)\) and \(r(t,u)\) are \(L^2\)-Carathéodory functions such that \(0\leq \gamma (t,u)\leq 1\) and \[ \lim _{| u| \to +\infty }\frac {r(t,u)}{| u| }=0. \] It is assumed that \(H_0\in \mathcal P^\ast \setminus \mathcal P\) and \(H_1\in \mathcal P\), with \(H_0(u)\leq H_1(u)\) for every \(u\in \mathbb R^2\), where \(\mathcal P^\ast\) is the set of nonnegative \(C^1\)-functions \(H\:\mathbb R^2\to \mathbb R\) with locally Lipschitz continuous gradient, which are positively homogeneous of degree 2 and \(\mathcal P\) are those functions from \(\mathcal P^\ast\) which are positive. In particular (1) contains the case of “double resonance” involving the first eigenvalue of the linear problem.
Let \(\varphi , \omega \) satisfy \(J\varphi '=\nabla H_1(\varphi )\), \(H_1(\varphi (t))=1/2\), \(\omega '=2\alpha (t)H_0(\varphi (t+\omega ))+\beta (t)-1\), where \(\alpha ,\beta \in L^2(0,T)\) are such that, for almost every \(t\in [0,T]\), \(\alpha (t)\geq 0,\beta (t)\geq 0,\alpha (t)+\beta (t)\leq 1\) and \(\alpha (t)+\beta (t)>0\).
The main result, Theorem 3.2 reads as follows:
Assume that, for almost every \(t\in [0,T]\) and every \(u\in \mathbb R^2\) with \(| u| \leq 1\), and for every \(\lambda \geq 1\), \(\langle F(t,\lambda u)| u\rangle \geq \eta (t)\) for a suitable \(\eta \in L^2(0,T)\). Moreover, suppose that the following two conditions hold.
(i) For every \(\xi \in S^1\) satisfying \(H_0(\xi )=0\), \[ \int _0^T\liminf _{(\lambda ,\eta )\to (+\infty ,\xi )}\langle F(t,\lambda \eta )| \eta \rangle \, dt>0. \]
(ii) For every \(\theta \in [0,T]\), \[ \int _0^T\limsup _{(\lambda ,\omega )\to (+\infty ,\theta )}\left [ \langle F(t,\lambda \varphi (t+\omega ))| \varphi (t+\omega )\rangle -2\lambda H_1(\varphi (t))\right ]\, dt<0. \] Then, problem (1) has a solution.
The proof is based on degree theory. This theorem can be seen as a complement to Theorem 2.1 in [A. Fonda and M. Garrione, J. Differ. Equations 250, No. 2, 1052–1082 (2011; Zbl 1227.34037)], where the case when \(H_0\) belongs to \(\mathcal P\) has been treated. Section 3 contains also a comparison of the main theorem with results obtained by H. Brézis and L. Nirenberg [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 225–326 (1978; Zbl 0386.47035)]. In the last section of the paper, two corollaries of the main theorem for the scalar second-order case of the form \(x''+g(t,x)=0\) are examined.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
47N20 Applications of operator theory to differential and integral equations
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