On noncoercive periodic systems with vector \(p\)-Laplacian (Q446226)

Consider the \(p\)-Laplacian systems NEWLINE\[NEWLINE\begin{aligned}-(|x'(t)|^{p-2}x'(t))'=\nabla F(t,x(t))&\qquad \text{a.e. on }T=[0,b],\\ x(0)=x(b), \quad x'(0)=x'(b),&\qquad 1<p<\infty, \end{aligned}\tag{PS}NEWLINE\]NEWLINE where \(|\cdot|\) stands for the Euclidean norm on \(\mathbb{R}^N\) and \(F:T\times \mathbb{R}^N\longrightarrow \mathbb{R}\) is a Carathéodory mapping such that \(F(t,\cdot)\) is of class \(C^1\) for almost every \(t\in T.\)NEWLINENEWLINEBy using a version of the local linking theorem due to S. J. Li and M. Willem, the authors obtain the existence of periodic solutions for \(p\)-Laplacian systems ({PS}). In the existence theorem, they assume that the potential function is \(p\)-superlinear, but in general it does not satisfy the AR-condition. Moreover, a multiplicity theorem is obtained. In the multiplicity theorem, the problem is strongly resonant with respect to the principal eigenvalue \(\lambda_0=0\). The main results are the following theorems.NEWLINENEWLINETheorem 1. Suppose that \(F:T\times \mathbb{R}^N\rightarrow \mathbb{R}\) satisfies the conditionsNEWLINENEWLINE(i) for all \(x\in \mathbb{R}^N\), \(t\rightarrow F(t,x)\) is measurable;NEWLINENEWLINE(ii) for almost all \(t\in T,\,x\rightarrow F(t,x)\) is \(C^1\) and \(F(t,0)=0\);NEWLINENEWLINE(iii) for almost all \(t\in T\) and all \(x\in \mathbb{R}^N\), NEWLINE\[NEWLINE|\nabla F(t,x)|\leq a(t)+c|x|^{r-1}NEWLINE\]NEWLINE with \(a\in L^1(T)_+\), \(c>0\) and \(p<r<\infty\);NEWLINENEWLINE(iv) \(\lim_{|x|\rightarrow\infty}(F(t,x)/|x|^p)=+\infty\) uniformly for almost all \(t\in T\), and there exists \(\mu>r-p\) such that NEWLINE\[NEWLINE\liminf\limits_{|x|\rightarrow\infty}\frac{(\nabla F(t,x),x)-pF(t,x)}{|x|^\mu}>0NEWLINE\]NEWLINE uniformly for almost all \(t\in T\);NEWLINENEWLINE(v) \(\limsup_{x\rightarrow0}(pF(t,x)/|x|^p)<1/b^p\) uniformly for almost all \(t\in T\), and there exists \(\delta>0\) such that \(F(t,x)\geq 0\) for almost all \(t\in T\) and all \(x\in \mathbb{R}^N\) with \(|x|\leq\delta\).NEWLINENEWLINEThen problem (PS) has a nontrivial solution \(x_0\in C^1(T;\mathbb{R}^N)\).NEWLINENEWLINETheorem 2. Suppose that \(F\) satisfies (i), (ii) and the conditionsNEWLINENEWLINE(vi) for almost all \(t\in T\) and all \(x\in \mathbb{R}^N\), NEWLINE\[NEWLINE|\nabla F(t,x)|\leq a_0(t)c_0(|x|)NEWLINE\]NEWLINE with \(a_0\in L^1(T)_+\), \(c_0\in C(\mathbb{R}_+)\), \(c_0\geq0\);NEWLINENEWLINE(vii) there exists a function \(F_\infty\in L^1(T)\) such that \(\int_0^bF_\infty(t)\,dt\leq0\) and NEWLINE\[NEWLINEF(t,x)\rightarrow F_\infty(t)\quad \text{for a.a. } t\in T,\text{ as } |x|\rightarrow\infty;NEWLINE\]NEWLINENEWLINENEWLINE(viii) there exists a function \(\eta\in L^1(T)_+\), \(\eta\neq0\), such that NEWLINE\[NEWLINE\liminf\limits_{x\rightarrow0}\frac{pF(t,x)}{|x|^p}\geq \eta(t)NEWLINE\]NEWLINE uniformly for almost all \(t\in T\);NEWLINE(ix) \(F(t,x)\leq \frac{1}{pb^p}|x|^p\) for almost all \(t\in T\) and all \(x\in \mathbb{R}^N\).NEWLINENEWLINEThen, problem (PS) has at least two nontrivial solutions \(x_0,\,u_0\in C^1(T;\mathbb{R}^N)\).