Ground state solutions for \(p\)-superlinear \(p\)-Laplacian equations (Q2877688)

In the paper under review, the authors study the existence of ground state solutions to the \(p\)-Laplacian equations, \(p>1\), i.e., NEWLINE\[NEWLINE \begin{cases} -\operatorname{div}(|\nabla u|^{p-2}\nabla u)+ V(x)|u|^{p-2}u=f(x,u), & x\in \mathbb{R}^N\\ u\in W^{1,p}(\mathbb{R}^N). \end{cases} NEWLINE\]NEWLINE Assuming that \(V(x)\) and \(f(x,t)\) are one-periodic in each direction of \(x\in \mathbb{R}^N\), the authors establish the existence of solutions to the above equation upon some new growth conditions on the function \(t\mapsto f(x,t)/|t|^{p-1}\), which weaken the Ambrosetti-Rabinowitz type condition and the monotonicity condition. In particular, both \(tf(x,t)\) and \(tf(x,t)-pF(x,t)\) are allowed to be sign-changing.