Existence of solutions for the double phase variational problems without AR-condition

Abstract

We consider the following double phase problem:

$$\begin{aligned} -\mathrm {div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)+V(x)|u|^{\alpha -2}u=f(x,u),\quad \mathrm {in}\ \mathbb {R}^{N}, \end{aligned}$$

where $N\ge 2,$ and $\frac{q}{p}<1+\frac{1}{N}, a:\mathbb {R}^{N}\rightarrow [0,+\infty )~\mathrm {is~Lipschitz~continuous.}$ The Ambrosetti–Rabinowitz type condition, that is so-called (AR) condition: there exist $L>0,\theta >q$ such that for $|t|\ge L$ and a.e. $x\in \mathbb {R}^{N}$,

$$\begin{aligned} 0<\theta F(x,t)\le t f(x,t), \end{aligned}$$

as well as the monotonicity of $f(x,t)/|t|^{q-1}$ are not assumed. Under appropriate assumptions on V and f, we prove that the above problem has at least a nontrivial solution.