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.