QUASILINEAR DOUBLE PHASE PROBLEMS ON THE ENTIRE SPACE $\mathbb{R}^N$

This study is concerned with the following double phase problem <p class="disp_formula">$ \begin{array}{ll} \quad-{\rm div}(|\nabla u|^{^{p-2}}\nabla u+\mu(x)|\nabla u|^{^{q-2}}\nabla u) +V(x)(|u|^{^{p-2}}u+\mu(x)|u|^{^{q-2}}u) \\ = \lambda f(x, u), \; x\in \mathbb{R}^N, \end{array} $ where $1&lt;p&lt;q&lt;N$, $\frac{q}{p}\leq 1+\frac{\alpha}{N}$, $\lambda$ is a real parameter, $0\leq\mu\in C{^{0,\alpha}}(\mathbb{R}^N)$ with $\alpha\in(0,1]$, $V(x)$ is an unbounded potential function and $f(x,u)$ is the reaction term. The aim is to determine the precise positive interval of $\lambda$ for which the problem admits at least one or two nontrivial solutions by applying abstract critical point results.