The existence of a nontrivial solution to the \(p{\&}q\)-Laplacian problem with nonlinearity asymptotic to \(u^{p - 1}\) at infinity in \(\mathbb R^N\)
Let \(m,n>0\), \(N\geq3\) and \(1<q<p<N\). Let \(\Delta_p u=\text{div}(|\nabla u|^{s-2}\nabla u)\) denote the \(p\)-Laplacian of \(u\). The nonlinear elliptic problem \[ -\Delta_p u+m| u|^{p-2}u-\Delta_q u+n| u|^{q-2}u=f(x,u),\quad x\in\mathbb R^N, \] is considered, where \(f:\mathbb R^N\times\mathbb R\to\mathbb R\) does not satisfy the usual requirement, i.e., the Ambrosetti-Rabinowitz condition \(0\leq\int_0^u f(x,t)\,dt\leq (p+\theta)^{-1}f(x,u)u\). Instead, it is assumed that \(f\) is a non-negative function satisfying the Carathéodory conditions, \(f(x,t)=0\) if \(t<0\), \(f(x,t)t^{1-p}\) tends to zero as \(t\to0_+\) and to a positive constant as \(t\to\infty\), both uniformly in \(x\in\mathbb R^N\), \(f(x,t)t^{1-p}\) is nondecreasing in \(t\) for every \(x\in\mathbb R^N\) and satisfies some additional conditions. The authors mainly follow the approach of \textit{G. Li} and \textit{H.-S. Zhou} [Proc. R. Soc. Edinb., Sect. A, Math. 130, No. 1, 81--105 (2000; Zbl 0942.35075)] to prove existence of a nontrivial solution \(u\in W^{1,p}(\mathbb R^N)\cap W^{1,q}(\mathbb R^N)\).
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