On a double resonant problem in \(\mathbb R^N\). (Q1850181)

The authors establish via variational methods the existence and multiplicity of solutions for a class of semilinear elliptic equations in \(\mathbb R^n\): \[ -\Delta u+b(x)u=f(x,u),\quad x\in\mathbb R^n\,, \] where \(n\geq 3\) and the potential \(b\) is a continuous function satisfying \(b(x)\geq b_0>0\) for all \(x\in\mathbb R^n\). The main goal of the paper is to explore the compactness provided by the condition on the potential, to study a class of double resonant problems under a local nonquadraticity condition. For the existence of solution the nonlinearity may satisfy a critical growth condition. Related results can be found [{\textit{P. H. Rabinowitz}}, Z. Angew. Math. Phys. 43, 270--291 (1992; Zbl 0763.35087)].