Double phase anisotropic variational problems and combined effects of reaction and absorption terms
The authors are interested in finding multiple solutions to quasilinear elliptic equations given by \[ -\text{div\,}A(x,\nabla u)+V(x)|u|^{\alpha(x)-2}u=f(x,u) \quad \text{in }\mathbb{R}^N, \] where \(A:\mathbb{R}^N\times \mathbb{R}^N\to \mathbb{R}^N\) and \(f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}\) satisfy sutiable structure conditions. The problem corresponds to double phase anisotropic phenomena in the sense that the differential operator has behaviors like \(|\xi|^{q(x)-2}\xi\) for small \(|\xi|\) and behaviors like \(|\xi|^{p(x)-2}\xi\) for large \(|\xi|\), where \(1<\alpha(\cdot)\leq p(\cdot)< q(\cdot)<N\). The aim of the authors is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz-Sobolev spaces with variable exponent. Their results extend previous works from cases where the exponents \(p\) and \(q\) are constant to the case of functions \(p(\cdot)\) and \(q(\cdot)\). In addition, the authors are also able to weaken some of the hypotheses in these papers and overcome the lack of compactness by using the Cerami compactness condition.
scientific article; zbMATH DE number 6938394
scientific article