AbstractThis paper deals with the existence of multiple solutions for the
quasilinear equation{-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text%
{in ${\mathbb{R}^{N}}$,}}which involves a general variable exponent elliptic operator {\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic
phenomena, in the sense that the differential operator has various types of behavior like {|\xi|^{q(x)-2}\xi} for small {|\xi|} and like {|\xi|^{p(x)-2}\xi} for large {|\xi|}, where {1<\alpha(\,\cdot\,)\leq p(\,\cdot\,)<q(\,\cdot\,)<N}. Our aim is to approach variationally the problem by using the tools of
critical points theory in generalized Orlicz–Sobolev spaces with variable
exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio,
Quasilinear elliptic equations in \mathbb{R}^{N} via variational methods and Orlicz–Sobolev embeddings,
Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu,
Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential,
Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the
exponents p and q are constant, to the case where {p(\,\cdot\,)} and {q(\,\cdot\,)} are functions. We also substantially weaken some of the
hypotheses in these papers and we overcome the lack of compactness by using
the weighting method.
On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics
