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.
ANISOTROPIC QUASILINEAR ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT
