Multiple solutions of double phase variational problems with variable exponent

Abstract

This 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 exponentspandqare 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.

Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves

“…The present paper complements our previous contributions related to double phase anisotropic variational integrals, see [34,36]. This paper also extends our recent results established in [3] to a mixed elliptic-hyperbolic setting.…”

Existence of Solution for Double-Phase Problem with Singular Weights

“…The existence and multiplicity of solutions of double-phase Dirichlet problems has been studied by several authors (see, e.g., [1][2][3][4][5][6][7][8]); in particular, for the eigenvalues of the double-phase operator, see [7]. For other double-phase problems with variable exponents, there are the works of Zhang and Radulescu [9], Shi et al [10], and Cencelj et al [11].…”

A weak solution for a $(p(x),q(x))$-Laplacian elliptic problem with a singular term

“…The differential operator p + q is known as the (p, q)-Laplacian operator, if p = q, where j , j > 1 denotes the j-Laplacian defined by j u := div(|∇u| j-2 ∇u). It is not homogeneous, thus some technical difficulties arise in applying the usual methods of the theory of elliptic equations (for further details, see [1,2,5,7,8,10,[12][13][14][15][16][19][20][21][22][23] and references therein).…”