Resonant double phase equations

The double phase Dirichlet problem when the lowest exponent is equal to 1

“…These different parts of the domain are usually called the p and the q-phase respectively and here they are determined by the function a; when a(x) = 0 we have the p-phase and when a(x) > 0 the q-phase respectively. Double phase problems were firstly introduced by Zhikov in [20] and in recent years they have become the focus of extensive research (see for example [3], [5], [6], [7], [9], [14], [15], [16], [18] and the references therein). Nevertheless with the exception of [7], [9] it seems that the case p = 1 has been overlooked and hence this paper is a step towards filling this gap.…”

Non‐autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

“…There are only interior regularity results, which are primarily due to Baroni et al [3] and Marcellini [10,20,21]. In fact, most of works dealt with double phase problems having unbalanced growth in bounded domains of ℝ 𝑁 , we refer the readers to [12][13][14][15][22][23][24] and references therein. However, there exist relatively few ones treating the problems in ℝ 𝑁 .…”

“…[3] and Marcellini [10, 20, 21]. In fact, most of works dealt with double phase problems having unbalanced growth in bounded domains of $$\mathbb {R}^N$$, we refer the readers to [12–15, 22–24] and references therein. However, there exist relatively few ones treating the problems in $$\mathbb {R}^N$$.…”

“…While double phase problems are set in $$\mathbb {R}^N$$, the condition $$\frac{p}{q}<1+\frac{1}{N}$$ can no longer be applicable to derive the compact embedding results, which leads to lack of compactness for the study. In this paper, such a condition is actually used to guarantee the regularity of solutions to (1.1) (see [8, 9]), which along with the maximum principle developed in [23, 24] can lead to the simplicity of eigenvalues, see Proposition 3.2.…”

A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces

“…This paper deals with generalized Orlicz spaces, also known as Musielak-Orlicz spaces. This is a very active field recently [2,5,13,15,18,28,30,31], boosted by work on the double phase problem by Baroni, Colombo and Mingione, e.g. [4,12].…”