Recent developments in problems with nonstandard growth and nonuniform ellipticity

Mingione, Giuseppe; Rădulescu, Vicenţiu D.

doi:10.1016/j.jmaa.2021.125197

Cited by 206 publications

(83 citation statements)

References 177 publications

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“…Therefore, (1.15) is the sharp differentiable version of (1.11), which is stronger than (1.11), but weaker than assuming that a(•) is Lipschitz. Lipschitz continuity of coefficients is typically assumed in the literature in the nonautonomous case (see for instance [61,66] and related references).…”

Section: Nonuniform Ellipticity At Polynomial Ratesmentioning

confidence: 99%

“…On the other hand, additional conditions ensuring the absence of the so-called Lavrentiev phenomenon are needed to build suitable approximation arguments, see for instance [31,38]. The radial structure is usually assumed in the vectorial case, otherwise singular minimizers might occur, even when data are smooth [66,72]. Again in the scalar case, we mention the recent, very interesting paper [54], where gradient regularity results are obtained for minimizers of functionals as in (1.25).…”

Section: Remarks and Organization Of The Papermentioning

confidence: 99%

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Lipschitz Bounds and Nonautonomous Integrals

Filippis

Mingione

2021

Arch Rational Mech Anal

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103

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We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

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Section: Nonuniform Ellipticity At Polynomial Ratesmentioning

confidence: 99%

Section: Remarks and Organization Of The Papermentioning

confidence: 99%

Lipschitz Bounds and Nonautonomous Integrals

Filippis

Mingione

2021

Arch Rational Mech Anal

Self Cite

103

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“…Let Ω ⊆ R N be a bounded domain (that is, a bounded connected set in the Ndimensional Euclidean space) with a smooth boundary ∂Ω. In this paper, we study the following double phase problem (P) −∆ a p u(z) − ∆ q u(z) = f (z, u(z)) in Ω, u| ∂Ω = 0, 1 < q < p < +∞ (see [1]), with the unknown u : Ω −→ R. If a ∈ L ∞ (Ω) \ {0}, a(z) 0 for almost all z ∈ Ω and 1 < r < +∞, then by ∆ a r we denote the weighted r-Laplacian differential operator defined by ∆ a r u = div (a(z)|Du| r−2 Du). If a ≡ 1, then we have the usual r-Laplacian.…”

Section: Introductionmentioning

confidence: 99%

“…For the nonlinear problems related to the nonlinear frequency shift phenomena we refer to Kalyabin et al [20] and Sadovnikov et al [21]. Finally we mention the two recent informative survey articles by Mingione-Rǎdulescu [1] and Rǎdulescu [22].…”

Section: Introductionmentioning

confidence: 99%

A Multiplicity Theorem for Superlinear Double Phase Problems

2021

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“…For fractional Kirchhoff problems with exponential growth, we refer to [24,25]. For the most recent trends and advances in variational problems exhibiting nonstandard growth conditions and/or nonuniform ellipticity we refer to the precious updated survey [23].…”

mentioning

confidence: 99%

Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

Mukherjee¹,

Pucci²,

Xiang³

2022

DCDS

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In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u>0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu>0 $\end{document}</tex-math></inline-formula> is a real parameter, <inline-formula><tex-math id="M6">\begin{document}$ \beta <{n/(n-s)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q\in (0,1) $\end{document}</tex-math></inline-formula>.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.

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Recent developments in problems with nonstandard growth and nonuniform ellipticity

Cited by 206 publications

References 177 publications

Lipschitz Bounds and Nonautonomous Integrals

Lipschitz Bounds and Nonautonomous Integrals

A Multiplicity Theorem for Superlinear Double Phase Problems

Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

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