Double-phase problems with reaction of arbitrary growth

Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.; Repovš, Dušan

doi:10.1007/s00033-018-1001-2

Cited by 75 publications

(38 citation statements)

References 34 publications

(82 reference statements)

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“…They assumed that the nonlinearity f (u) has superlinear growth in a neighborhood of u = 0 and then obtained the number of signed and sign-changing solutions which are dependent on the parameter λ. The idea in [32] has been applied to some different problems, for example, [33] and [35] for quasilinear elliptic problems with p-Laplacian operator, [34] for an elliptic problem with fractional Laplacian operator, [36] for Schrödinger equations, [11] for Neumann problem with nonhomogeneous differential operator and critical growth, and [38] for quasilinear Schrödinger equations. Especially, in [12], Li and Su investigated the Kirchhoff-type equations…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian

2020

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In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann-Liouville fractional derivatives and a parameter λ. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution u λ under some local superquadratic conditions for each given large parameter λ. We get a concrete lower bound of the parameter λ, and then obtain two estimates of weak solutions u λ . We also obtain that u λ → 0 if λ tends to ∞. Finally, we present an example as an application of our results. φ p (s) := |s| p−2 s, ∇F (t, x) is the gradient of F with respect to x = (x 1 , · · · , x N ) ∈ R N , that is, ∇F (t, x) = ( ∂F ∂x1 , · · · , ∂F ∂xN ) τ , and F : [0, T ] × R N → R satisfies the following condition:(H0) there exists a constant δ > 0 such that F (t, x) is continuously differentiable in x ∈ R N with |x| ≤ δ for a.e.t ∈ [0, T ], measurable in t for every x ∈ R N with |x| ≤ δ, and there exist a ∈ C(R + ,for all x ∈ R N with |x| ≤ δ and a.e. t ∈ [0, T ].When α = 1, the operator t D α T ( 0 D α t u(t)) reduces to the usual second order differential operator −d 2 /dt 2 . Hence, if α = 1, p = 2, N = 1, λ = 1 and V (t) = 0 for a.e. t ∈ [0, T ], system (1.1) becomes the equation with

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian

2020

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“…For this reason models as those in (1.2) are particularly useful to describe strongly anisotropic media. We refer to the papers [2,7,8,16,17,54,65] for more results, different directions and related topics. Another, softer instance of functional with non-standard growth used to describe anisotropic models [1,65,66,67] is the variable exponent one…”

Section: Introductionmentioning

confidence: 99%

Manifold Constrained Non-uniformly Elliptic Problems

Filippis

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2019

J Geom Anal

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We consider the problem of minimizing variational integrals defined on nonlinear Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a nonuniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed at proving partial regularity of minima and providing techniques for treating larger classes of similar constrained non-uniformly elliptic variational problems. In order to give estimates for the singular sets we use a general family of Hausdorff type measures following the local geometry of the integrand. A suitable comparison is provided with respect to the naturally associated capacities.

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“…where c 6 , c 7 > 0. Next, with an argument similar as the same developed in the first part of this proof, relation (19) implies that for some c 8 > 0 (20) E(u n ) − 1 s E ′ (u n ), u n c 8 ( ∇u n q p,q + u n α ) for all n 1.…”

Section: Proof Of Theoremmentioning

confidence: 87%

“…Refined regularity results are proved in [13], by using an approximation technique relying on estimates obtained through a careful use of difference quotients. Other recent works dealing with nonlinear problems with unbalanced growth (either isotropic or anisotropic) are the papers by Bahrouni, Rȃdulescu and Repovš [5], Cencelj, Rȃdulescu and Repovš [11], and Papageorgiou, Rȃdulescu and Repovš [19]. The differential operator defined in (1) and which is generated by a potential with variable growth was introduced by Azzollini et al [2,3] in relationship with wide classes of nonlinear PDEs with a variational structure.…”

Section: Introductionmentioning

confidence: 99%