Quasilinear Elliptic Equations with Degenerations and Singularities

Drábek, Pavel; Kufner, Alois; Nicolosi, Francesco

doi:10.1515/9783110804775

Cited by 257 publications

(237 citation statements)

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“…In that context we refer to the works of Sharapudinov [37], Tsenov [39] and Zhikov [41,42]. For deep results in weighted Sobolev spaces with applications to partial differential equations and nonlinear analysis we refer to the excellent monographs by Drabek, Kufner and Nicolosi [13], Hyers, Isac and Rassias [21], Kufner and Persson [25], and Precup [34]. We also refer to the recent works by Diening [11], Ruzicka [36] and Chen, Levine and Rao [8] for applications of Sobolev spaces with variable exponent in the study of electrorheological fluids or in image restoration.…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

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Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

Dinu¹

2006

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. We establish the existence of an entire solution for a class of stationary Schrödinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows-up at infinity. The abstract framework is related to Lebesgue-Sobolev spaces with variable exponent. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the Mountain Pass Lemma without the PalaisSmale condition for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz [35] on the existence of ground-state solutions of the nonlinear Schrödinger equation.

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Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

“…So, by relation (4) p.104 in Chang [7] and by the definition of (−F ) 0 , we deduce that By density, this hemivariational inequality holds for all ϕ ∈ E and this means that u 0 solves Problem (13).…”

Section: Proof Of Theoremmentioning

confidence: 85%

Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

Dinu¹

2006

Nonlinear Analysis: Theory, Methods & Applications

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“…Furthermore, we prove an useful multiplicative inequality for boundary integrals. Section 3 handles the constant exponent case (i.e., V = W 1,p (Ω) with 1 < p < ∞) where we will apply Moser's iteration following the ideas of Drábek-Kufner-Nicolosi [12]. In the last section we extend our results to the variable exponent case (i.e.…”

Section: Introductionmentioning

confidence: 89%

On the Boundedness of Solutions to Elliptic Variational Inequalities

Winkert

2014

Set-Valued Var. Anal

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Abstract. In this paper we present global a priori bounds for a class of variational inequalities involving general elliptic operators of second-order and terms of generalized directional derivatives. Based on Moser's and De Giorgi's iteration technique we prove the boundedness of solutions of such inequalities under certain criteria on the set of constraints. In our proofs we also use the localization method with a certain partition of unity and a version of a multiplicative inequality estimating the boundary integrals. Some sets of constraints satisfying the required conditions are stated as well.

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“…Consequently, μ( ) = 0 or h For more details, we refer to [3,7]. The proof of the following lemma follows as an easy combination of Hölder's inequality with the Sobolev embeddings and it is omitted.…”

Section: Fourth-order Elliptic Equations Involving Sobolev Exponents 135mentioning

confidence: 99%

Existence and Bifurcation Results for Fourth-Order Elliptic Equations Involving Two Critical Sobolev Exponents

2009

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Abstract. Let be a smooth bounded domain in R N , with N ≥ 5. We provide existence and bifurcation results for the elliptic fourth-order equation 2 u − p u = f (λ, x, u) in , under the Dirichlet boundary conditions u = 0 and ∇u = 0. Here λ is a positive real number, 1 < p ≤ 2 # and f (., ., u) has a subcritical or a critical growth s, 1 < s ≤ 2 * , where 2 * := 2N N−4 and 2 # := 2N N−2 . Our approach is variational, and it is based on the mountain-pass theorem, the Ekeland variational principle and the concentration-compactness principle.AMS Subject Classification. 35J35, 35B33, 35G20, 35B32. 1.Introduction. An approach for confronting second-order critical semilinear elliptic equations in a bounded domain in R N was introduced in [2], where it was shown that the Palais-Smale compactness condition holds for certain levels of the associated functional. Therefore, under the appropriate assumptions, the mountainpass theorem could be applied to yield a solution to the critical problem.The existence of solutions of fourth-order critical elliptic problems can also be proved by using this approach, see [4,5,8,11,15] and the references therein.In this paper, we study problems of the form

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Quasilinear Elliptic Equations with Degenerations and Singularities

Cited by 257 publications

References 0 publications

Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

On the Boundedness of Solutions to Elliptic Variational Inequalities

Existence and Bifurcation Results for Fourth-Order Elliptic Equations Involving Two Critical Sobolev Exponents

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