Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in RNwith Critical Sobolev Exponent

Drábek, Pavel; Huang, Yin Xi

doi:10.1006/jdeq.1997.3306

Cited by 92 publications

(42 citation statements)

References 18 publications

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“…Then it was widely used in the study of partial differential equations with critical growth (e.g. Egnell 1988;Drabek & Huang 1997;Goncalves & Alves 1998;Li & Zhang 2009, and the references therein). Fu (2009) established a principle of concentration compactness in Sobolev space W 1, p(x) (Ω), where Ω ⊂ R N is a bounded domain, and then discussed the existence of solutions for a class of p(x)-Laplacian equations with critical growth.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiple solutions for a class of p ( x )-Laplacian equations in involving the critical exponent

Zhang

2010

Proc. R. Soc. A.

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In this paper, we first establish a principle of concentration compactness in W 1,p(x) (R N ). Then, based on this concentration compactness principle, we study the existence of solutions for a class of p(x)-Laplacian equations in R N involving the critical exponent. Under suitable assumptions, we obtain a sequence of radially symmetric solutions associated with a sequence of positive energies going towards infinity.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiple solutions for a class of p ( x )-Laplacian equations in involving the critical exponent

Zhang

2010

Proc. R. Soc. A.

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“…Solutions bifurcate from (λ 1 , 0) and branch to the left or right depending on the sign of g(x, λ 1 )|φ 1 (x)| p * dx. The evidence in [8] found that, depending on the sign of g(x, λ 1 )|φ 1 (x)| p * dx, there may be two solutions in a right neighbourhood of λ 1 , or one solution in a left neighbourhood of λ 1 and none in a right neighbourhood. For g ≡ 1, Azorero and Alonso [1] use the variational approach to find that at each 0 < λ < λ 1 , there is a nontrivial solution to the problem.…”

Section: Concluding Remarks and Examplesmentioning

confidence: 99%

Asymptotic Bifurcation Results for Quasilinear Elliptic Operators

Chabrowski¹,

Drábek²,

Tonkes³

2005

Glasgow Math. J.

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Abstract.We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.2000 Mathematics Subject Classification. 47J15, 35J60, 35J65, 47J10.

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“…as ε → 0 (see, e.g., Drábek and Huang [10]). Consider the critical level c defined at the beginning of this section with u 0 = v ε .…”

Section: Proofs Of Theorems 12 and 13mentioning

confidence: 99%

On a class of critical (p, q)-Laplacian problems

Candito

Marano

Perera

2015

Nonlinear Differ. Equ. Appl.

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We obtain nontrivial solutions of a critical (p, q)-Laplacian problem in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical Sobolev exponents, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.

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Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in RNwith Critical Sobolev Exponent

Cited by 92 publications

References 18 publications

Multiple solutions for a class of p ( x )-Laplacian equations in involving the critical exponent

Multiple solutions for a class of p ( x )-Laplacian equations in involving the critical exponent

Asymptotic Bifurcation Results for Quasilinear Elliptic Operators

On a class of critical (p, q)-Laplacian problems

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