On some degenerate quasilinear equations involving variable exponents

In the present paper, we are concerned with some degenerate quasilinear equations involving variable exponents. Using various (variational and nonvariational) techniques, we prove existence, nonexistence and multiplicity results.     

On a biharmonic equation involving nearly critical exponent

This paper is concerned with a biharmonic equation under the Navier boundary condition , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in , n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P _−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x ₀ ∈Ω as ε → 0, moreover x ₀ is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x ₀ of the Robin’s function, there exist solutions of (P _−ε) concentrating around x ₀ as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P _−ε), the supercritical problem (P _+ε) has no solutions which concentrate around a point of Ω as ε → 0. Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena Pacella for their constant support.     

Multiplicity results for a class of quasilinear eigenvalue problems on unbounded domains

Using a recent result of Ricceri [10] we prove a multiplicity result for a class of quasilinear eigenvalue problems with nonlinear boundary conditions on an unbounded domain. Our paper completes previous results obtained by Carstea and Rădulescu [4], Chabrowski [1], [2], Kandilakis and Lyberopoulos [6] and Pflüger [7]. Received: 17 April 2007     

Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces

The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz-Sobolev space.     

Existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions

In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.     

Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent

In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class of approximating functions, due to Gazzola and Ruf, with the concentrating functions of the best Sobolev constant.     

On superlinear problems without the Ambrosetti and Rabinowitz condition

Existence and multiplicity results are obtained for superlinear p-Laplacian equations without the Ambrosetti and Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. Our results extend the recent results of Miyagaki and Souto [O. Miyagaki, M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008) 3628-3638].     

Renormalized solutions for nonlinear elliptic equations with variable exponents and L data

We prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and L¹$L^1$ data. The functional setting involves Lebesgue–Sobolev space with variable exponents W^1,p(⋅)(Ω)$W^{1,p(\cdot )}\left(\Omega \right)$.     

Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces

In this paper, we are interested in the existence of infinitely many weak solutions for a non-homogeneous eigenvalue Dirichlet problem. By using variational methods, in an appropriate Orlicz–Sobolev setting, we determine intervals of parameters such that our problem admits either a sequence of non-negative weak solutions strongly converging to zero provided that the non-linearity has a suitable behaviour at zero or an unbounded sequence of non-negative weak solutions if a similar behaviour occurs at infinity.     

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

On a ‐Kirchhoff equation with critical exponent and an additional nonlocal term via truncation argument

We study existence and multiplicity of solutions of the following nonlocal ‐Kirchhoff equation with critical exponent, via truncation argument on the Sobolev space with variable exponent, where Ω is a bounded smooth domain of , , M, f are continuous functions, , and are real parameter.     

On the numerical condition of a generalized Hankel eigenvalue problem

The generalized eigenvalue problem with H a Hankel matrix and the corresponding shifted Hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments, the determination of abscissa of quadrature formulas, of poles of Padé approximants, or of the unknown powers of a sparse black box polynomial in computer algebra. In many of these applications, the entries of the Hankel matrix are only known up to a certain precision. We study the sensitivity of the nonlinear application mapping the vector of Hankel entries to its generalized eigenvalues. A basic tool in this study is a result on the condition number of Vandermonde matrices with not necessarily real abscissas which are possibly row-scaled. B. Beckermann was supported in part by INTAS research network NaCCA 03-51-6637. G. H. Golub was supported in part by DOE grant DE-FC-02-01ER41177. G. Labahn was supported in part by NSERC and MITACS Canada grants.     

On resonant Neumann problems: Multiplicity of solutions

We consider a semilinear Neumann problem with an asymptotically linear reaction term. We assume that resonance occurs at infinity. Using variational methods based on the critical point theory, together with the reduction technique and Morse theory, we show that the problem has at least four nontrivial smooth solutions.     

Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC ²-function at both isolated critical point and infinity.     

On the fundamental eigenvalue ratio of the p-Laplacian

It is shown that the fundamental eigenvalue ratio of the p-Laplacian is bounded by a quantity depending only on the dimension N and p.     

Morse Index and Critical Groups for p-Laplace Equations with Critical Exponents

In this work we consider a class of Euler functionals defined in Banach spaces, associated to quasilinear elliptic problems involving the critical Sobolev exponent. We perform critical groups estimates via the Morse index. Dedicated to the memory of Professor Aldo Cossu The research of the authors was supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena” (PRIN 2005).     

Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions

The aim of this paper is to establish the existence of an unbounded sequence of weak solutions for a class of differential equations with p(x)$p\left(x\right)$-Laplacian and subject to small perturbations of nonhomogeneous Neumann conditions. The approach is based on variational methods.     

On a class of critical singular quasilinear elliptic problem with indefinite weights

In this paper, the eigenvalue problem for a class of quasilinear elliptic equations involving critical potential and indefinite weights is investigated. We obtain the simplicity, strict monotonicity and isolation of the first eigenvalue λ₁. Furthermore, because of the isolation of λ₁, we prove the existence of the second eigenvalue λ₂. Then, using the Trudinger-Moser inequality, we obtain the existence of a nontrivial weak solution for a class of quasilinear elliptic equations involving critical singularity and indefinite weights in the case of 0<λ<λ₁ by the Mountain Pass Lemma, and in the case of λ₁≤λ<λ₂ by the Linking Argument Theorem.