Spectrum of the Ap-Laplacian Operator

This work deals with the nonlinear boundary eigenvalue problem(V:P(Gammaho;I)):-A_p u = lambda ho(x)|u|^{p-2}u in I =], b[,u(a) = u(b) = 0,where A_p is called the A_p-Laplacian operator and defined by A_p u = (Gamma(x) |u'|^{p-2}u'),p > 1, lambda is a real parameter, ho is an indefinite weight, a, b are real numbers and Gamma in C^1(I) cap C^0(overline{I}) and it is nonnegative on overline{I}.We prove in this paper that the spectrum of the A_p-Laplacian operator is given by a sequence of eigenvalues. Moreover, each eigenvalue is simple, isolated andverifies the strict monotonicity property with respect to the weight ho and the domain I. The k¡th eigenfunction corresponding to the k-th eigenvalue has exactly k-1 zeros in (a,b). Finally, we give a simple variational formulation of eigenvalues

Spectre d'ordre supérieur et problèmes de non résonance

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Higher order spectrum and quasilinear elliptic problems

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Existence and multiplicity of solutions for a Steklov eigenvalue problem involving the p(x)-Laplacian-like operator

Using the variational method, we prove the existence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained

Existence and multiplicity results for elliptic problems with Nonlinear Boundary Conditions and variable exponents

abstract: By applying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem: is a bounded domain of smooth boundary ∂Ω and ν is the outward normal vector on ∂Ω. p : Ω → R, a : Ω × R N → R N , f : Ω × R → R and g : ∂Ω × R → R are fulfilling appropriate conditions