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We prove some properties of the first eigenvalue for the elliptic system −Δpu=λ|u|α|v|βv in Ω, −Δqv=λ|u|α|v|βu in Ω, (u,v)∈W01,p(Ω)×W01,q(Ω). In particular, the first eigenvalue is shown to be simple. Moreover, the stability with respect to (p,q) is established

Abstract. In this paper, we consider a nonlinear elliptic problem involving the p-Laplacian with perturbation terms in the whole R N . Via variational arguments, we obtain existence and regularity of nontrivial solutions.

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This work is devoted to the study of the spectrum for p-biharmonic operator with an indefinite weight in a bounded domain.

We study the following nonlinear eigenvalue problem with nonlinear Robin boundary conditionWe successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λ n ∞ . To this end we endow W 1,p (Ω) with a norm invoking the trace and use the duality mapping on W 1,p (Ω) to apply mini-max arguments on C 1 -manifold. Classification (2010). 35J35, 35J40, 35J66, 58E05. Mathematics Subject

AbstractIn this work, we investigate the spectrum denoted by Λ for the {p(x)}-biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that {\sup\Lambda=+\infty}. Moreover, we prove that {\inf\Lambda>0} if and only if the domain Ω satisfies the {p(x)}-Hardy inequality.

We show some new Sobolev's trace embedding that we apply to prove that the fourth-order nonlinear boundary conditionsΔp2u+|u|p−2u=0inΩand−(∂/∂n)(|Δu|p−2Δu)=λρ|u|p−2uon∂Ωpossess at least one nondecreasing sequence of positive eigenvalues.