Generalized Pohožaev and Pucci-Serrin identities and non-existence results for p(x)-Laplacian type equations

Abstract

In this article, generalizations of the well-known Pohožaev and Pucci-Serrin identities and (as consequences of them) some non-existence results for Dirichlet problems with p (x )-Laplacian are obtained.The main ideas within this paper as well as the most important results obtained during the study are presented in Section 1.In Section 2, there are recalled the main results concerning the generalized Sobolev spaces (also known as Sobolev spaces with variable exponent) which will be used throughout this paper.In Sections 3 and 4, we will give some necessary technical results regarding the differential calculus in generalized Sobolev spaces and the differentiability properties of superposition operators between generalized Sobolev spaces (in the Marcus-Mizel direction).In Sections 5 and 6, a Pohožaev-type identity and a non-existence result obtained by Pucci and Serrin for a Dirichlet problem with p-Laplacian are generalized to a Dirichlet problem with p (x )-Laplacian. The Pohožaev identity and the non-existence Pohožaev's result for Dirichlet problems with Lapla-

An isoperimetric inequality for a nonlinear eigenvalue problem

“…If on the contrary we have N − p > 0, then (4.2) is satisfied exatly for q ≤ p * as in our hypotheses. We finally recall the following generalization of the Pohozaev identity, established in [24] and [8].…”

REGULARIDAD ESCONDIDA DE LA SOLUCIÓN PARA UNA ECUACIÓN DE ONDA CON EL OPERADOR p-LAPLACIANO

“…En estas condiciones podemos aplicar los resultados en Dinca-Isaia [4], por lo que los cálculos con integración por partes son válidos. Sea u j = u se multiplican ambos lados de (22), por m i ∂u ∂x i e integrando sobre Q, obtenemos después de algunas simplificaciones…”

Pohozaev-type identities for differential operators driven by homogeneous vector fields

“…The streamline has been definitely interested in extending Pohozaev's results to more general equations, such as quasi-linear elliptic equations, polyharmonic equations and fractional differential equations. Without any attempt of completeness, we refer to [23,15] for the case of the p-laplacian, to [29] for higher order differential operators and to [18,36] for more recent contributions dealing with nonlocal operators. We must however remind that there has been a certain interest also in studying non-existence results in case of more general domains, see e.g.…”