Existence of infinitely many weak solutions for some quasilinear $\vec {p}(x)$-elliptic Neumann problems

Ahmed, Ahmed ; Ahmedatt, Taghi ; Hjiaj, Hassane ; Touzani, Abdelfattah

Mathematica Bohemica, Tome 142 (2017) no. 3, pp. 243-262

Abstract (VO)

We consider the following quasilinear Neumann boundary-value problem of the type $$ \begin {cases} -\displaystyle \sum _{i=1}^{N}\frac {\partial }{\partial x_{i}}a_{i}\Big (x,\frac {\partial u}{\partial x_{i}}\Big ) + b(x)|u|^{p_{0}(x)-2}u = f(x,u)+ g(x,u) \text {in} \ \Omega , \\ \quad \dfrac {\partial u}{\partial \gamma } = 0 \text {on} \ \partial \Omega . \end {cases} $$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.

Détail

DOI : 10.21136/MB.2017.0037-15

Classification : 35D30, 35J20, 35J25, 35J62
Keywords: Neumann problem; quasilinear elliptic equation; weak solution; variational principle; anisotropic variable exponent Sobolev space

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