Weak solutions for fractional \(p(x,\cdot)\)-Laplacian Dirichlet problems with weight. (English) Zbl 1487.35389

Ait Hammou, Mustapha

Analysis, München 42, No. 2, 121-132 (2022).

Summary: The main purpose of this paper is to show the existence of weak solutions for a problem involving the fractional \(p(x,\cdot\,)\)-Laplacian operator of the following form: \[ \begin{cases} \begin{aligned} (-\Delta_{p(x,\cdot\,)})^su(x)+w(x) |u|^{\bar{p}(x)-2}u & =\lambda f(x,u) &&\text{in }\Omega,\\ u&=0&&\text{in }\mathbb{R}^{N}\setminus\Omega, \end{aligned}\end{cases} \] The main tool used for this purpose is the Berkovits topological degree.

MSC:

35R11	Fractional partial differential equations
35J25	Boundary value problems for second-order elliptic equations
35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35S15	Boundary value problems for PDEs with pseudodifferential operators
47H11	Degree theory for nonlinear operators

Keywords:

fractional \(p(\cdot,\cdot)\)-Laplacian operator; weighted variable exponent space; topological degree

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