Existence of infinitely many weak solutions for some quasilinear \(\vec{p}(x)\)-elliptic Neumann problems. (English) Zbl 06770144

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

Math. Bohem. 142, No. 3, 243-262 (2017).

Summary: We consider the following quasilinear Neumann boundary-value problem of the type \[ \begin{cases} -\sum_{i=1}^N\frac{\partial}{\partial x_i}a_i\Big( x,\frac{\partial u}{\partial x_i}\Big) +b(x)|u|^{p_0(x)-2}u=f(x,u)+g(x,u) &\text{in} \;\Omega ,\\ \quad\dfrac{\partial u}{\partial\gamma}=0 &\text{on} \;\partial\Omega.\end{cases} \] We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.

MSC:

35J20	Variational methods for second-order elliptic equations
35J62	Quasilinear elliptic equations

Keywords:

Neumann problem; quasilinear elliptic equation; weak solution; variational principle; anisotropic variable exponent Sobolev space