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