EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR NONLOCAL $\overrightarrow{p}(x)$-LAPLACIAN PROBLEM

Abstract

In this paper, we study the nonlocal anisotropic $\overrightarrow{p}(x)$-Laplacian problem of the following form \begin{gather*} - \sum_{i=1}^N M_{i} \Big( \int_{\Omega} \frac{|\partial_{x_i} u|^{p_i(x)}}{p_i(x)} dx \Big) \partial_{x_i} \Big( |\partial_{x_i} u|^{p_i(x)-2} \partial_{x_i} u \Big) = f(x,u) \quad \text{in } \Omega, \\ u=0 \quad \text{on } \partial \Omega. \end{gather*} By means of a direct variational approach and the theory of the anisotropic variable exponent Sobolev space, we obtain the existence and multiplicity of weak energy solution. Moreover, we get much better results with $f$ in a special form.