On a nonlocal \({\vec p}(.)\)-Laplacian equations via genus theory (Q2811557)

This paper is concerned with the study of anisotropic elliptic equations of the form NEWLINE\[NEWLINE -\sum_{i=1}^N M_i(I_i(u))\big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\big)=f(x,u)\big(F(x,u)\big)^r NEWLINE\]NEWLINE in a smooth and bounded open set \(\Omega\subset\mathbb R^N\), \(N\geq 3\), complemented by homogeneous Dirichlet boundary conditions. Here \(p_i\) are continuous functions on \(\overline\Omega\), \(r>0\) is a real parameter, \(M_i:[0,\infty)\to [0,\infty)\), \(f:\Omega\times\mathbb R\to \mathbb R\) and \(F(x,t)=\int_0^t f(x,s)ds\). Also, NEWLINE\[NEWLINE I_i(u)=\int_{\Omega} \frac{|\partial_{x_i}u|^{p_i(x)}}{p_i(x)}dx. NEWLINE\]NEWLINE Such problems are extensions of \(p\)-Kirchhoff equations. Under some extra condition on the growth of \(M_i\) and \(f\), the author obtains the existence and multiplicity of solutions by means of genus theory.