Existence and multiplicity of solutions for the nonlocal p(x)-Laplacian equations in $R^N$

• Main Author: Chao Ji
• Format: Article
• Language: English
• Published: University of Szeged 2012-08-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: nonlocal p(x)-laplacian equation; variational method; variable exponent spaces
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=1789

This work deals with the nonlocal $p(x)$-Laplacian equations in $R^{N}$ with non-variational form \begin{align*} \left\{\begin{aligned} &A(u)\big(-\Delta_{p(x)}u+|u|^{p(x)-2}u\big)=B(u)f(x,u) \text{in}R^{N},\\ &u\in W^{1, p(x)}(R^{N}), \end{aligned} \right.\end{align*} and with the variational form \begin{align*} \left\{\begin{aligned} & a\Big(\int_{R^{N}}\frac{\vert \nabla u\vert^{p(x)}+\vert u\vert^{p(x)}}{p(x)}dx\Big)(-\Delta_{p(x)}u+|u|^{p(x)-2}u)&\\ &=B\Big(\int_{R^{N}}F(x, u)dx \Big)f(x, u) \text{in} R^{N},&\\ &u\in W^{1, p(x)}(R^{N}), \end{aligned}\right. \end{align*} where $F(x,t)=\int_{0}^{t}f(x,s)ds$, and $a$ is allowed to be singular at zero. Using $(S_{+})$ mapping theory and the variational method, some results on existence and multiplicity for the problems in $R^{N}$ are obtained.