Some existence results for operator equations involving duality mappings on Sobolev spaces with variable exponent (Q637046)

The authors consider duality equations \(J_{\varphi }u=h\) and \(J_{\varphi }u=N_{f}u\) in variable exponent Sobolev spaces on some domain \(\varOmega \), where \(J_{\varphi }\) and \(N_{f}\) is a duality mapping with gauge \(\varphi \) and the Nemytskii operator generated by a Carathéodory function \(f\), respectively. Surjectivity results are established under suitable additional assumptions on the smoothness of \(\varOmega \) and growth conditions on \(f\) linked with the variable exponents in question. The major tools, also established here, are smoothness properties of the norm. A formula for the gradient of the norm is proved.