Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathbb R^N\) (Q2877619)

In the paper under review, the authors study the existence of solutions to \(p(x)\)-Laplacian equations, i.e., NEWLINE\[NEWLINE -\Delta_{p(x)}u+|u|^{p(x)-2}u=K(x)f(u)\text{ in }\mathbb R^N,NEWLINE\]NEWLINE where \(u\in W^{1,p(x)}(\mathbb R^{N})\), \(p(x)=p(|x|)\in C(\mathbb {R}^N)\) with NEWLINE\[NEWLINE2\leq N<\inf_{\mathbb {R}^N} p(x)\leq \sup_{\mathbb {R}^N} p(x)<\infty,NEWLINE\]NEWLINE \(K:{\mathbb {R}^N}\to \mathbb {R}\) is a measurable function and \(f\in C(\mathbb {R};\mathbb {R})\). The authors introduce a revised Ambrosetti-Rabinowitz condition, under which they show that the equation has a nontrivial solution and has infinitely many solutions if, further, \(f\) is an even function.