A multiplicity result for double phase problem in the whole space

Abstract

<abstract><p>In the present paper, we discuss the solutions of the following double phase problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\rm div}(|\nabla u|^{^{p-2}}\nabla u+ \mu(x) |\nabla u|^{^{q-2}}\nabla u)+ |u|^{^{p-2}}u+\mu(x)|u|^{^{q-2}}u = f(x, u), \;x\in \mathbb{R}^N, $\end{document} </tex-math></disp-formula></p> <p>where $ N \geq2 $, $ 1 < p < q < N $ and $ 0\leq\mu\in C^{^{0, \alpha}}(\mathbb{R}^N), \; \alpha\in(0, 1] $. Based on the theory of the double phase Sobolev spaces $ W^{^{1, H}}(\mathbb{R}^N) $, we prove the existence of at least two non-trivial weak solutions.</p></abstract>