LAUSR

This paper deals with the existence solution for the nonlinear curl–curl Kirchhoff problem: $$\begin{aligned} M \left( \displaystyle \int \limits _{{{\mathrm{{\mathbb {R}}}}}^3} \left( \left| \nabla \times U \right| ^{2}+V(x)\left| U\right| ^{2} \right) dx \right) \left( \nabla \times \nabla \times U+V(x) U\right) = \Gamma (x) \left| U \right| ^{p-1} U,\quad {{\mathrm{{\mathbb {R}}}}}^3.\qquad (\hbox {I}) \end{aligned}$$M∫R3∇×U2+V(x)U2dx∇×∇×U+V(x)U=Γ(x)Up-1U,R3.(I)Our approach relies on the subspace (defocusing case) of $$ H({{\mathrm{curl}}},{{\mathrm{{\mathbb {R}}}}}^3) $$H(curl,R3) by variational method from mountain pass theorem, where $$ V(x) \in L^{\infty } ({{\mathrm{{\mathbb {R}}}}}^3),~V(x) \geqslant 1 $$V(x)∈L∞(R3),V(x)⩾1 (a.e.) and $$ p>1 $$p>1 and $$ M: {{\mathrm{{\mathbb {R}}}}}^+ \longrightarrow {{\mathrm{{\mathbb {R}}}}}^+ $$M:R+⟶R+ is a continuous and increasing function.