Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $u_1(0) = u_1(1),u_2(0) = u_2(1)$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \lambda_2|u_2|u_2$$ $\lambda_1\ne\lambda_2,\lambda_1,\lambda_2 \in (0,\infty)$ . I need to prove/disprove that $\int\limits_0^1|u_1|u_1u_2dx + \int\limits_0^1|u_2|u_2u_1dx =0 $

Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $$u_1(0) = u_1(1),u_1'(0)= u_1'(1)$$ and $$u_2(0) = u_2(1),u_2'(0)=u_2'(1)$$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \lambda_2|u_2|u_2$$ $\lambda_1\ne\lambda_2,\lambda_1,\lambda_2 \in (0,\infty)$ . I need to prove/disprove that $\int\limits_0^1|u_1|u_1u_2dx + \int\limits_0^1|u_2|u_2u_1dx =0 $

Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $u_1(0) = u_1(1),u_2(0) = u_2(1)$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \lambda_2|u_2|u_2$$ $\lambda_1\ne\lambda_2,\lambda_1,\lambda_2 \in (0,\infty)$ . I need to prove/disprove that $\int\limits_0^1u_1^2u_2 + \int\limits_0^1u_2^2u_1 =0 $

Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $u_1(0) = u_1(1),u_2(0) = u_2(1)$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \lambda_2|u_2|u_2$$ $\lambda_1\ne\lambda_2,\lambda_1,\lambda_2 \in (0,\infty)$ . I need to prove/disprove that $\int\limits_0^1|u_1|u_1u_2dx + \int\limits_0^1|u_2|u_2u_1dx =0 $