This article concerns the nonhomogeneous Klein-Gordon equation coupled with a Born-Infeld type equation,$$\displaylines{- \Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u)+h(x), \quad x\in \mathbb{R}^3,\cr \Delta \phi+\beta\Delta_4\phi=4\pi(\omega+\phi)u^2, \quad x\in \mathbb{R}^3, }$$ where \(\omega\)  is a positive constant. We obtain the existence of two solutions using the Mountain Pass Theorem, and the Ekeland's variational principle in critical point theory.

Since Pb is a non-biodegradable inorganic pollutant and a non-essential metal, its long-term presence in soil poses a great threat to the environment. Iris lactea Pall. var. chinensis (Fisch.) Koidz., a perennial dense bush herb with high resistance of Pb and wide adaptability, was used in pot experiments to study the effects of exogenous nitrate N (NO -3-N) on the absorption and transportation of Pb and plant growth under different Pb concentrations. Then, the mechanism of NO -3-N affecting Pb and nutrient uptake and transport was explored. The concentration of Pb in the experiment ranged from 0 to 1600 mg/kg, and the added concentration of NO -3-N was 0.0-0.3 g/kg. The results showed that I. lactea was highly tolerant to Pb, and the shoot fraction was more sensitive to varied Pb concentrations in the soil than the root fraction. This protective function became more pronounced under the condition of raised Pb concentration in the soil. When the concentration of Pb in the soil reached 800 mg/kg, the highest Pb content of I. lactea was found under the condition of 0.1 g/kg of NO -3 -N addition. When Pb concentration in the soil increased to 1600 mg/kg, the increase in NO -3-N addition promoted Pb uptake by the root. To ensure the well growth of I. lactea and the effect of remediation of Pb-contaminated soil, the recommended concentration of NO -3-N in the soil is 0.1 g/kg. This result provides a theoretical basis for exogenous N regulation of phytoremediation of Pb-contaminated soil.

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In this paper, we study the following nonhomogeneous Klein-Gordon equation with Born-Infeld theory −∆u + λV (x)u − K(x)(2ω + φ)φu = f (x, u) + h(x), x ∈ R 3 , ∆φ + β∆4φ = 4πK(x)(ω + φ)u 2 , x ∈ R 3 , where ω > 0 is a constant, λ > 0 is a parameter and 4φ = div(||φ| 2 φ). Under some suitable assumptions on V, K, f and h, the existence of multiple solutions is proved by using the Linking theorem and the Ekeland’s variational principle in critical point theory. Especially, the potential V is allowed to be sign-changing. Mathematics Subject Classification (2010). Primary 35B33; Secondary 35J65.