Conjecture on the pointwise decay of Euler–Lagrange solutions

Let $d$ be the dimension, let $p_0(d)$ be the exponent used in the generalized Euler–Lagrange equation, and let $\upsilon$ be the weight

$$\upsilon(x)=\min\big(1,|x'|^{-d},|x_d-|x'|^2|^{-d}\big).$$

Let $\lambda\in\mathbb C$ and let $f\in L^{p_0(d)}(\mathbb R^d)$ solve the generalized Euler–Lagrange equation. Decay conjecture. (i) There exists $C=C(f,\lambda)<\infty$ such that, for almost every $x\in\mathbb R^d$, $f(x)\leq C\upsilon(x)$. (ii) If $\lambda>0$, $f$ is nonnegative, and $\|f\|_{p_0(d)}>0$, then there exists $c=c(f,\lambda)>0$ such that, for almost every $x\in\mathbb R^d$, $f(x)\geq c\upsilon(x)$. The conjecture describes the expected sharp pointwise behavior of extremizers or Euler–Lagrange solutions; the supplied text does not establish its status or provide a proof.