On semilinear Grushin--Schr\"odinger equation in $\mathbb{R}^N$

TL;DR

This paper proves the existence of nonnegative solutions to a semilinear Schrödinger equation involving the Grushin operator in a space with variable potentials, and establishes their regularity properties.

Contribution

It introduces new existence results for solutions to a Grushin--Schrodinger equation with variable potentials and analyzes their regularity.

Findings

Existence of nontrivial nonnegative weak solutions.

Embedding of the solution space into weighted Lebesgue spaces.

Regularity results for solutions.

Abstract

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -\Delta_\gamma u + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $\Delta_\gamma $ denotes the so-called Grushin-type operator in $\mathbb{R}^N$. The potentials $V$ and $Q$ are assumed to be controlled below and above, respectively, by functions of type $(1+|z|)^a$, $a\in\mathbb{R}$. The main result is the embedded of the space $E_V^\gamma$ into the weighted Lebesgue space $L_Q^p(\mathbb{R}^N)$, under suitable conditions. Finally, we derive regularity results for the obtained weak solutions.