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On semilinear Grushin--Schrödinger equation in $\mathbb{R}^N$

Jônison Carvalho, Arlúcio Viana

Abstract

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -Δ_γu + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $Δ_γ$ 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 $E_V^γ$ into the weighted Lebesgue space $L_Q^p(\mathbb{R}^N)$, under suitable conditions. Finally, we derive regularity results for the obtained weak solutions.

On semilinear Grushin--Schrödinger equation in $\mathbb{R}^N$

Abstract

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -Δ_γu + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where denotes the so-called Grushin-type operator in . The potentials and are assumed to be controlled below and above, respectively, by functions of type , . The main result is the embedded of the space into the weighted Lebesgue space , under suitable conditions. Finally, we derive regularity results for the obtained weak solutions.
Paper Structure (6 sections, 8 theorems, 109 equations)

This paper contains 6 sections, 8 theorems, 109 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['imersao']}
  3. Proof of Theorem \ref{['existencia_de_solucao']}
  4. Variational structure
  5. Weak nonnegative solution for \ref{['P']}
  6. Regularity

Key Result

Theorem 1.1

Assume that V and Q hold. Then the embedding $E_V^\gamma \hookrightarrow L^p_Q(\mathbb{R}^N)$ is continuous if one of the following conditions holds: Furthermore, the embedding is compact whenever:

Theorems & Definitions (16)

  • Theorem 1.1: Weighted Sobolev Embedding
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['imersao']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 6 more