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Note on elliptic equations on closed manifolds with singular nonlinearities

Bartosz Bieganowski, Adam Konysz

Abstract

We consider a general elliptic equation $$ -Δ_g u+V(x)u=f(x,u)+g(x,u^2)u $$ on a closed Riemannian manifold $(M, g)$ and utilize a recent variational approach by Hebey, Pacard, Pollack to show the existence of a nontrivial solution under general assumptions on nonlinear terms $f$ and $g$.

Note on elliptic equations on closed manifolds with singular nonlinearities

Abstract

We consider a general elliptic equation on a closed Riemannian manifold and utilize a recent variational approach by Hebey, Pacard, Pollack to show the existence of a nontrivial solution under general assumptions on nonlinear terms and .
Paper Structure (5 sections, 6 theorems, 62 equations)

This paper contains 5 sections, 6 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. The epsilon-perturbed problem and the Mountain Pass Theorem
  3. Regularity of solutions to epsilon-perturbed problem (\ref{['eq:eps']})
  4. Proof of Theorem \ref{['th:main']}
  5. Brezis-Kato result on a compact Riemannian manifold

Key Result

Theorem 1.2

Suppose that (F1)--(F3), (G1)--(G4), (V), (GF) are satisfied. Then, there exists a nontrivial, positive weak solution $u \in H^1 (M)$ of eq:einstein-lichnerowicz, namely for any $\varphi \in H^1(M)$, $\int_{M} g(x, u^2) \left| u \varphi \right| \, dv_g < \infty$ and

Theorems & Definitions (12)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • ...and 2 more