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Multi-bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

Jun Wang, Zhaoyang Yin

Abstract

In this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity \begin{equation*} -(\nabla+iA(x))^2u+λV(x)u =|u|^{q-2}u+u\log |u|^2,\ u\in H^1(\mathbb{R}^N,\mathbb{C}), \end{equation*} where the magnetic potential $A \in L_{l o c}^2\left(\mathbb{R}^N, \mathbb{R}^N\right)$, $2<q<2^*,\ λ>0$ is a parameter and the nonnegative continuous function $V: \mathbb{R}^N \rightarrow \mathbb{R}$ has the deepening potential well. Using the variational methods, we obtain that the equation has at least $2^k-1$ multi-bump solutions when $λ>0$ is large enough.

Multi-bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

Abstract

In this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity \begin{equation*} -(\nabla+iA(x))^2u+λV(x)u =|u|^{q-2}u+u\log |u|^2,\ u\in H^1(\mathbb{R}^N,\mathbb{C}), \end{equation*} where the magnetic potential , is a parameter and the nonnegative continuous function has the deepening potential well. Using the variational methods, we obtain that the equation has at least multi-bump solutions when is large enough.
Paper Structure (6 sections, 18 theorems, 208 equations)

This paper contains 6 sections, 18 theorems, 208 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary
  3. Auxiliary problem
  4. Minimax level
  5. Uniform estimate
  6. The proof of Theorem \ref{['t1.1']}

Key Result

Theorem 1.1

Assume that $N\geq1$ and $(V_1)-(V_3)$ hold. Then for any non-empty subset $\Gamma$ of $\{1,2, \ldots, k\}$, there exists $\lambda^*>0$ such that for all $\lambda \geqslant \lambda^*$, the problem eq1.1 has a nontrivial solution $u_\lambda$. Moreover, the family $\left\{u_\lambda\right\}_{\lambda \g where $\Omega_{\Gamma}=\bigcup\limits_{j \in \Gamma} \Omega_j$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 22 more