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The existence of solutions of Schrödinger equations with essence resonance

Chong Li

Abstract

The current paper investigates a class of asymptotically linear Schrodinger equations. The Palais-Smale condition fails to hold in this case. Especially under the hypothesis (V2), the lack of compactness occurs at the interaction between nonlinear term and continuum spectrum. For this reason, we introduce a bootstrap iteration approach for elliptic equation on RN. The iteration is self-contained and can be regarded as a generalization of Agmon-Douglis-Nirenberg theorem. The proof characterizes iteration steps independent of the choice of the parameter, which are indeed manipulated by intrinsic natures of potentials and nonlinear terms, and furthermore presents precise estimates for asymptotically linear functions or continuous nonlinear terms restricted on a bounded domain in RN. Additionally, a comparison theorem for the spectrum of Schrodinger operator is also established in this paper. With above preparations, we can get a nontrivial solution without mountain pass geometry, and more importantly make an explicit description of nondegeneracy of solutions with monotonicity hypothesis.

The existence of solutions of Schrödinger equations with essence resonance

Abstract

The current paper investigates a class of asymptotically linear Schrodinger equations. The Palais-Smale condition fails to hold in this case. Especially under the hypothesis (V2), the lack of compactness occurs at the interaction between nonlinear term and continuum spectrum. For this reason, we introduce a bootstrap iteration approach for elliptic equation on RN. The iteration is self-contained and can be regarded as a generalization of Agmon-Douglis-Nirenberg theorem. The proof characterizes iteration steps independent of the choice of the parameter, which are indeed manipulated by intrinsic natures of potentials and nonlinear terms, and furthermore presents precise estimates for asymptotically linear functions or continuous nonlinear terms restricted on a bounded domain in RN. Additionally, a comparison theorem for the spectrum of Schrodinger operator is also established in this paper. With above preparations, we can get a nontrivial solution without mountain pass geometry, and more importantly make an explicit description of nondegeneracy of solutions with monotonicity hypothesis.
Paper Structure (6 sections, 11 theorems, 204 equations)

This paper contains 6 sections, 11 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. The global regularity of solutions of Schrödinger equations with Kato-Rellich potentials and a generalization of Agmon-Douglis-Nirenberg theorem
  3. Compactness for the nonresonance case
  4. A comparison theorem for the spectrum of Schrödinger operator
  5. Proof of Theorem1.1
  6. Appendix

Key Result

Theorem 1.1

Let $V$ be a K-R potential and suppose $N\neq 2$ for the case $V_1\neq 0$. Given $\lambda =\sigma _0$, under the hypotheses $\left( V_1\right) \left( V_2\right) \left( g_1\right)$-$\left( g_7\right)$, if $\sigma _0+g_0\leq \mu _i<\mu _k$ or $\sigma _0+g_0<\mu _i$, then $\left( eq1\right)$ possesses

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 4.1
  • Theorem 4.2
  • Proposition 4.3
  • Example 5.1
  • ...and 2 more