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<p>We focus on the existence of the multi-bump solutions to the nonlinear fractional Schrödinger equation</p><p><Equation ID="Equa"> <EquationSource Format="TEX">\(h^{2s}(-\Delta)^sv-V(x)|v|^{2\sigma}v={-\lambda}v,\;\;\;x\in\mathbb{R}^N\)</EquationSource> </Equation></p><p>satisfying an <i>L</i><sup>2</sup>-constraint <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\|v \|^2_{L^2(\mathbb{R}^N)}=m \hbar^\alpha\)</EquationSource> </InlineEquation> in the <i>L</i><sup>2</sup>-subcritical case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma \in(0, {{2s}\over{{N}}})\)</EquationSource> </InlineEquation> and <i>L</i><sup>2</sup>-supcritical case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\sigma \in({{2s}\over{{N}}}, {{2_s^*}\over{{N}}})\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2_s^*={{2 N}\over{{(N-2s)^{+}}}}\)</EquationSource> </InlineEquation> is the usual critical Sobolev exponent, <i>ħ</i> is a small positive parameter, 0 &lt; <i>s</i> &lt; 1, (−Δ)<sup><i>s</i></sup> is the fractional Laplacian operator, <i>m</i> &gt; 0, <i>N</i> ≥ 1 and <i>V</i>(<i>x</i>) &gt; 0 admits several local maximum points. Here, λ ∈ ℝ arises as a Lagrange multiplier.</p><p>Combining the variational gluing arguments and a penalization technique, we can construct normalized multi-bump solutions concentrating at a finite set of local maximum points of <i>V</i>. A feature of this analysis is that it requires neither nondegeneracy assumptions on <i>V</i> nor uniqueness for the limit system. To the best of our knowledge, this paper is the first work with a variational approach dealing with the normalized multi-bump solutions for fractional Schrödinger equations.</p>

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Normalized multi-bump solutions of nonlinear fractional Schrödinger equations via variational approach

  • 期刊论文

  • 发表日期：

    2026年5月28日

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  • Zongyan Lv,
  • Zhongwei Tang,
  • Yichen Zhang

摘要

We focus on the existence of the multi-bump solutions to the nonlinear fractional Schrödinger equation

\(h^{2s}(-\Delta)^sv-V(x)|v|^{2\sigma}v={-\lambda}v,\;\;\;x\in\mathbb{R}^N\)

satisfying an L2-constraint \(\|v \|^2_{L^2(\mathbb{R}^N)}=m \hbar^\alpha\) in the L2-subcritical case \(\sigma \in(0, {{2s}\over{{N}}})\) and L2-supcritical case \(\sigma \in({{2s}\over{{N}}}, {{2_s^*}\over{{N}}})\) , where \(2_s^*={{2 N}\over{{(N-2s)^{+}}}}\) is the usual critical Sobolev exponent, ħ is a small positive parameter, 0 < s < 1, (−Δ)s is the fractional Laplacian operator, m > 0, N ≥ 1 and V(x) > 0 admits several local maximum points. Here, λ ∈ ℝ arises as a Lagrange multiplier.

Combining the variational gluing arguments and a penalization technique, we can construct normalized multi-bump solutions concentrating at a finite set of local maximum points of V. A feature of this analysis is that it requires neither nondegeneracy assumptions on V nor uniqueness for the limit system. To the best of our knowledge, this paper is the first work with a variational approach dealing with the normalized multi-bump solutions for fractional Schrödinger equations.