We consider the constrained nonlinear Schrödinger problem in <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R}^{N}$</EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </math></EquationSource> <EquationSource Format="TEX">$N\ge 3$</EquationSource> </InlineEquation>) where <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo>,</mo> <mi>m</mi> <mo><</mo> <mn>2</mn> <mo>+</mo> <mfrac> <mn>4</mn> <mi>N</mi> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$2< p,m<2+\tfrac{4}{N}$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$2^{*}=\tfrac{2N}{N-2}$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>a</mi> <mo>,</mo> <mi>μ</mi> <mo>,</mo> <mi>κ</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$a,\mu ,\kappa >0$</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>γ</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\gamma >0$</EquationSource> </InlineEquation> is a parameter.We prove that there exists an explicit threshold <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mn>0</mn> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>μ</mi> <mo>,</mo> <mi>κ</mi> <mo>,</mo> <mi>N</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\gamma _{0}=\gamma _{0}(a,\mu ,\kappa ,N,p,m)>0$</EquationSource> </InlineEquation>, determined by the variational inequalities and Sobolev embedding constants, such that for every <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mn>0</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <msub> <mi>γ</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$0<\gamma <\gamma _{0}$</EquationSource> </InlineEquation>, problem <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>a</mi> </msub> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(P_{a})$</EquationSource> </InlineEquation> admits a positive normalized ground state <InlineEquation ID="IEq10"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\lambda ,u)\in \mathbb{R}\times H^{1}(\mathbb{R}^{N})$</EquationSource> </InlineEquation> with <InlineEquation ID="IEq11"> <EquationSource Format="MATHML"><math> <mi>λ</mi> <mo><</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\lambda <0$</EquationSource> </InlineEquation>.The proof is variational and relies on a careful analysis of the energy functional on the <InlineEquation ID="IEq12"> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$L^{2}$</EquationSource> </InlineEquation>-sphere, the construction of a Pohozaev manifold as a natural constraint, and concentration–compactness arguments adapted to handle the Sobolev-critical term. The double-subcritical structure (<InlineEquation ID="IEq13"> <EquationSource Format="MATHML"><math> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$|u|^{p-2}u$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="MATHML"><math> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>m</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$|u|^{m-2}u$</EquationSource> </InlineEquation>) competes with the critical term, and the smallness of γ ensures compactness of minimizing sequences.

Normalized ground states for a nonlinear Schrödinger equation with double critical nonlinearities

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2026年1月7日

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Salah Boulaaras,
Rafik Guefaifia

摘要

We consider the constrained nonlinear Schrödinger problem in $R^{N}$ $\mathbb{R}^{N}$ ( $N \geq 3$ $N\ge 3$ ) where $2 < p, m < 2 + \frac{4}{N}$ $2< p,m<2+\tfrac{4}{N}$ , $2^{*} = \frac{2 N}{N - 2}$ $2^{*}=\tfrac{2N}{N-2}$ , $a, μ, κ > 0$ $a,\mu ,\kappa >0$ , and $γ > 0$ $\gamma >0$ is a parameter.

We prove that there exists an explicit threshold $γ_{0} = γ_{0} (a, μ, κ, N, p, m) > 0$ $\gamma _{0}=\gamma _{0}(a,\mu ,\kappa ,N,p,m)>0$ , determined by the variational inequalities and Sobolev embedding constants, such that for every $0 < γ < γ_{0}$ $0<\gamma <\gamma _{0}$ , problem $(P_{a})$ $(P_{a})$ admits a positive normalized ground state $(λ, u) \in R \times H^{1} (R^{N})$ $(\lambda ,u)\in \mathbb{R}\times H^{1}(\mathbb{R}^{N})$ with $λ < 0$ $\lambda <0$ .

The proof is variational and relies on a careful analysis of the energy functional on the $L^{2}$ $L^{2}$ -sphere, the construction of a Pohozaev manifold as a natural constraint, and concentration–compactness arguments adapted to handle the Sobolev-critical term. The double-subcritical structure ( $| u |^{p - 2} u$ $|u|^{p-2}u$ and $| u |^{m - 2} u$ $|u|^{m-2}u$ ) competes with the critical term, and the smallness of γ ensures compactness of minimizing sequences.