We study concavity properties of positive solutions to the Logarithmic Schrödinger equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$-\Delta u=u\, \log u^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mspace width="0.166667em" /> <mo>log</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane–Emden problems <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$-\Delta u = \sigma \, (u^q-u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>σ</mi> <mspace width="0.166667em" /> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mi>q</mi> </msup> <mo>-</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and build, for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, solutions <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u_q^{(1-q)/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>u</mi> <mi>q</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation> is convex. By choosing <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma _q=2/({q-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>q</mi> </msub> <mo>=</mo> <mn>2</mn> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>q</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and letting <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$q \rightarrow 1^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> we eventually construct a solution u of the Logarithmic Schrödinger equation such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\log u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> is concave. This seems one of the few attempts in studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.

Power law convergence and concavity for the logarithmic Schrödinger equation

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2026年4月7日

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Marco Gallo,
Sunra Mosconi,
Marco Squassina

摘要

We study concavity properties of positive solutions to the Logarithmic Schrödinger equation $-\Delta u=u\, \log u^2$ $- Δ u = u log u^{2}$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane–Emden problems $-\Delta u = \sigma \, (u^q-u)$ $- Δ u = σ (u^{q} - u)$ and build, for any $\sigma >0$ $σ > 0$ and $q>1$ $q > 1$ , solutions $u_q$ $u_{q}$ such that $u_q^{(1-q)/2}$ $u_{q}^{(1 - q) / 2}$ is convex. By choosing $\sigma _q=2/({q-1})$ $σ_{q} = 2 / (q - 1)$ and letting $q \rightarrow 1^+$ $q \to 1^{+}$ we eventually construct a solution u of the Logarithmic Schrödinger equation such that $\log u$ $log u$ is concave. This seems one of the few attempts in studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.