<p>We study the existence and multiplicity of positive solutions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^1(\mathbb {R}^N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, with prescribed <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm, for the (stationary) nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this paper, we consider the same problem, in presence of a weakly attractive, possibly irregular, potential, wondering (i) whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and (ii) if the existence of a mountain-pass solution persists. We provide positive answers, depending on suitable assumptions on the potential and on the mass value. Moreover, by the Hopf-Cole transform, we give some applications of our results to the existence of multiple solutions to ergodic Mean Field Games systems with potential and quadratic Hamiltonian.</p>

Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case

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2026年2月17日

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Gianmaria Verzini,
Junwei Yu

摘要

We study the existence and multiplicity of positive solutions in $H^1(\mathbb {R}^N)$ $H^{1} (R^{N})$ , $N\ge 3$ $N \geq 3$ , with prescribed $L^2$ $L^{2}$ -norm, for the (stationary) nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$ $L^{2}$ -sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this paper, we consider the same problem, in presence of a weakly attractive, possibly irregular, potential, wondering (i) whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and (ii) if the existence of a mountain-pass solution persists. We provide positive answers, depending on suitable assumptions on the potential and on the mass value. Moreover, by the Hopf-Cole transform, we give some applications of our results to the existence of multiple solutions to ergodic Mean Field Games systems with potential and quadratic Hamiltonian.