Schrödinger–Newton equations in dimension two via a Pohozaev–Trudinger log-weighted inequality

Abstract

<jats:p>We study the following Choquard type equation in the whole plane <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} (C)\\quad -\\Delta u+V(x)u=(I_2*F(x,u))f(x,u),\\quad x\\in \\mathbb {R}^2 \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mtable>\n <mml:mtr>\n <mml:mtd>\n <mml:mrow>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>C</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mspace />\n <mml:mo>-</mml:mo>\n <mml:mi>Δ</mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mi>V</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mi>u</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mrow />\n <mml:mo>∗</mml:mo>\n <mml:mi>F</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>,</mml:mo>\n <mml:mspace />\n <mml:mi>x</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:msup>\n <mml:mrow>\n <mml:mi>R</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msup>\n </mml:mrow>\n </mml:mtd>\n </mml:mtr>\n </mml:mtable>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:disp-formula>where <jats:inline-formula><jats:alternatives><jats:tex-math>$$I_2$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n </mml:math></jats:alternatives></jats:inline-formula> is the Newton logarithmic kernel, <jats:italic>V</jats:italic> is a bounded Schrödinger potential and the nonlinearity <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>,\xa0<jats:italic>u</jats:italic>), whose primitive in <jats:italic>u</jats:italic> vanishing at zero is <jats:italic>F</jats:italic>(<jats:italic>x</jats:italic>,\xa0<jats:italic>u</jats:italic>), exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (<jats:italic>C</jats:italic>).</jats:p>