Type: Article
Publication Date: 2015-12-17
Citations: 52
DOI: https://doi.org/10.1515/anona-2015-0123
Abstract We study the following nonlinear Choquard equation: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>Δ</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:msup> <m:mrow> <m:mo>|</m:mo> <m:mi>x</m:mi> <m:mo>|</m:mo> </m:mrow> <m:mi>μ</m:mi> </m:msup> </m:mfrac> <m:mo>∗</m:mo> <m:mi>F</m:mi> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>f</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo> </m:mo> <m:mrow> <m:mtext>in </m:mtext> <m:mo></m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> $-\Delta u+V(x)u=\biggl{(}\frac{1}{|x|^{\mu}}\ast F(u)\biggr{)}f(u)\quad\text{% in }\mathbb{R}^{N},$ where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>μ</m:mi> <m:mo><</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> ${0<\mu<N}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> ${N\geq 3}$ , V is a continuous real function and F is the primitive function of f . Under some suitable assumptions on the potential V , which include the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>V</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>∞</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${V(\infty)=0}$ , that is, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>V</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>→</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${V(x)\to 0}$ as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>x</m:mi> <m:mo>|</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:mrow> </m:math> ${|x|\to+\infty}$ , we prove the existence of a nontrivial solution for the above equation by the penalization method.