Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

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> -Δ u+V(x)u=(1|x|μ F(u))f(u)% in RN, 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<μ<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≥ 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(∞)=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)→ 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|→+∞ , we prove the existence of a nontrivial solution for the above equation by the penalization method.