Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
Yiqing Li, Binlin Zhang, Xiumei Han
Abstract
Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> <m:mo>∗</m:mo> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mspace width="1em"/> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> -{\varepsilon }^{2}\Delta u+V\left(x)u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }\ast | u{| }^{2})u=\lambda | u{| }^{p-1}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> \varepsilon ,\lambda \gt 0 are parameters, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>⩾</m:mo> <m:mn>2</m:mn> </m:math> N\geqslant 2 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>+</m:mo> <m:mn>6</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mspace width="0.1em"/> <m:mtext>/</m:mtext> <m:mspace width="0.1em"/> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:math> \left(\alpha +6)\hspace{0.1em}\text{/}\hspace{0.1em}\left(\alpha +2)\lt p\lt {2}^{\ast }-1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> {I}_{\alpha } is the Riesz potential with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mi>N</m:mi> </m:math> 0\lt \alpha \lt N , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> <m:mo>∈</m:mo> <m:mi class="MJX-tex-caligraphic" mathvariant="script">C</m:mi> <m:mrow>