Normalized solutions for the <i>p</i> -Laplacian equation with a trapping potential
Chao Wang, Juntao Sun
Abstract
Abstract In this article, we are concerned with normalized solutions for the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -Laplacian equation with a trapping potential and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msup> </m:math> {L}^{r} -supercritical growth, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mi>p</m:mi> </m:math> r=p or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>2</m:mn> <m:mo>.</m:mo> </m:math> 2. The solutions correspond to critical points of the underlying energy functional subject to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msup> </m:math> {L}^{r} -norm constraint, namely, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <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:mrow> </m:msub> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:mi>c</m:mi> </m:math> {\int }_{{{\mathbb{R}}}^{N}}| u{| }^{r}{\rm{d}}x=c for given <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>c</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> <m:mo>.</m:mo> </m:math> c\gt 0. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> </m:math> r=p, we show that such problem has a ground state with positive energy for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>c</m:mi> </m:math> c small enough. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> </m:math> r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.