Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities

Abstract

Abstract In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <m:mtr> <m:mtd> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>y</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mi>μ</m:mi> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mtd> <m:mtd> <m:mtext> in </m:mtext> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mtd> <m:mtd> <m:mtext> on </m:mtext> <m:mi mathvariant="normal">∂</m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> <m:mo fence="true" stretchy="true"/> </m:mrow> </m:math> \begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u &amp; \text { in } \Omega, \\ u=0 &amp; \text { on } \partial \Omega,\end{cases} where Ω is an open bounded domain in ℝ N , which contains some points (0, z *), <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mi>μ</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>q</m:mi> <m:mo>&lt;</m:mo> <m:mn>2</m:mn> <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 stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mo stretchy="false">(</m:mo> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:math> \mu&gt;0,1&amp;#x003C;q&amp;#x003C;2,2^*(t)=\frac{2(N-t)}{N-2} , 0 ≤ t &lt; 2, x = ( y , z ) ∈ ℝ k × ℝ N−k , 2 ≤ k ≤ N . We prove that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mi>N</m:mi> <m:mo>&gt;</m:mo> <m:mn>2</m:mn> <m:mfrac> <m:mrow> <m:mi>q</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> <m:mo>+</m:mo>