<i>p</i> -Laplacian Equations in ℝ ^<i>N</i> with Finite Potential via the Truncation Method

Abstract

Abstract We consider the problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo fence="true" 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:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:math> {-\Delta_{p}u+a(x)\lvert u\rvert^{p-2}u=\lvert u\rvert^{q-2}u} in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {\mathbb{R}^{N}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {1&lt;p&lt;N} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi>q</m:mi> <m:mo>&lt;</m:mo> <m:msup> <m:mi>p</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>N</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>-</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:math> {p&lt;q&lt;p^{*}=\frac{Np}{N-p}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\Delta_{p}} is the p -Laplacian operator, and the potential function a is positive, bounded and verifies suitable decay assumptions. The existence of infinitely many solutions of the equation is proved via the truncation method.