Existence and concentration of positive solutions for a critical <i>p</i> & <i>q</i> equation
Gustavo S. A. Costa, Giovany M. Figueiredo
Abstract
Abstract We show existence and concentration results for a class of p & q critical problems given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo>−</m:mo> <m:mi>d</m:mi> <m:mi>i</m:mi> <m:mi>v</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>a</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:msup> <m:mi>ϵ</m:mi> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi mathvariant="normal">∇</m:mi> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:msup> <m:mi>ϵ</m:mi> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi mathvariant="normal">∇</m:mi> <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>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">∇</m:mi> <m:mi>u</m:mi> </m:mrow> </m:mfenced> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>b</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:mrow> <m:mo>|</m:mo> </m:mrow> <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>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mi>q</m:mi> <m:mrow> <m:mo>⋆</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mspace width="thinmathspace"/> <m:mtext>in</m:mtext> <m:mspace width="thinmathspace"/> <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> $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+|u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ where u ∈ W 1, p (ℝ N ) ∩ W 1, q (ℝ N ), ϵ > 0 is a small parameter, 1 < p ≤ q < N , N ≥ 2 and q * = Nq /( N − q ). The potential V is positive and f is a superlinear function of C 1 class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino & Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.