Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. III.
This is a third part of a series of very interesting papers [Part I, \textit{F. Gazzola} and \textit{J. Serrin}, Ann. Inst. H. Poincaré, Anal. Non Lineaire 19, 477--504 (2002; Zbl 1013.35031); Part II: \textit{F. Gazzola}, \textit{L. A. Peletier}, \textit{P. Pucci}, \textit{J. Serrin}, Ann. Inst. H. Poincaré, Anal. Non Linéaire 20, 947--974 (2003; Zbl 1086.35041)] devoted to the study of asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. Precisely, set \(\Delta_mu=\text{ div\,}(| \nabla u| ^{m-2}\nabla u)\) for the \(m\)-Laplacian, and let \(m^*=nm/(n-m)\) be the critical Sobolev exponent for the embedding \(D^{1,m}({\mathbb R}^n)\subset L^{m^*}({\mathbb R}^n)\). Consider the quasilinear degenerate elliptic equation \[ -\Delta_mu=-\delta u^{q-1}+u^{p-1}\quad \text{ in}\;{\mathbb R}^n \eqno{(1)} \] with \(\delta>0,\) \(n>m>1\) and \(1<q<p<m^*\). A ground state of \((1)\) is defined as a \(C^1({\mathbb R}^n)\) nonnegative nontrivial radial distribution solution which vanishes at infinity. The equation \((1)\) admits a unique ground state for all \(\delta,p,q\) in the given range while, if either \(p=m^*\) and \(\delta>0\) or \(\delta=0\) and \(p\in(q,m^*)\) then there are no ground states of \((1)\). Finally, if both \(\delta=0\) and \(p=m^*\) then \((1)\) admits an one-parameter family of ground states given by the formula \[ U_d(x)=d\left(1+D\left(d^{m/(n-m)}| x| \right)^{m/(m-1)}\right)^{(m-n)/m},\quad d>0 \eqno{(2)} \] where \(D=D_{n,m}=(m-1)/(n-m)n^{1/(m-1)}\) and \(U_d(0)=d.\) The main goal of the paper under review is the study of behavior of ground states in the borderline cases when \(p\uparrow m^*\) and/or \(\delta\downarrow 0.\) Also \(q=m\) is a limit case because, as proven in [\textit{B. Franchi}, \textit{E. Lanconelli} and \textit{J. Serrin}, Adv. Math. 118, 177--243 (1996; Zbl 0853.35035)], if \(q<m\) then the ground state of \((1)\) has compact support while if \(q\geq m\) it remains positive in \({\mathbb R}^n.\) As shown in Parts I and III, the case \(q=m\) presents a new effect related to discontinuous dependence of the asymptotic behavior on the parameters \(m\) and \(n.\) Considering \(q\) as a free parameter, the authors succeed to interpret this discontinuous dependence in terms of two critical exponents: \(m_R={{n(m-1)}\over {n-m}}\) (known as critical remainder exponent) and \(m_*={{m(n-1)}\over {n-m}}\) (known as Serrin's exponent) which satisfy \(1<m_R<m_*<m^*\). The main results of the paper can be summarized as follows. \(\bullet\) The Serrin exponent \(m_*\) is a borderline between existence and nonexistence for the ``coercive'' problem \(\Delta_mW=W^{q-1}.\) Precisely, this equation possesses a unique nonnegative radial solution on \({\mathbb R}^n\setminus\{0\}\) blowing-up at the origin like the fundamental solution if and only if \(q<m_*\). \(\bullet\) Maintaining \(\delta>0\) fixed and letting \(p\uparrow m^*,\) it is shown that the ground state \(u\) of \((1)\) converges to a Dirac measure with mass at the origin and \(u(0)\) blows up with different rates when \(q>m_R,\) \(q=m_R\) and \(q<m_R\). Moreover, since the ground state is of compact support (a ball) when \(q<m,\) it is proved that the radius of the ball vanishes as \(p\uparrow m^*\) exhibiting this way a concentration phenomenon. \(\bullet\) If \(p\) is fixed and \(\delta\downarrow0\), the authors prove \(u\to0\) uniformly in \({\mathbb R}^n\) and determine the precise rate of convergence. In the particular case \(q<m\), it is shown that the radius of the ball supporting the ground state diverges to infinity, which is to say that the ground state spreads out as \(\delta\downarrow0\). \(\bullet\) If both \(\delta\downarrow0\) and \(p\uparrow m^*\), maintaining the ``equilibrium'' \(\delta=d(m^*-p)\) then the ground state does not concentrate nor spread out, and it converges uniformly on \({\mathbb R}^n\) to one of the functions in (2).
degenerate quasilinear elliptic operators
scientific article; zbMATH DE number 2062715