Critical growth quasilinear elliptic problems with shifting subcritical perturbation
Let \(\Omega\subset \mathbb{R}^n\) be a bounded smooth domain. The following quasilinear critical growth problem with subcritical nonautonomous perturbations is investigated: \[ -\Delta_p u= g(x)[(u- k)^+]^{q- 1}+ u^{p^*- 1},\quad u> 0,\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega.\tag{P\(_k\)} \] Here, \(-\Delta_p u= \text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplace operator, \(1< p< n\), \(p^*= np/(n- p)\) is the critical Sobolev exponent, \(g\) is a positive function satisfying some integrability and smallness conditions and \(k> 0\) is a parameter. The admissible range for the subcritical parameter depends on whether or not \(n\geq p^2\). The formal limiting problem for \(k\to\infty\) is the Dirichlet problem for the purely critical equation \[ -\Delta_p u= u^{p^*-1},\quad u> 0,\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega,\tag{P\(_\infty\)} \] and it is expected that the limiting behaviour of solutions of \((\text{P}_k)\) for \(k\to\infty\) is governed by \((\text{P}_k)\) Besides other interesting topics, this limit is discussed rigorously in the present paper. In particular, it is shown that \((\text{P}_\infty)\) has no mountain pass solution, while all problems \((\text{P}_k)\) have; and the concentration behaviour of the latter for \(k\to\infty\) is described precisely.
scientific article; zbMATH DE number 1832829