On the asymptotics of ground states for a boundary value problem for the equation $-\epsilon {\mathrm{\Delta }}_{p}u=a\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u-b\mathrm{\mid }u{\mathrm{\mid }}^{\gamma -2}u$

We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, $$-\epsilon {\mathrm{\Delta }}_{p}u=a(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u-b(x)\mathrm{\mid }u{\mathrm{\mid }}^{\gamma -2}u,u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0,$$ where $1<p<q<\gamma <p^{\ast }$, $a\ge 0$, $b\ge {\sigma }_b>0$, and $\epsilon >0$ is small. By means of the nonlinear Rayleigh quotient method, we introduce two critical parameter values, ${\epsilon }^{\ast }$ and ${\epsilon }_e^{\ast }$, related respectively to the Nehari manifold and to the zero energy level. We prove the nonexistence of nontrivial weak solutions for $\epsilon >{\epsilon }^{\ast }$, and the existence of at least two positive weak solutions for $0<\epsilon <{\epsilon }_e^{\ast }$; one of them is a ground state. The main result describes the asymptotic behaviour of ground states as $\epsilon \to 0^{+}$. If, in addition, $a\ge {\sigma }_a>0$, then every family of positive ground states $u_{\epsilon }$ converges in measure in $\mathrm{\Omega }$ to the explicit profile $${\bar{u}}_0(x)={\left(\frac{a(x)}{b(x)}\right)}^{1\mathrm{/}(\gamma -q)}\mathrm{.}$$ Moreover, $$u_{\epsilon }\to {\bar{u}}_0\text{strongly in }{L}^r(\mathrm{\Omega }),1\le r<\gamma ,$$ and $$u_{\epsilon }\rightharpoonup {\bar{u}}_0\text{weakly in }{L}^r(\mathrm{\Omega }),1<r\le \gamma \mathrm{.}$$