On ground states asymptotics for boundary value problem for equation $-\varepsilon \Delta_p u = a|u|^{q-2}u - b|u|^{\gamma-2}u$

Abstract

We study a singularly perturbed Dirichlet boundary value problem for the equation $$ -\varepsilon \Delta_p u = a(x)|u|^{q-2}u - b(x)|u|^{\gamma-2}u, $$ where $\Delta_p$ is the $p$-Laplacian, $\Omega\subset\mathbb{R}^N$ is a bounded domain, $1<p<q<\gamma<p^*$,   $a,b\in L^\infty(\Omega)$, $a(x)\geq0$, $b(x)\geq\sigma_b>0$, and $\varepsilon>0$ is a small parameter.   Using nonlinear Rayleigh quotient method, we introduce two critical values of the parameter, $\varepsilon^*$ and $\varepsilon_e^*$,  associated respectively with the Nehari manifold and the zero energy level. We prove that for $\varepsilon>\varepsilon^*$ the problem has no nontrivial weak solutions, while for $0<\varepsilon<\varepsilon_e^*$ there exist at least two positive weak solutions; one of them is the ground state.   The main asymptotic result describes the behaviour of the ground states in the singular limit $\varepsilon\to0^+$. Under the additional condition $a(x)\geq\sigma_a>0$ we prove that each family of ground states $u_\varepsilon$ converges in measure in $\Omega$ to the explicitly given limit function $$ \bar u_0(x) = \left(\frac{a(x)}{b(x)}\right)^{\frac{1}{\gamma-q}}. $$ Moreover, $u_\varepsilon\to\bar u_0$ strongly in $L^r(\Omega)$, $1\leq r<\gamma$, and $u_\varepsilon\rightharpoonup\bar u_0$ weakly in $L^\gamma(\Omega)$.