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

Abstract

We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, \[ -\varepsilon \Delta_p u = a(x)|u|^{q-2}u - b(x)|u|^{\gamma-2}u, \qquad u|_{\partial\Omega}=0, \] where $1 0$, and $\varepsilon>0$ is small. By means of the nonlinear Rayleigh quotient method, we introduce two critical parameter values, $\varepsilon^*$ and $\varepsilon_e^*$, related respectively to the Nehari manifold and to the zero energy level. We prove the nonexistence of nontrivial weak solutions for $\varepsilon>\varepsilon^*$, and the existence of at least two positive weak solutions for $0<\varepsilon<\varepsilon_e^*$; one of them is a ground state. The main result describes the asymptotic behaviour of ground states as $\varepsilon\to 0^+$. If, in addition, $a\geq\sigma_a>0$, then every family of positive ground states $u_\varepsilon$ converges in measure in $\Omega$ to the explicit profile \[ \bar u_0(x) = \left(\frac{a(x)}{b(x)}\right)^{1/(\gamma-q)}. \] Moreover, \[ u_\varepsilon\to\bar u_0 \quad\text{strongly in }L^r(\Omega), \qquad 1\le r<\gamma, \] and \[ u_\varepsilon\rightharpoonup\bar u_0 \quad\text{weakly in }L^r(\Omega), \qquad 1<r\le\gamma . \]