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

Abstract

We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, \[ -\varepsilon Δ_p u = a(x)|u|^{q-2}u - b(x)|u|^{γ-2}u, \qquad u|_{\partialΩ}=0, \] 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 $\Omega$ to the explicit profile \[ \bar u_0(x) = \left(\frac{a(x)}{b(x)}\right)^{1/(γ-q)}. \] Moreover, \[ u_\varepsilon\to\bar u_0 \quad\text{strongly in }L^r(Ω), \qquad 1\le r<γ, \] and \[ u_\varepsilon\rightharpoonup\bar u_0 \quad\text{weakly in }L^r(Ω), \qquad 1<r\leγ. \]

28 pages, 1 figure