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 $1<p<q<\gamma<p^*$, $a\geq 0$, $b\geq\sigma_b>0$, 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 \[ u_\varepsilon\rightharpoonup\bar u_0 \quad\text{weakly in }L^r(\Omega), \qquad 1<r\le\gamma . \]
| Comments: | 28 pages, 1 figure |
| Subjects: | Analysis of PDEs (math.AP); Mathematical Physics (math-ph) |
| Cite as: | arXiv:2605.24482 [math.AP] |
| (or arXiv:2605.24482v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24482 arXiv-issued DOI via DataCite (pending registration) |
From: Yavdat Il'yasov [view email]
[v1]
Sat, 23 May 2026 09:11:02 UTC (23 KB)
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