Multi-bump solutions for sublinear elliptic equations with nonsymmetric coefficients

TL;DR

This work addresses the existence of nonnegative multibump solutions to a sublinear elliptic equation with a nonsymmetric potential $K$, showing that if $\|K-1\|_{L^p_{\mathrm{loc}}}$ is small, one can construct solutions with arbitrarily many bumps and, via a limiting argument, with infinitely many bumps. The authors develop a constrained, region-wise stability framework around the compactly supported ground state of the limiting problem, using a truncated functional space to tightly control bump supports and their interactions. A finite-bump variational scheme on localized domains is complemented by a delicate analysis of stability and Lagrange multipliers as $K\to1$, enabling a passage from finite to infinite bump configurations. The resulting results reveal a robust, symmetry-free mechanism for multi-bump solutions and provide precise asymptotics for bump locations and supports, with potential implications for blow-up profiles in related sublinear problems.

Abstract

We investigate the existence of nonnegative bump solutions to the sublinear elliptic equation \[ \begin{cases} -Δv - K(x)v + |v|^{q-2}v = 0 & \text{in } \mathbb{R}^N, \\ v(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q \in (1,2)$, $ N \geq 2$, and the potential $K \in L^p_{\mathrm{loc}}(\mathbb{R}^N)$ with $p > N/2$ is a function without any symmetry assumptions. Under the condition that $\|K - 1\|_{L^p_{\mathrm{loc}}}$ is sufficiently small, we construct infinitely many solutions with arbitrarily many bumps. The construction is challenged by the sensitive interaction between bumps, whose limiting profiles have compact support. The key to ensuring their effective separation lies in obtaining sharp estimates of the support sets. Our method, based on a truncated functional space, provides precisely such control. We derive qualitative local stability estimates in region-wise maximum norms that govern the size of each bump's essential support, confining its core to a designated region and minimizing overlap. Crucially, these estimates are uniform in the number of bumps, which is the pivotal step in establishing the existence of solutions with infinitely many bumps.

Theorems & Definitions (64)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof