Past and recent contributions to indefinite sublinear elliptic problems

Abstract

We review the indefinite sublinear elliptic equation $-Δu=a(x)u^{q}$ in a smooth bounded domain $Ω\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in sufficient and necessary conditions on $a$ and $q$ for the existence of positive solutions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.

Figures (4)

• Figure 1: (i) The indefinite weight $a_{\frac{1}{2}}$; (ii) $\mathcal{S}(a_{\frac{1}{2}})$; (iii) The positive solution $u\not \gg 0$ for $a_{\frac{1}{2}}$.
• Figure 2: The curve of positive solutions emanating from $(0,\mathcal{S}(a))$: Cases (i) $\mu_{\mathcal{D}}(a)=1$, (ii) $\mu_{\mathcal{D}}(a)>1$, (iii) $\mu _{\mathcal{D}}(a)<1$.
• Figure 3: Bifurcating solutions $u\gg0$ (i) from $\Gamma_{1}$ at $\left( 1, t_{\mathcal{N}}^{\ast }\, \phi_{\mathcal{N}} \right)$ in case $\mu_{\mathcal{N}}(a)=1$; (ii) from zero in case $\mu_{\mathcal{N}}(a)>1$; (iii) from infinity in case $\mu_{\mathcal{N}}(a)<1$.
• Figure 4: The bifurcation curve of the unique nontrivial solution in the case $\mu_{\mathcal{D}}(a)=1$, assuming that $\Omega_{+}$ is connected, satisfies (A.2), and includes a tubular neighborhood of $\partial\Omega$. Here the full curve represents $u_{\mathcal{D}}(q)\gg0$, whereas the dotted curve represents solutions vanishing somewhere in $\Omega$.

Theorems & Definitions (26)

• Theorem 1
• Corollary 2
• Remark 2.1
• Example 3
• Theorem 4
• Remark 2.2
• Theorem 5
• Remark 2.3
• Theorem 4.1
• Theorem 4.2