Sharp Blow-Up Profiles of Positive Solutions for a Class of Semilinear Elliptic Problems

This paper analyzes the behavior of the positive solution θ ε \theta_{\varepsilon} of the perturbed problem { - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - [ a ε ⁢ ( x ) + b ε ⁢ ( x ) ] ⁢ u p = 0 in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} {}{-\Delta u}&=\lambda m(x)u-[a_{\varepsilon}(x)+b_{\varepsilon}(x)]u^{p}=0&&\text{in}\ \Omega,\\ Bu&=0&&\text{on}\ \partial\Omega,\end{aligned}\right. as ε ↓ 0 \varepsilon\downarrow 0 , where a ε ⁢ ( x ) ≈ ε α ⁢ a ⁢ ( x ) a_{\varepsilon}(x)\approx\varepsilon^{\alpha}a(x) and b ε ⁢ ( x ) ≈ ε β ⁢ b ⁢ ( x ) b_{\varepsilon}(x)\approx\varepsilon^{\beta}b(x) for some α ≥ 0 \alpha\geq 0 and β ≥ 0 \beta\geq 0 , and some Hölder continuous functions a ⁢ ( x ) a(x) and b ⁢ ( x ) b(x) such that a ⪈ 0 a\gneq 0 (i.e., a ≥ 0 a\geq 0 and a ≢ 0 a\not\equiv 0 ) and min Ω ¯ ⁡ b > 0 \min_{\bar{\Omega}}b>0 . The most intriguing and interesting case arises when a ⁢ ( x ) a(x) degenerates, in the sense that Ω 0 ≡ int ⁡ a - 1 ⁢ ( 0 ) \Omega_{0}\equiv\operatorname{int}a^{-1}(0) is a non-empty smooth open subdomain of Ω, as in this case a “blow-up” phenomenon appears due to the spatial degeneracy of a ⁢ ( x ) a(x) for sufficiently large 𝜆. In all these cases, the asymptotic behavior of θ ε \theta_{\varepsilon} will be characterized according to the several admissible values of the parameters 𝛼 and 𝛽. Our study reveals that there may exist two different blow-up speeds for θ ε \theta_{\varepsilon} in the degenerate case.