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

Abstract

Abstract 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} \displaystyle{}{-\Delta u}&\displaystyle=\lambda m(x)u-% [a_{\varepsilon}(x)+b_{\varepsilon}(x)]u^{p}=0&&\displaystyle\text{in}\ \Omega% ,\\ \displaystyle Bu&\displaystyle=0&&\displaystyle\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.