This paper deals with the semilinear elliptic problem

where \(p>1\), \(\lambda >0\), \(m,a,b\in C({\bar{\Omega }})\), with \(a \gneq 0\), \(b\gneq 0\), \(\Omega \) is a bounded \(C^{2}\) domain of \({\mathbb {R}}^N\) (\(N\ge 1\)), B is a general classical mixed boundary operator, and \(\varepsilon \ge 0\). Thus, a(x) and b(x) can vanish on some subdomain of \(\Omega \) and the weight function m(x) can change sign in \(\Omega \). Through this paper we are always considering classical solutions. First, we characterize the existence of positive solutions of this problem in the special case when \(\varepsilon =0\). Then, we investigate the sharp patterns of the positive solutions when \(\varepsilon \downarrow 0\) and \(\varepsilon \uparrow \infty \). Our study reveals how the existence of sharp profiles is determined by the behavior of b(x).

本文讨论半线性椭圆问题

其中\（p> 1 \），\（\ lambda> 0 \），\（m，a，b \ in C（{\ bar {\ Omega}}）\），带有\（a \ gneq 0 \），\（b \ gneq 0 \） ，\（\欧米茄\）是一个有界\（C ^ {2} \）的域\（{\ mathbb {R}} ^ N \） （\（N \ GE 1 \）），B是一般的经典混合边界算子，和\（\ varepsilon \ ge 0 \）。因此，a（x）和b（x）可以在\（\ Omega \）的某个子域上消失，并且权重函数m（x）可以更改符号\（\ Omega \）。通过本文，我们始终在考虑经典解决方案。首先，在特殊情况下，当\（\ varepsilon = 0 \）时，我们描述了此问题的正解的存在性。然后，我们研究\（\ varepsilon \ downarrow 0 \）和\（\ varepsilon \ uparrow \ infty \）时正解的尖锐模式。我们的研究揭示了尖锐轮廓的存在是如何由b（x）的行为决定的。