Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity

Let $\Omega$ be a bounded domain in $\mathbf{R}^n,$ $n \ge 3,$ with a boundary $\partial \Omega \in C^2.$ We consider the following singularly perturbed nonlinear elliptic problem on $\Omega$:

$$\varepsilon^2 \Delta u - u + f(u) = 0, u > 0 \textrm{ on }\Omega, \quad u = 0 \textrm{ on } \partial \Omega,$$

where the nonlinearity $f$ is of subcritical growth. Under rather strong conditions on $f,$ it has been known that for small $\varepsilon > 0,$ there exists a mountain pass solution $u_\varepsilon$ of above problem which exhibits a spike layer near a maximum point of the distance function $d$ from $\partial \Omega$ as $\varepsilon \to 0.$ In this paper, we construct a solution $u_\varepsilon$ of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on $f$, which we believe to be almost optimal.