LAUSR

Let $$\varOmega $$Ω be a smooth bounded domain in $$\mathbb {R}^N$$RN, $$N\ge 3$$N≥3. We consider the following singularly perturbed nonlinear elliptic problem on $$\varOmega $$Ω, $$\begin{aligned} \varepsilon ^2\varDelta v-v+f(v)=0,\quad v>0\ \text {on}\ \varOmega ,\qquad \frac{\partial v}{\partial \nu }=0\quad \text {on}\ \partial \varOmega , \end{aligned}$$ε2Δv-v+f(v)=0,v>0onΩ,∂v∂ν=0on∂Ω,where $$\nu $$ν is an exterior unit normal vector to $$\partial \varOmega $$∂Ω and a nonlinearity f satisfies subcritical growth condition. It has been known that for any $$l_0, l_1 \in \mathbb {N} \cup \{ 0 \}$$l0,l1∈N∪{0}, $$l_0+l_1>0$$l0+l1>0, there exists a solution $$v_\varepsilon $$vε of the above problem which exhibits $$l_0$$l0-boundary peaks and $$l_1$$l1-interior peaks for small $$\varepsilon >0$$ε>0 under rather strong conditions on f, such as the linearized non-degeneracy condition for a limiting problem. In this paper, we extend the previous result to more general class of f satisfying Berestycki–Lions conditions which we believe to be almost optimal.