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\begin{document}$$\varOmega $$\end{document} be a smooth bounded domain in RN\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^N$$\end{document}, N≥3\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 3$$\end{document}. We consider the following singularly perturbed nonlinear elliptic problem on Ω\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega $$\end{document}, ε2Δv-v+f(v)=0,v>0onΩ,∂v∂ν=0on∂Ω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varepsilon ^2\varDelta v-v+f(v)=0,\quad v>0\ \text {on}\ \varOmega ,\qquad \frac{\partial v}{\partial \nu }=0\quad \text {on}\ \partial \varOmega , \end{aligned}$$\end{document}where ν\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document} is an exterior unit normal vector to ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \varOmega $$\end{document} and a nonlinearity f satisfies subcritical growth condition. It has been known that for any l0,l1∈N∪{0}\documentclass[12pt]{minimal}
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\begin{document}$$l_0, l_1 \in \mathbb {N} \cup \{ 0 \}$$\end{document}, l0+l1>0\documentclass[12pt]{minimal}
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\begin{document}$$l_0+l_1>0$$\end{document}, there exists a solution vε\documentclass[12pt]{minimal}
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\begin{document}$$v_\varepsilon $$\end{document} of the above problem which exhibits l0\documentclass[12pt]{minimal}
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\begin{document}$$l_0$$\end{document}-boundary peaks and l1\documentclass[12pt]{minimal}
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\begin{document}$$l_1$$\end{document}-interior peaks for small ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document} 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.
