In the present paper, we prove the existence of concentrated solution for the following non-local and non-variational singularly perturbed problem, 0.1-ε2A(ε-n||u||Lqq)Δu+V(x)u=|u|p-1u,x∈Rn,0<u∈H1(Rn),lim|x|→∞u(x)=0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\varepsilon ^2A(\varepsilon ^{-n}||u||^q_{L^q})\Delta u+ V(x)u=|u|^{p-1}u, &{}x\in \mathbb {R}^n,\\ 0<u\in H^1(\mathbb {R}^n), \lim \limits _{|x|\rightarrow \infty }u(x)=0, \end{array} \right. \end{aligned}$$\end{document}where n≥1\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 1$$\end{document}, 1<p<2∗-1\documentclass[12pt]{minimal}
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\begin{document}$$1<p<2^*-1$$\end{document}, 2≤q<2∗\documentclass[12pt]{minimal}
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\begin{document}$$2\le q<2^*$$\end{document}, A:(0,+∞)→[a,+∞)\documentclass[12pt]{minimal}
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\begin{document}$$A: (0, +\infty )\rightarrow [a, +\infty )$$\end{document}, V(x):Rn→R\documentclass[12pt]{minimal}
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\begin{document}$$V(x): \mathbb {R}^n\rightarrow \mathbb {R}$$\end{document} are two continuous functions, a>0\documentclass[12pt]{minimal}
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\begin{document}$$a>0$$\end{document} and ε\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon $$\end{document} is a positive and small parameter. Problem (0.1) can be seen as a model to study the vibration of nonlinear string or bacteria’s density balance law in Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^n$$\end{document}. The existence is based on the well-known Lyapunov–Schmidt reduction method. In order to make Lyapunov–Schmidt reduction method work well, existence and so-called non-degeneracy result of positive solutions for following problem will be needed.
0.2-A(‖U‖Lqq)ΔU(y)+αU(y)=Up(y),y∈Rn,0<U∈H1(Rn).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -A(\Vert U\Vert ^q_{L^q})\Delta U(y)+ \alpha U(y)=U^{p}(y), &{}y\in \mathbb {R}^n,\\ 0<U\in H^1(\mathbb {R}^n). \end{array} \right. \end{aligned}$$\end{document}In Sect. 2, existence and non-degeneracy results for positive solutions of problem (0.2) are proved. Compared to the standard Schrödinger equation, our results imply that the non-local term A(‖U‖Lqq)\documentclass[12pt]{minimal}
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\begin{document}$$A(\Vert U\Vert _{L^q}^q)$$\end{document} has no effect on the non-degeneracy result of positive solutions of problem (0.2) when the solution U satisfies ‖U‖LqqA′(‖U‖Lqq)≠2nA(‖U‖Lqq)\documentclass[12pt]{minimal}
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\begin{document}$$\Vert U\Vert ^q_{L^q}A'(\Vert U\Vert ^q_{L^q})\ne \frac{2}{n}A(\Vert U\Vert ^q_{L^q})$$\end{document} and that the non-local term A(‖U‖Lqq)\documentclass[12pt]{minimal}
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\begin{document}$$A(\Vert U\Vert _{L^q}^q)$$\end{document} has significant effect on the existence result. Since the linearized operator L[U] for problem (0.2) is non-self-adjoint, our non-degeneracy result is a result about the Kernel space of the adjoint operator of linearized operator L[U] for problem (0.2) rather than the Kernel space of L[U] itself, which is essentially different from common non-local and self-adjoint equations, such as Kirchhoff equation, fractional Laplace equations and Choquard equations. Moreover, in our non-degeneracy result, we only assume that the solution U satisfies ‖U‖LqqA′(‖U‖Lqq)≠2nA(‖U‖Lqq)\documentclass[12pt]{minimal}
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\begin{document}$$\Vert U\Vert _{L^q}^qA'(\Vert U\Vert _{L^q}^q)\ne \frac{2}{n}A(\Vert U\Vert _{L^q}^q)$$\end{document}. In our existence result of the concentrated solution, we only assume that A∈Cloc1,1\documentclass[12pt]{minimal}
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\begin{document}$$A\in C^{1,1}_{loc}$$\end{document}. These two assumptions imply that A can be a non-monotonous or unbounded function.
