Concentrated solution for some non-local and non-variational singularly perturbed problems

Abstract

In the present paper, we prove the existence of concentrated solution for the following non-local and non-variational singularly perturbed problem, 0.1 $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\varepsilon ^2A(\varepsilon ^{-n}||u||^q_{L^q})\Delta u+ V(x)u=|u|^{p-1}u, &{}x\in \mathbb {R}^n,\\ 0<u\in H^1(\mathbb {R}^n), \lim \limits _{|x|\rightarrow \infty }u(x)=0, \end{array} \right. \end{aligned}$$ where $$n\ge 1$$ , $$1<p<2^*-1$$ , $$2\le q<2^*$$ , $$A: (0, +\infty )\rightarrow [a, +\infty )$$ , $$V(x): \mathbb {R}^n\rightarrow \mathbb {R}$$ are two continuous functions, $$a>0$$ and $$\varepsilon $$ 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 $$\mathbb {R}^n$$ . 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 $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -A(\Vert U\Vert ^q_{L^q})\Delta U(y)+ \alpha U(y)=U^{p}(y), &{}y\in \mathbb {R}^n,\\ 0<U\in H^1(\mathbb {R}^n). \end{array} \right. \end{aligned}$$ In Sect. 2, existence and non-degeneracy results for positive solutions of problem (0.2) are proved. Compared to the standard Schrodinger equation, our results imply that the non-local term $$A(\Vert U\Vert _{L^q}^q)$$ has no effect on the non-degeneracy result of positive solutions of problem (0.2) when the solution U satisfies $$\Vert U\Vert ^q_{L^q}A'(\Vert U\Vert ^q_{L^q})\ne \frac{2}{n}A(\Vert U\Vert ^q_{L^q})$$ and that the non-local term $$A(\Vert U\Vert _{L^q}^q)$$ 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 $$\Vert U\Vert _{L^q}^qA'(\Vert U\Vert _{L^q}^q)\ne \frac{2}{n}A(\Vert U\Vert _{L^q}^q)$$ . In our existence result of the concentrated solution, we only assume that $$A\in C^{1,1}_{loc}$$ . These two assumptions imply that A can be a non-monotonous or unbounded function.