On the number of interior peaks of solutions to a non-autonomous singularly perturbed Neumann problem.(English)Zbl 1162.35009

Let $$N\geq2$$ and suppose that $$\Omega\subseteq\mathbb{R}^N$$ is a bounded smooth domain, $$a\in C^\infty(\overline{\Omega})$$, and $$1<p<2^*-1$$, where $$2^*$$ is the usual critical Sobolev exponent that corresponds to this problem. It is never stated but implicitly assumed that $$a$$ is positive. For a small positive parameter $$\varepsilon$$ consider the elliptic problem $\begin{cases} -\varepsilon^2\Delta u+u=a(x)u^p&\text{in }\Omega\\ u>0&\text{in }\Omega\\ \frac{\partial u}{\partial\nu}=0&\text{on }\partial\Omega. \end{cases}\tag{1}$ Here $$\partial/\partial\nu$$ denotes the partial derivative in the normal direction on $$\partial\Omega$$.
If $$Q_0\in\Omega$$ is a strict local minimum of $$a$$ then it is shown that there are $$\varepsilon_0>0$$ and $$K_0(\varepsilon)>0$$, $$K_0(\varepsilon)\to\infty$$ as $$\varepsilon\to0$$, such that for $$\varepsilon\in(0,\varepsilon_0]$$ and $$K\leq K_0(\varepsilon)$$ there is a multipeak solution of $$(1)$$ with exactly $$K$$ local maximum points. These maxima tend to $$Q_0$$ as $$\varepsilon\to0$$.
In the result of [F.-H. Lin, W.-M. Ni and J.-C. Wei, Commun. Pure Appl. Math. 60, No. 2, 252–281 (2007; Zbl 1170.35424)], a similar statement is made for the case $$a\equiv1$$, with asymptotics $$K_0(\varepsilon)\sim\varepsilon^{-N}|\log\varepsilon|^{-N}$$. In contrast, in the paper under review the asymptotics are $$K_0(\varepsilon) \sim\varepsilon^{-M}|\log\varepsilon|^{-N}$$ for some $$M\in(0,1)$$ that depends on the order of the zero of $$\nabla a$$ in $$Q_0$$.

MSC:

 35B25 Singular perturbations in context of PDEs 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations

Zbl 1170.35424
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