Super-position of spikes for a slightly super-critical elliptic equation in \(\mathbb R^N\)
The authors are interested in radial solutions of the problem \[ -\Delta u + V(|x|)u = u^{\frac{N+2}{N-2}+\varepsilon}\quad\text{in } \mathbb{R^N},\qquad u>0\quad \text{in }\mathbb{R}^N,\qquad u \in H^1(\mathbb{R}^N), \] where \(N \geq 5\), \(\varepsilon>0\) is small and \(V \in L^{\infty}(\mathbb{R}^N) \cap L^{\frac{N}{2}}(\mathbb{R}^N)\). They prove that the radial property of \(V\), together with the presence of the super-critical exponent yields to the existence of solutions with a richer structure than the one previously found. They build solutions of a scaled (and equivalent) problem whose shape resembles a super-position of spikes around the origin 0 with different rate of concentration. The main result of the paper is the following: Let \(N \geq 5\) and \(V \in L^{N/2}(\mathbb{R}^N)\). If \(V(0)<0\) then, given an integer \(k \geq 1\), there exists \(\varepsilon_0>0\) such that, if \(\varepsilon<\varepsilon_0\), there are positive constants \(\alpha_1,\ldots,\alpha_k\), which depend on \(k,N\) and \(V(0)\), and a solution \(u_{\varepsilon}\) of the problem of the form \[ u_{\varepsilon}(x) = \gamma \sum_{j=1}^k \left( \frac{1}{1+\alpha_j^{\frac{4}{N-2}}\varepsilon^{-\frac{4}{N-2}(j-1)-1}|x|^2} \right)^{\frac{N-2}{2}} \alpha_j \varepsilon^{-(j-1)-\frac{N-2}{4}}(1+o(1)), \] where \(\gamma = (N(N-2))^{\frac{N-2}{4}}\) and \(o(1) \to 0\) uniformly on compact sets of \(\mathbb{R}^N\). The proof uses a variant of the so-called Emden-Fowler transformation, that reduces the problem to a non-autonomous ordinary differential equation, followed by a Lyapunov-Schmidt reduction, which provides the solutions they are looking for.
scientific article; zbMATH DE number 2177460