Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki-Lions condition
Consider the singularly perturbed Neumann problem \[ \varepsilon^2\Delta v-v+f(v)=0\text{ in }\Omega,\quad v>0\text{ in }\Omega, \] \[ \frac{\partial v}{\partial \nu}=0\text{ on }\partial \Omega, \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), with \(N\geq 3\), \(\nu\) is the outward unit normal to \(\partial \Omega\), and \(f\in C^1(\mathbb{R})\). In the paper, the authors study, for small \(\varepsilon>0\), the existence of solutions to problem \((P)\) having both interior and boundary peaks. The main result is the following theorem. Assume that 1) there exists \(q_0\in(0,1)\) such that \ \(\displaystyle{f(0)=\lim_{t\rightarrow 0}\frac{f(t)}{t^{1+q_0}}=0}\); 2) \({\limsup_{t\rightarrow +\infty}\frac{|f(t)|}{t^\beta}<\infty}\), for some \(\beta \in (1,\frac{N+2}{N-2})\); 3) there exists \(T>0\) such that \(\displaystyle{\int_0^Tf(t)dt>\frac{T^2}{2}}\). Then, for each \(l_0,l_1\in N\cup\{0\}\), with \(l_0+l_1>0\), there exists \(\varepsilon_0>0\) such that, for each \(\varepsilon \in (0,\varepsilon_0)\) there exists a solution \(v_\varepsilon\) to problem \((P)\) which has \(l_0\)-boundary peaks and \(l_1\) interior peaks. After transforming problem \((P)\) into an equivalent problem by the change of variable \(x\rightarrow \varepsilon x\), the authors find a solution as a critical point of the associated energy functional in a neighborhood of an approximate solution expressed by a sum of translates of least energy solutions of the problem \(\Delta V-V+f(V)=0\), in \(\mathbb{R}^N\), \(V>0\) in \(\mathbb{R}^N\), \(\lim_{|x|\rightarrow \infty}V(x)=0\), \(V(0)=\max_{\mathbb{R}^N}V\). This result improves previous existence results where additional assumptions on the nonlinearity \(f\) were assumed. The proof, quite long and involved, is based on variational arguments.
scientific article; zbMATH DE number 6690578
scientific article