Infinitely many bound states for some nonlinear scalar field equations
Existence of infinitely many \(H^1(\mathbb{R}^N)\) solutions to the equation \[ -\Delta u+ a(x)u= |u|^{p-2}u,\quad N\geq 2,\;p> 2,\;p<{2N\over N-2},\;\text{ when }N> 2,\tag{1} \] is proved without symmetry assumptions on \(a(x)\). The solutions are critical points of the energy functional \[ I(u)= {1\over 2} \int_{\mathbb{R}^N} (|\nabla u|^2+ a(x) u^2)\,dx- {1\over p} \int_{\mathbb{R}^N} |u|^p\,dx. \] Variational methods are useful to study (1) when considered on a bounded domain in \(\mathbb{R}^N\). But on \(\mathbb{R}^N\), without assuming symmetry of \(a(x)\), variational methods can not be used. To overcome this difficulty, the authors assume \(a\in C^1(\mathbb{R}^N)\), \(\liminf_{|x|\to\infty}a(x)= a_\infty> 0\) and \[ {\partial a\over\partial\xi}(x) e^{\alpha|x|}\to\infty,\quad |\nabla_{\tau_x}a(x)|\leq \overline c{\partial a\over\partial \xi}(x),\quad |x|>\overline c>1,\quad \xi= {x\over|x|}. \] Then the authors consider a sequence of balls \(B_{\rho_n}= \{|x|< \rho_n\}\) in \(\mathbb{R}^N\) and the sequence \(\{u_n\}\) of solutions of (1) on \(B_{\rho_n}\) with the boundary condition \(u_n= 0\) on \(\partial B_{\rho_n}\). Precompactness of this sequence is shown by proving a local Pokhozhaev type inequality (Lemma 4.2 and 4.3). Then taking a suitable subsequence of \(\{u_n\}\), existence of infinitely many solutions to (1) is proved (Th.1.1. Proof is given in \S5). Useful facts and preliminary results are summarized in \S2. \S3 gives preliminary results to \S4. The authors remark that the same result holds on \(\mathbb{R}^N\setminus\Omega\), where \(\Omega\) is a bounded smooth open set in \(\mathbb{R}^N\).
scientific article; zbMATH DE number 2167753