Multiplicity of positive and nodal solutions for nonlinear elliptic problems in \(\mathbb{R}^ N\)
Consider the problem \[ -\Delta u+\mu u=Q(x)u|u|^{p-2},\quad x\in \mathbb{R}^n,\quad u\in H^1_0(\mathbb{R}^n),\tag{\(*\)} \] where \(N\geq 3\), \(2<p<2N/(N-2)\), \(\mu>0\), and \(Q\in C(\mathbb{R}^N)\). The authors prove that if \(Q(x)\geq 0\) in \(\mathbb{R}^N\) and there are points \(a^1,\dots,a^k\) in \(\mathbb{R}^N\) at which \(Q(a^j)=\max\{Q(x):x\in \mathbb{R}^N\}\), \(j=1,\dots,k\), then there exists a \(\mu_0>0\) such that \((*)\) has at least \(k\) positive solutions and \(k\) nodal solutions (solutions change sign) for each \(\mu\geq\mu_0\). This result is established by using the ``shape'' of the graph of \(Q(x)\) and the concentration-compactness principle of \textit{P. L. Lions} [Ann. Inst. H. Poincaré, Anal. Nonlin. 109-145 (1984; Zbl 0541.49009)] and \textit{I. Ekeland's} variational principle [J. Math. Anal. Appl. 47, 324-353 (1974; Zbl 0286.49015)].
scientific article; zbMATH DE number 936274
scientific article