Combined effects of concave and convex nonlinearities in elliptic equation on \(\mathbb R^N\)
The author proves that, for some \(\lambda^*>0\), the concave-convex type problem \[ \begin{cases} -\Delta u + u =a(x)u^{p-1}+ \lambda b(x)u^{q-1} & \text{ in } \mathbb{R}^N, \\ u \in H^1(\mathbb{R}^N), \end{cases} \] where \(1<p<2<q<2^*\) and \(N\geq 3\), admits two positive solutions if \(0<\lambda<\lambda^*\). These solutions are obtained as minimizers of the associated functional \[ I_{\lambda}(u)=\frac{1}{2}\int (|\nabla u|^2 + |u|^2) - \frac{1}{p} \int a(x)|u|^p- \frac{\lambda}{q}\int b(x)|u|^q \] restricted to the Nehari manifold \[ \Lambda_{\lambda}=\{u \in H^1(\mathbb{R}^N): \langle I'_{\lambda}(u),u \rangle =0 \}. \] To this aim, the author follows the procedure originally used in [\textit{G. Tarantello}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9, No.~3, 281--304 (1992; Zbl 0785.35046)]. The assumptions on \(a\) and \(b\) are such that \(I_{\lambda}\) satisfies the Palais-Smale condition at the appropriate levels.
scientific article; zbMATH DE number 5563832