A semilinear elliptic equation with convex and concave nonlinearities
In the paper the author studies the problem (P) of finding positive solutions \(u\in D^{1,2}(\mathbb{R}^N)\) for the class of semilinear elliptic equations of the form \(-\Delta u-\lambda g(x)u=k(x)|u|^{q-2}u-h(x)|u|^{p-2}u\) where it is assumed that (E) \(N\geq 3\) and \(1<q<2<p\leq 2^{*}\). By \textit{K. J. Brown, C. Cosner} and \textit{J. Fleckinger} [Proc. Am. Math. Soc. 109, 147-155 (1990; Zbl 0726.35089)], assuming (G) \(g\in L^{N\over 2}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})\) is indefinite in sign, then the principal positive and negative eigenvalues of the linear part are well definite: \(0<\lambda_{1}=(\sup_{\|u\|_{D^{1,2}}=1}\int_{\mathbb{R}^{N}}g(x)u(x)^{2} dx)^{-1}\), \(0<\lambda_{-1}=(\sup_{\|u\|_{D^{1,2}}=1}\int_{\mathbb{R}^{N}}(-g(x))u(x)^{2} dx)^{-1}\). Setting \(p_{0}=2N/(2N-pN+2p)\) if \(p<2^{*}\), \(p_{0}=+\infty\) if \(p=2^{*}\) and \(q_{0}=2N/(2N-qN+2q)\), the problem (P) is studied under the further assumption (HK) \(h\in L^{p_{0}}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})\) and \(g\in L^{q_{0}}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})\) \noindent and the following is proved: Theorem. Assume (E), (G), (HK) and that \(\lambda\in (-\lambda_{-1},\lambda_{1})\). Then a) if \(p<2^{*}\), \(k\geq 0\), \(k\not= 0\) and \(h\) changes sign then (P) admits infinitely many solutions at negative energy level; b) if \(p<2^{*}\), \(h\leq 0\), \(h\not= 0\) and \(k\not=0\) then (P) admits an unbounded (in \(D^{1,2}(\mathbb{R}^{N})\)) sequence of solutions; c) if \(p=2^{*}\), \(k\geq 0\), \(h\geq 0\), \(h\not= 0\) then (P) admits a sequence of solutions \(\{ u_{n}\}\) at negative energy level and such that \(\|u_{n}\|_{D^{1,2}}\to 0\); d) if \(p=2^{*}\), \(k\geq 0\), \(h\) is indefinite in sign and \(0<\|k\|_{q_{0}}\) is sufficiently small then (P) admits a sequence of solutions \(\{ u_{n}\}\) at negative energy level.
scientific article; zbMATH DE number 1441650