Existence of positive solutions to a superlinear elliptic problem (Q2703366)

The authors study the existence of global positive solutions of the semilinear elliptic problem NEWLINE\[NEWLINE-\varepsilon^2\Delta u+ V(z)u= f(u),\quad x\in\mathbb{R}^N\quad (N\geq 2),\tag{\(*\)}NEWLINE\]NEWLINE where \(V\) and \(f\) are nonnegative continuous functions which satisfyNEWLINENEWLINENEWLINE\({f(t)\over t}\) is nondecreasing for \(t>0\), \(\lim_{t\to 0^+} {f(t)\over t}= 0\), \(\lim_{t\to\infty} {f(t)\over t}= \infty\), andNEWLINENEWLINENEWLINE\(\theta \int^t_0 f(s) ds\leq tf(t)\) for \(t> 0\) and some \(\theta> 2\);NEWLINENEWLINENEWLINE\(V\) is radially symmetric and bounded away from zero in a neighborhood of the origin and infinity, and vanishes in an annulus centered at the origin.NEWLINENEWLINENEWLINEThe main result proves that \((*)\) has a radially symmetric classical solution \(u_\varepsilon\in H^1(\mathbb{R}^N)\) which vanishes at infinity. This is proved by applying the Mountain Pass Lemma to establish the existence of a stationary point of the functional NEWLINE\[NEWLINEI_\varepsilon(u)= \int_{\mathbb{R}^n} {1\over 2} (\varepsilon^2|\nabla u|^2+ Vu^2) dz- \int_{\mathbb{R}^N} F(u) dzNEWLINE\]NEWLINE on the space of radially symmetric functions in \(H^1(\mathbb{R}^N)\). Here, \(F\) is the antiderivative of \(f\) which vanishes at the origin.