# A dimension-depending multiplicity result for a perturbed Schrödinger equation

 Title A dimension-depending multiplicity result for a perturbed Schrödinger equation Publication Type Journal Article Authors Kristály, Alexandru, Gheorghe Morosanu, and Donal O'Regan Journal title Dynamic Systems and Applications Year 2013 Pages 325-335 Volume 22 Issue 2-3 Abstract We consider the Schrodinger equation $$\Delta u + V (x)u = \lambda K(x)f(u) + \mu L(x)g(u) \mbox{ in } R^N; \ u\in H^1(R^N), \eqno{(P)}$$ where $N\ge  2$, $\lambda , \mu  \ge 0$ are parameters, $V,K,L : R^N\rightarrow R$ are radially symmetric potentials, $f : R\rightarrow R$ is a continuous function with sublinear growth at in finity, and $g : R\rightarrow R$ is a continuous sub-critical function. We first prove that for $\lambda$ small enough no non-zero solution exists for $(P)$, while for $\lambda$ large and $\mu$ small enough at least two distinct non-zero radially symmetric solutions do exist for $(P)$. By exploiting a Ricceri-type three-critical points theorem, the principle of symmetric criticality and a group-theoretical approach, the existence of at least $N-3$ ($N$ mod 2) distinct pairs of non-zero solutions is guaranteed for $(P)$ whenever $\lambda$ is large and  $\mu$ is small enough, $N\neq 3$, and $f, g$ are odd. ISSN 1056-2176 Language English
Unit:
Department of Mathematics and its Applications