We consider periodic solutions for superlinear second order non-autonomous dynamical systems including both kinetic and potential terms. We study the existence of nontrivial and ground state solutions.

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# Your search: "author:"Schechter, M""

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## Scholarly Works (10 results)

We study the existence of periodic solutions for a second order non-autonomous dynamical system. We allow both sublinear and superlinear problems. We obtain ground state solutions.

We study the existence of homoclinic solutions for a second-order nonautonomous dynamical system including both the kinetic and potential terms. We assume very little concerning the kinetic term, just enough to make the essential spectrum of the linear operator the same as that for free particles. Our theorems cover all cases and allow both sublinear and superlinear problems. We obtain ground state solutions.

We study the existence of semi-classical bound states of the nonlinear Schrödinger equation \begin{linenomath} -\varepsilon2\Delta u+V(x)u=f(u),\quad x\in {\bf R}^N,\end{linenomath} where N ≥ 3;, Ï is a positive parameter; V:R N → [0, ∞) satisfies some suitable assumptions. We study two cases: if f is asymptotically linear, i.e., if lim|t| → ∞ f(t)/t=constant, then we get positive solutions. If f is superlinear and f(u)=|u| p-2 u+|u| q-2 u, 2* > p > q > 2, we obtain the existence of multiple sign-changing semi-classical bound states with information on the estimates of the energies, the Morse indices and the number of nodal domains. For this purpose, we establish a mountain cliff theorem without the compactness condition and apply a new sign-changing critical point theorem. © 2013 Cambridge Philosophical Society.

We study the existence of periodic solutions for a second order non-autonomous dynamical system containing variable kinetic energy terms. Our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other. We study sublinear problems and the existence of non-constant solutions. © 2014 Elsevier Inc.

We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983):
$$-\Delta u=\lambda u+ |u|^{2^\ast-2}u, \quad u\in H_0^1(\Omega),$$where Ω is a bounded smooth domain of R
N (N ≧ 7) and 2* is the critical Sobolev exponent. We show that, for each fixed λ > 0, this problem has infinitely many sign-changing solutions. In particular, if λ ≧ λ1, the Brézis–Nirenberg problem has and only has infinitely many sign-changing solutions except zero. The main tool is the estimates of Morse indices of nodal solutions.

We solve elliptic semilinear boundary value problems in which the nonlinear term is superlinear. By weakening the hypotheses, we are able to include more equations than hitherto permitted. In particular, we do not require the superquadracity condition imposed by most authors, and it is not assumed that the region is bounded.

The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via linking theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualified solutions.

The Sandwich Pair theorems have presented very efficient ways to determine the existence of critical points or critical sequences for nonlinear differentiable functionals. In this paper, under rather weak hypotheses new relationships are established between sign-changing critical points and Sandwich Pairs or Linking Sandwich Pairs. The abstract results are demonstrated by applications on semi-linear elliptic equations. © 2013 Elsevier Ltd. All rights reserved.