Multi-bump solutions for a strongly indefinite semilinear Schrödinger equation without symmetry or convexity assumptions

Multi-bump Solutions for a Strongly Indefinite Semilinear Schrödinger Equation Without Symmetry or convexity Assumptions

In this paper, we study the following semilinear Schrödinger equation with periodic coefficient: −△u + V (x)u = f(x, u), u ∈ H1(RN). The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term f(x, t) satisfies some superlinear growth conditions and need not be odd or increasing strictly in t. Using a new variational reduction method and a generaliz...

Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems

Solutions without Any Symmetry for Semilinear Elliptic Problems

— We prove the existence of infinitely many solitary waves for the nonlinear KleinGordon or Schrödinger equation ∆u− u + u = 0, in R, which have finite energy and whose maximal group of symmetry reduces to the identity.

Existence of a Nontrivial Solution to a Strongly Indefinite Semilinear Equation

Under general hypotheses, we prove the existence of a nontrivial solution for the equation Lu = N(u), where u belongs to a Hilbert space H , L is an invertible continuous selfadjoint operator, and N is superlinear. We are particularly interested in the case where L is strongly indefinite and N is not compact. We apply the result to the Choquard-Pekar equation -Au(x)+p(x)u(x) = u(x) f U <'y\ dy,...

Positive Solutions for a Periodic Boundary Value Problem without Assumptions of Monotonicity and Convexity

In the case of not requiring the nonlinear term to be monotone or convex, we study the existence of positive solutions for second-order periodic boundary value problem by using the first eigenvalue of the corresponding linear problem and fixed point index theory. The work significantly improves and generalizes the main results of J. Graef et al. [A periodic boundary value problem with vanishing...