Ground State Solutions for Quasilinear Schrödinger Equations with Periodic Potential (original) (raw)
On superlinear Schrödinger equations with periodic potential
Calculus of Variations and Partial Differential Equations, 2011
We obtain ground state solutions for a wide class of superlinear Schrödinger equations with periodic potential. The result improves a recent result of Szulkin and Weth [A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009) 3802-3822]. The main ingredient is the observation that even in the strongly indefinite case, all Cerami sequences for the energy functional are bounded.
Proceedings of the American Mathematical Society
We study the Schrödinger equations −∆u + V (x)u = f (x, u) in R N and −∆u − λu = f (x, u) in a bounded domain Ω ⊂ R N. We assume that f is superlinear but of subcritical growth and u → f (x, u)/|u| is nondecreasing. In R N we also assume that V and f are periodic in x1,. .. , xN. We show that these equations have a ground state and that there exist infinitely many solutions if f is odd in u. Our results generalize those in [11] where u → f (x, u)/|u| was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis.
Existence of solutions to asymptotically periodic Schrodinger equations
Electronic Journal of Differential Equations, 2017
We show the existence of a nonzero solution for the semilinear Schrodinger equation −Deltau+V(x)u=f(x,u)-\Delta u+V(x)u=f(x,u)−Deltau+V(x)u=f(x,u). The potential V is periodic and 0 belongs to a gap of sigma(−Delta+V)\sigma(-\Delta +V)sigma(−Delta+V). The function f is superlinear and asymptotically periodic with respect to x variable. In the proof we apply a new critical point theorem for strongly indefinite functionals proved in [3].
On a Schrodinger equation with periodic potential and spectrum point zero
Indiana University Mathematics Journal, 2003
The main purpose of this paper is to establish the existence of a solution of the semilinear Schrödinger equation −∆u + V (x)u = f (u), in R 2 where V is a 1-periodic functions with respect to x, 0 lies in a gap of the spectrum of −∆ + V , and f (s) behaves like ± exp(αs 2) when s → ±∞.
On a periodic Schrödinger equation involving periodic and nonperiodic nonlinearities in R2
Journal of Mathematical Analysis and Applications, 2018
We study the existence of solutions for the nonlinear Schrödinger equation −Δu + V (x)u = f (x, u) in R 2 , where the potential V is 1-periodic, 0 lies in a spectral gap from the spectrum of the Schrödinger operator S = −Δ + V and the nonlinearity f (x, t) has exponential growth in the sense of Trudinger-Moser. The main feature here is that f (x, t) is allowed to be both periodic and nonperiodic in the x variable. Our proofs rely on a linking theorem and the Lions concentration compactness principle.
Schrödinger equations with asymptotically periodic terms
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
We study the existence of non-trivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in ℝN. By combining variational methods and the concentration-compactness principle, we obtain a non-trivial solution for the asymptotically periodic case and a ground state solution for the periodic one. In the proofs we apply the mountain pass theorem and its local version.
Ground states of degenerate quasilinear Schrödinger equation with vanishing potentials
Nonlinear Analysis, 2019
In this paper we study the existence of nontrivial ground state solutions for the following class of p-Laplacian type equation −div (a(x, ∇u)) + V (x)|x| −αp * |u| p−2 u = K(x)|x| −αp * f (u) in R N , where 1 < p < N , N ≥ 3, −∞ < α < N −p p , α ≤ e ≤ α + 1, d = 1 + α − e, p * := p * (α, e) = N p N −dp (critical Hardy-Sobolev exponent); f has a quasicritical growth; V and K are nonnegative potentials; the function a satisfies |a(x, ∇u)| ≤ c 0 |x| −αp h 0 (x)|∇u| p−1 + c 0 (1 + |x| −αp)h 1 (x)|∇u| p−1 for any ξ ∈ R N , a.e. x ∈ R N , for any two positive functions h 1 ∈ L ∞ loc (R N), h 0 ∈ L p p−1 α (R N), with α = αp p * .
Solutions of Gross–Pitaevskii Equation with Periodic Potential in Dimension Two
Analysis as a Tool in Mathematical Physics, 2020
Quasi-periodic solutions of the Gross-Pitaevskii equation with a periodic potential in dimension three are studied. It is proven that there is an extensive "non-resonant" set G ⊂ R 3 such that for every k ∈ G there is a solution asymptotically close to a plane wave Ae i k, x as | k| → ∞, given A is sufficiently small.
A note on quasilinear Schrödinger equations with singular or vanishing radial potentials
Differential and Integral Equations
In this note we complete the study of [3], where we got existence results for the quasilinear elliptic equation −∆w + V (|x|) w − w ∆w 2 = K(|x|)g(w) in R N , with singular or vanishing continuous radial potentials V (r), K(r). In [3] we assumed, for technical reasons, that K(r) was vanishing as r → 0, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables w = f (u) and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in R N \ {0}. The nonlinearity g has a double-power behavior, whose standard example is g(t) = min{t q 1 −1 , t q 2 −1 } (t > 0), recovering the usual case of a singlepower behavior when q1 = q2.