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3 Open Mathematics

3 2017

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lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

100%

Li P. , Shang Y.

Open Mathematics

2017

tom 15

nr 1

578-586

Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.

Multiplicity solutions of a class fractional Schrödinger equations

100%

Jia L.-J. , Ge B. , Cui Y.-X. , Sun L.-L.

Open Mathematics

2017

tom 15

nr 1

1010-1023

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (−Δ)su+V(x)u=λf(x,u)inRN, $$ {( - \Delta )^s}u + V(x)u = \lambda f(x,u)\,\,{\rm in}\,\,{\mathbb{R}^N}, $$ where [...] (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN $ {( - \Delta )^s}u(x) = 2\lim\limits_{\varepsilon \to 0} \int_ {{\mathbb{R}^N}\backslash {B_\varepsilon }(X)} {{u(x) - u(y)} \over {|x - y{|^{N + 2s}}}}dy,\,\,x \in {\mathbb{R}^N} $ is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

Triple solutions for a Dirichlet boundary value problem involving a perturbed discretep(k)-Laplacian operator

88%

Khaleghi Moghadam M. , Henderson J.

Open Mathematics

2017

tom 15

nr 1

1075-1089

Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.