1,129 research outputs found

    Multiple solutions for a class of nonhomogeneous fractional Schr\"odinger equations in RN\mathbb{R}^{N}

    Full text link
    This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where s∈(0,1)s\in (0,1), N>2sN> 2s, (−Δ)s(-\Delta)^{s} is the fractional Laplacian, kk is a bounded positive function, h∈L2(RN)h\in L^{2}(\mathbb{R}^{N}), h≢0h\not \equiv 0 is nonnegative and ff is either asymptotically linear or superlinear at infinity.\\ By using the ss-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that ∣h∣2|h|_{2} is sufficiently small

    Multiple solutions for a fractional pp-Laplacian equation with sign-changing potential

    Full text link
    We use a variant of the fountain Theorem to prove the existence of infinitely many weak solutions for the following fractional p-Laplace equation (-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u) in R^N, where s∈(0,1)s \in (0,1),p≥2 p \geq 2,N≥2 N \geq 2, (−Δ)ps(-\Delta)^{s}_{p} is the fractional pp-Laplace operator, the nonlinearity f is pp-superlinear at infinity and the potential V(x) is allowed to be sign-changing

    Ground states for superlinear fractional Schr\"odinger equations in \R^{N}

    Full text link
    In this paper we study ground states of the following fractional Schr\"odinger equation (- \Delta)^{s} u + V(x) u = f(x, u) \, \mbox{ in } \, \R^{N}, u\in \H^{s}(\R^{N}) where s∈(0,1)s\in (0,1), N>2sN>2s and ff is a continuous function satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. We consider the cases when the potential V(x)V(x) is 11-periodic or has a bounded potential well

    Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition

    Full text link
    The purpose of this paper is to study TT-periodic solutions to [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P) u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N where s∈(0,1)s\in (0,1), N>2sN>2s, T>0T>0, m>0m> 0 and f(x,u)f(x,u) is a continuous function, TT-periodic in xx and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator (−Δx+m2)s(-\Delta_{x}+m^{2})^{s} can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder ST=(0,T)N×(0,∞)\mathcal{S}_{T}=(0,T)^{N}\times (0,\infty). By using a variant of the Linking Theorem, we show that the extended problem in ST\mathcal{S}_{T} admits a nontrivial solution v(x,ξ)v(x,\xi) which is TT-periodic in xx. Moreover, by a procedure of limit as m→0m\rightarrow 0, we also prove the existence of a nontrivial solution to (P) with m=0m=0

    Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness

    Full text link
    We study the following Kirchhoff equation −(1+b∫R3∣∇u∣2dx)Δu+V(x)u=f(x,u), x∈R3.- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u), \ x \in \mathbb{R}^3. A special feature of this paper is that the nonlinearity ff and the potential VV are indefinite, hence sign-changing. Under some appropriate assumptions on VV and ff, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem
    • …
    corecore