431 research outputs found

    Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schr\"odinger equations with exterior conditions

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    We consider Dirichlet exterior value problems related to a class of non-local Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain existence and uniqueness of weak solutions. Next we prove a refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also, we prove a weak anti-maximum principle in the sense of Cl\'ement-Peletier, valid on compact subsets of the domain, and a full anti-maximum principle by restricting to fractional Schr\"odinger operators. Furthermore, we show a maximum principle for narrow domains, and a refined elliptic ABP-type estimate. Finally, we obtain Liouville-type theorems for harmonic solutions and for a class of semi-linear equations. Our approach is probabilistic, making use of the properties of subordinate Brownian motion.Comment: 35 pages, Liouville-type theorems for semi-linear equations adde

    On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

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    We consider the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity ε2sM([u]s,Aε2)(−Δ)Aεsu+V(x)u=\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-\Delta)_{A_\varepsilon}^su + V(x)u = ∣u∣2s∗−2u+h(x,∣u∣2)u,|u|^{2_s^\ast-2}u + h(x,|u|^2)u,   x∈RN,\ \ x\in \mathbb{R}^N, where u(x)→0 u(x) \rightarrow 0 as ∣x∣→∞,|x| \rightarrow \infty, and (−Δ)Aεs(-\Delta)_{A_\varepsilon}^s is the fractional magnetic operator with 0<s<10<s<1, 2s∗=2N/(N−2s),2_s^\ast = 2N/(N-2s), M:R0+→R+M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+} is a continuous nondecreasing function, V:RN→R0+,V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0, and A:RN→RNA: \mathbb{R}^N \rightarrow \mathbb{R}^N are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε<E\varepsilon < \mathcal {E}; and (ii) for any m∗∈Nm^\ast \in \mathbb{N}, has m∗m^\ast pairs of solutions if ε<Em∗\varepsilon < \mathcal {E}_{m^\ast}, where E\mathcal {E} and Em∗\mathcal {E}_{m^\ast} are sufficiently small positive numbers. Moreover, these solutions uε→0u_\varepsilon \rightarrow 0 as ε→0\varepsilon \rightarrow 0

    Liouville properties and critical value of fully nonlinear elliptic operators

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    We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation

    Nonlinear equations involving the square root of the Laplacian

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    In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian A1/2A_{1/2} in a smooth bounded domain Ω⊂Rn\Omega\subset \mathbb{R}^n (n≥2n\geq 2) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega. \end{array}\right. \end{equation*} The existence of at least two non-trivial L∞L^{\infty}-bounded weak solutions is established for large value of the parameter λ\lambda requiring that the nonlinear term ff is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method

    A new critical curve for a class of quasilinear elliptic systems

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    We study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms. The main model cases are given by the pp-Laplacian operator as well as the mean curvature operator in non parametric form. We prove that if the exponents lie under a certain curve, then the system has only the trivial solution. These results hold without any restriction provided the possible solutions are more regular. The underlying framework is the classical Euclidean case as well as the Carnot groups setting.Comment: 28 page

    Periodic solutions for critical fractional problems

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    We deal with the existence of 2π2\pi-periodic solutions to the following non-local critical problem \begin{equation*} \left\{\begin{array}{ll} [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in} (-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N}, \quad i=1, \dots, N, \end{array} \right. \end{equation*} where s∈(0,1)s\in (0,1), N≥4sN \geq 4s, m≥0m\geq 0, 2s∗=2NN−2s2^{*}_{s}=\frac{2N}{N-2s} is the fractional critical Sobolev exponent, W(x)W(x) is a positive continuous function, and f(x,u)f(x, u) is a superlinear 2π2\pi-periodic (in xx) continuous function with subcritical growth. When m>0m>0, the existence of a nonconstant periodic solution is obtained by applying the Linking Theorem, after transforming the above non-local problem into a degenerate elliptic problem in the half-cylinder (−π,π)N×(0,∞)(-\pi,\pi)^{N}\times (0, \infty), with a nonlinear Neumann boundary condition, through a suitable variant of the extension method in periodic setting. We also consider the case m=0m=0 by using a careful procedure of limit. As far as we know, all these results are new.Comment: Calculus of Variations and Partial Differential Equations (2018
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