65,018 research outputs found

    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), N4sN \geq 4s, m0m\geq 0, 2s=2NN2s2^{*}_{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

    Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

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    We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page

    Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems

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    We prove the existence of positive periodic solutions for the second order nonlinear equation u"+a(x)g(u)=0u" + a(x) g(u) = 0, where g(u)g(u) has superlinear growth at zero and at infinity. The weight function a(x)a(x) is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.Comment: 41 page

    Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case

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    We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where g ⁣:[0,+[[0,+[g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[} is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when 0Ta(t) ⁣dt<0\int_{0}^{T} a(t) \!dt < 0 and λ>0\lambda > 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.Comment: 26 page
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