## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (192.320 recursos)

Differential Integral Equations

1. #### Nodal solutions to problem with mean curvature operator in Minkowski space

Dai, Guowei; Wang, Jun
This paper is devoted to investigate the existence and multiplicity of radial nodal solutions for the following Dirichlet problem with mean curvature operator in Minkowski space \begin{eqnarray} \begin{cases} -\text{div} \Big (\frac{\nabla v}{\sqrt{1-\vert \nabla v\vert^2}} \Big ) = \lambda f(\vert x\vert,v)\,\, &\text{in}\,\, B_R(0),\\ v=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial B_R(0). \end{cases} \nonumber \end{eqnarray} By bifurcation approach, we determine the interval of parameter $\lambda$ in which the above problem has two or four radial nodal solutions which have exactly $n-1$ simple zeros in $(0,R)$ according to linear/sublinear/ superlinear nonlinearity at zero. The asymptotic behaviors of radial nodal solutions as $\lambda \to +\infty$ and $n \to +\infty$ are also studied.

2. #### Pullback dynamics of non-autonomous wave equations with acoustic boundary condition

Ma, To Fu; Souza, Thales Maier
This paper is concerned with a class of wave equations with acoustic boundary condition subject to non-autonomous external forces. Under some general assumptions, the problem generates a well-posed evolution process. Then, we establish the existence of a minimal pullback attractor within a universe of tempered sets defined by the forcing terms. We also, study the upper semicontinuity of attractors as the non-autonomous perturbation tends to zero. Our results allow unbounded external forces and nonlinearities with critical growth.

3. #### Uniform bounds for solutions to volume-surface reaction diffusion systems

Sharma, Vandana; Morgan, Jeff
We consider reaction-diffusion systems where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish criteria guaranteeing that solutions are uniformly bounded in time.

4. #### Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

Bhakta, Mousomi; Mukherjee, Debangana
In this paper, we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $\mathcal{L}_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{align*} \mathcal{L}_{K} u + \mu |u|^{q-1}u + \lambda |u|^{p-1}u &= 0 \quad\mbox{in}\quad \Omega, \\ u&=0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N > 2s$, $s\in(0, 1)$, $0 < q < 1 < p\leq \frac{N+2s}{N-2s}$. Moreover, when $\mathcal{L}_K$ reduces to the fractional laplacian operator $-(-\Delta)^s$, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2} (\frac{N+2s}{N-2s}) < q < 1$, $N > 6s$, $\lambda =1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign...

5. #### Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

Green, William R.; Toprak, Ebru
We investigate dispersive estimates for the Schrödinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion \begin{align*} e^{itH}\chi(H) P_{ac}(H) & =O(\frac 1{\log t}) A_0+ O(\frac 1 t )A_1+O((t\log t)^{-1})A_2 \\ & + O(t^{-1}(\log t)^{-2})A_3. \end{align*} Here, $A_0,A_1:L^1(\mathbb R^4)\to L^\infty (\mathbb R^4)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions,...

6. #### Bifurcation of Space Periodic Solutions in Symmetric Reversible FDEs

Balanov, Zalman; Wu, Hao-Pin
In this paper, we propose an equivariant degree based method to study bifurcation of periodic solutions (of constant period) in symmetric networks of reversible FDEs. Such a bifurcation occurs when eigenvalues of linearization move along the imaginary axis (without change of stability of the trivial solution and possibly without $1:k$ resonance). Physical examples motivating considered settings are related to stationary solutions to PDEs with non-local interaction: reversible mixed delay differential equations (MDDEs) and integro-differential equations (IDEs). In the case of $S_4$-symmetric networks of MDDEs and IDEs, we present exact computations of full equivariant bifurcation invariants. Algorithms and computational procedures used in this paper are also included.

7. #### Bifurcation of Space Periodic Solutions in Symmetric Reversible FDEs

Balanov, Zalman; Wu, Hao-Pin
In this paper, we propose an equivariant degree based method to study bifurcation of periodic solutions (of constant period) in symmetric networks of reversible FDEs. Such a bifurcation occurs when eigenvalues of linearization move along the imaginary axis (without change of stability of the trivial solution and possibly without $1:k$ resonance). Physical examples motivating considered settings are related to stationary solutions to PDEs with non-local interaction: reversible mixed delay differential equations (MDDEs) and integro-differential equations (IDEs). In the case of $S_4$-symmetric networks of MDDEs and IDEs, we present exact computations of full equivariant bifurcation invariants. Algorithms and computational procedures used in this paper are also included.

8. #### On ordinary differential inclusions with mixed boundary conditions

Bonanno, Gabriele; Iannizzotto, Antonio; Marras, Monica
By means of nonsmooth critical point theory, we prove existence of three weak solutions for an ordinary differential inclusion of Sturm-Liouville type involving a general set-valued reaction term depending on a parameter, and coupled with mixed boundary conditions. As an application, we give a multiplicity result for ordinary differential equations involving discontinuous nonlinearities.

9. #### On ordinary differential inclusions with mixed boundary conditions

Bonanno, Gabriele; Iannizzotto, Antonio; Marras, Monica
By means of nonsmooth critical point theory, we prove existence of three weak solutions for an ordinary differential inclusion of Sturm-Liouville type involving a general set-valued reaction term depending on a parameter, and coupled with mixed boundary conditions. As an application, we give a multiplicity result for ordinary differential equations involving discontinuous nonlinearities.

10. #### On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

Ciesielski, Jakub; Janczewska, Joanna; Waterstraat, Nils
We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in \mathbb R$, $q\in\mathbb R^n$, $a\colon\mathbb R\to\mathbb R$ is a continuous positive bounded function, $G\colon\mathbb R^n\to\mathbb R$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb R\to\mathbb R^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.

11. #### On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

Ciesielski, Jakub; Janczewska, Joanna; Waterstraat, Nils
We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in \mathbb R$, $q\in\mathbb R^n$, $a\colon\mathbb R\to\mathbb R$ is a continuous positive bounded function, $G\colon\mathbb R^n\to\mathbb R$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb R\to\mathbb R^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.

12. #### Positive semiclassical states for a fractional Schrödinger-Poisson system

Murcia, Edwin G.; Siciliano, Gaetano
We consider a fractional Schrödinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e., in the semiclassical limit'') the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential.

13. #### Positive semiclassical states for a fractional Schrödinger-Poisson system

Murcia, Edwin G.; Siciliano, Gaetano
We consider a fractional Schrödinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e., in the semiclassical limit'') the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential.

14. #### Three solutions for an elliptic system near resonance with the principal eigenvalue

Massa, Eugenio; Rossato, Rafael Antônio
We consider an elliptic system of Hamiltonian type with linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least three solutions when the linear part is near resonance with the principal eigenvalue, either from above or from below. For two of these solutions, we also obtain information on the sign of its components. The system is associated to a strongly indefinite functional and the solutions are obtained trough saddle point theorem, after truncating the nonlinearity.

15. #### Three solutions for an elliptic system near resonance with the principal eigenvalue

Massa, Eugenio; Rossato, Rafael Antônio
We consider an elliptic system of Hamiltonian type with linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least three solutions when the linear part is near resonance with the principal eigenvalue, either from above or from below. For two of these solutions, we also obtain information on the sign of its components. The system is associated to a strongly indefinite functional and the solutions are obtained trough saddle point theorem, after truncating the nonlinearity.

16. #### The focusing cubic NLS with inverse-square potential in three space dimensions

Killip, Rowan; Murphy, Jason; Visan, Monica; Zheng, Jiqiang
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS [7, 11]. We also prove failure of uniform space-time bounds at the

17. #### The focusing cubic NLS with inverse-square potential in three space dimensions

Killip, Rowan; Murphy, Jason; Visan, Monica; Zheng, Jiqiang
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS [7, 11]. We also prove failure of uniform space-time bounds at the

18. #### Multiple positive solutions for the $m$-Laplacian and a nonlinearity with many zeros

Iturriaga, Leonelo; Lorca, Sebastián; Massa, Eugenio
In this paper, we consider the quasilinear elliptic equation $-\Delta_m u=\lambda f(u)$, in a bounded, smooth and convex domain. When the nonnegative nonlinearity $f$ has multiple positive zeros, we prove the existence of at least two positive solutions for each of these zeros, for $\lambda$ large, without any hypothesis on the behavior at infinity of $f$. We also prove a result concerning the behavior of the solutions as $\lambda\to\infty$.

19. #### An infinite number of solutions for an elliptic problem with power nonlinearity

We consider the following nonlinear elliptic equation $$\tag{0.1} A_\frac{1}{2} u = K(x) |u|^{p-1}u \hbox{ in } \Omega, \;\; u=0 \hbox{ on } \partial\Omega,$$ where $\Omega$ is a bounded domain of $\mathbb{R}^n, n\geq 1, K(x)$ is a given function, $A_\frac{1}{2}$ represents the square root of $-\Delta$ in $\Omega$ with zero Dirichlet boundary condition and $1 < p < \frac{n+1}{n-1}$, $(p > 1$ if $n=1$). We apply the Brouwer's fixed point theorem to prove that (0.1) has infinitely many distinct solutions.

20. #### Ground states for a fractional scalar field problem with critical growth

Ambrosio, Vincenzo
We prove the existence of a ground state solution for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u= g(u) \mbox{ in } \mathbb R^{N} \end{align*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, and $g\in C^{1, \beta}( \mathbb R, \mathbb R)$ is an odd function satisfying the critical growth assumption.

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