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Solutions of the Schrodinger-Poisson problem concentrating on spheres, part I: necessary conditions
No full textIn this paper we study a coupled nonlinear Schrodinger-Poisson problem with radial functions. This system has been introduced as a model describing standing waves for the nonlinear Schrodinger equations in the presence of the electrostatic field.
We provide necessary conditions for concentration on sphere for the solutions of this kind of problem extending the results already known
Quasi-radial solutions for the Lane–Emden problem in the ball
Full text linkWe consider the Lane-Emden problem in the unit ball B of R^2 centered at the origin with Dirichlet boundary conditions and exponent ∈(1,+∞) of the power nonlinearity. We prove the existence of sign-changing solutions having 2 nodal domains, whose nodal line does not touch ∂ and which are non-radial. We call these solutions quasi-radial. The result is obtained for any p sufficiently large, considering least energy nodal solutions in spaces of functions invariant under suitable dihedral groups of symmetry and proving that they fulfill the required qualitative properties. We also show that these symmetric least energy solutions are instead radial for p close enough to 1, thus displaying a breaking of symmetry phenomenon in dependence on the exponent p. We then investigate the nonradial bifurcation at certain values of p from the sign-changing radial least energy solution of.. The bifurcation result gives again, with a different approach and for values of p close to the ones at which the bifurcations appear, the existence of non-radial but quasi-radial nodal solutions
On concentration of positive bound states for the Schrödinger-Poisson problem with potentials
No full textWe study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration
Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem
Full text linkWe consider the Schroedinger–Poisson–Slater (SPS) system in R3 and a nonlocal SPS type equation in balls of R3 with Dirichlet boundary conditions. We show that for every k ∈ N each problem considered admits a nodal radially symmetric solution which changes sign exactly k times in the radial variable.
Moreover, when the domain is the ball of R3 we obtain the existence of radial global solutions for the associated nonlocal parabolic problem having k + 1 nodal regions at every time
Local and global solutions for some parabolic nonlocal problems
Full text linkWe study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a ‘‘Newtonian’’ nonlocal term
Solutions of the Schrödinger-Poisson problem concentrating on spheres. II. Existence
No full textThe SchroedingerPoisson system describes standing waves for the nonlinear Schroedinger equation interacting with the electrostatic field.
We deal with the semiclassical states for this system and prove the existence of radial solutions concentrating on spheres in the presence of an external potential and with a non-constant density charge.
In particular, we show that the necessary conditions obtained in Part I are also sufficient if suitable non-degeneracy conditions are assumed.
We use a perturbation technique in a variational setting
Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit
No full textWe study the following system of equations known as Schrödinger–Poisson problem (Formula Presented) where ε>0 is a small parameter, f:R→R is given, N ≥ 3 , aN is the surface measure of the unit sphere in RN and the unknowns are υ,φ:RN→R. We construct non-radial sign-changing multi-peak solutions in the semiclassical limit. The peaks are displaced in suitable symmetric configurations and collapse to the same point as ε→ 0. The proof is based on the Lyapunov–Schmidt reduction
Ground and bound states for a static Schrödinger-Poisson-Slater problem
Full text linkWe study a static version of the Schroedinger–Poisson–Slater equation, with a power nonlinear term. The case when the power p < 2 being already studied, we consider here p ≥ 2. For p > 2 we study both the existence of ground and bound states. It turns out that p = 2 is critical in a certain sense, and will be studied separately. Finally, we prove that radial solutions satisfy a point-wise exponential decay at infinity for p > 2
Prescribed Gauss curvature problem on singular surfaces
No full textWe study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface Σ admitting conical singularities of orders αi ’s at points pi ’s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min–max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity χ(Σ)+∑iαi approaches a positive even integer, where χ(Σ) is the Euler characteristic of the surface Σ
Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods
No full textWe consider positive solutions of a fractional Lane–Emden-type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1
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