Nils Ackermann | Goethe-Universität Frankfurt am Main (original) (raw)
Papers by Nils Ackermann
On the multiplicity of sign changing solutions to nonlinear periodic Schrodinger equations Nils A... more On the multiplicity of sign changing solutions to nonlinear periodic Schrodinger equations Nils Ackermann Mathematisch. es Institut, Universitat Giessen Amdtstr. 2, 35392 Giessen Germany Abstract. Under weak non degeneracy assumptions we show that for certain superlinear, ...
12 smooth functions the result follows. 9 Alternatively, one uses directly in the variations-of-c... more 12 smooth functions the result follows. 9 Alternatively, one uses directly in the variations-of-constants formula thatA is resolvent positive. 84 Strong monotonicity has striking consequences for the qualitative be-havior of the flow. It is used, for example, in the study of convergence of ...
Communications in Contemporary Mathematics, 2005
ABSTRACT We prove the existence of infinitely many geometrically distinct two bump solutions of p... more ABSTRACT We prove the existence of infinitely many geometrically distinct two bump solutions of periodic superlinear Schrödinger equations of the type -Δu+V(x)u=f(x,u), where x∈ℝ N and lim |x|→∞ u(x)=0. The solutions we construct change sign and have exactly two nodal domains. The usual multibump constructions for these equations rely on strong non-degeneracy assumptions. We present a new approach that only requires a weak splitting condition. In the second part of the paper we exhibit classes of potentials V for which this splitting condition holds.
Calculus of Variations and Partial Differential Equations, 1998
Abstract. For a smooth domain Ω with compact boundary we investigate the problem −d2∆u + u = f (u... more Abstract. For a smooth domain Ω with compact boundary we investigate the problem −d2∆u + u = f (u) with Neumann boundary conditions, where f has superlinear but subcritical growth. Provided that d > 0 is sufficiently small we show the existence of at least cat(∂Ω) positive solutions ...
We are concerned with the properties of weak solutions of the stationary Schrödinger equation... more We are concerned with the properties of weak solutions of the
stationary Schrödinger equation −Deltau+Vu=f(u)-\Delta u + Vu = f(u)−Deltau+Vu=f(u), $u\in
H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where VVV is Hölder
continuous and infV>0\inf V>0infV>0. Assuming fff to be continuous and
bounded near 000 by a power function with exponent larger than 111
we provide precise decay estimates at infinity for solutions in
terms of Green's function of the Schrödinger operator. In some
cases this improves known theorems on the decay of solutions. If
fff is also real analytic on (0,infty)(0,\infty)(0,infty) we obtain that the set of
positive solutions is locally path connected. For a periodic
potential VVV this implies that the standard variational functional
has discrete critical values in the low energy range and that a
compact isolated set of positive solutions exists, under additional
assumptions.
Journal of Differential Equations, 2013
We consider the supercritical problem
Milan Journal of Mathematics, 2011
Let M be a smooth k-dimensional closed submanifold of R N , N ≥ 2, and let Ω R be the open tubula... more Let M be a smooth k-dimensional closed submanifold of R N , N ≥ 2, and let Ω R be the open tubular neighborhood of radius 1 of the expanded manifold M R := {Rx : x ∈ M }. For R sufficiently large we show the existence of positive multibump solutions to the problem
is Borel measurable, define for σ -finite positive Borel measures µ, ν on R n the bilinear integr... more is Borel measurable, define for σ -finite positive Borel measures µ, ν on R n the bilinear integral expression
In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vect... more In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vector fields that are asymptotically additive in a well-defined sense. This result is used to obtain multibump solutions for two basic types of periodic stationary Schrödinger equations with superlinear nonlinearity. The nonlinear term may be of convolution type. If the superquadratic term in the energy functional is convex, our results also apply in certain cases if 0 is in a gap of the spectrum of the Schrödinger operator.
For a domain Ω ⊂ R N we consider the equation
We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems... more We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain, posed with homogeneous Dirichlet boundary conditions. The distinguishing feature of this class is the indefinite superlinear (but subcritical) growth of the nonlinearity at infinity. We present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria. In addition, we can prove the existence of connecting orbits in many cases.
Let Γ denote a smooth simple curve in R N , N ≥ 2, possibly with boundary. Let Ω R be the open no... more Let Γ denote a smooth simple curve in R N , N ≥ 2, possibly with boundary. Let Ω R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ := {Rx | x ∈ Γ ∂Γ}. Consider the superlinear problem −∆u + λu = f (u) on the domains Ω R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd.
We consider the singularly perturbed semilinear parabolic problem u t − d 2 ∆u + u = f (u) with h... more We consider the singularly perturbed semilinear parabolic problem u t − d 2 ∆u + u = f (u) with homogeneous Neumann boundary conditions on a smoothly bounded domain Ω ⊆ R N . Here f is superlinear at 0 and ±∞ and has subcritical growth. For small d > 0 we construct a compact connected invariant set X d in the boundary of the domain of attraction of the asymptotically stable equilibrium 0. The main features of X d are that it consists of positive functions that are pairwise non-comparable, and that its topology is at least as rich as the topology of ∂Ω in a certain sense. If the number of equilibria in X d is finite this implies the existence of connecting orbits within X d that are not a consequence of a well known result by Matano. * Supported by DFG Grants BA 1009/15-1, BA 1009/15-2. † Supported by the research project MSM 0021620839, financed by MSMT, and partly supported by the GACR project 201/06/0352.
We consider the supercritical problem
We consider the Choquard-Pekar equation
The time-independent superlinear Schrödinger equation with spatially periodic and positive potent... more The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials.
We investigate the dynamics of the semiflow ϕ induced on H 1 0 ( ) by the Cauchy problem of the s... more We investigate the dynamics of the semiflow ϕ induced on H 1 0 ( ) by the Cauchy problem of the semilinear parabolic equation
We prove a variant of the Brézis-Lieb Lemma that applies to more general nonlinear superposition ... more We prove a variant of the Brézis-Lieb Lemma that applies to more general nonlinear superposition operators within a certain range of growth exponents, at the expense of stronger conditions on the admissible sequences of functions. This new set of conditions is well adapted to second order semilinear elliptic partial differential equations on R N . The proof rests on the uniform continuity of superposition operators on bounded subsets of Sobolev space, which we obtain from an application of the concentration compactness method.
On the multiplicity of sign changing solutions to nonlinear periodic Schrodinger equations Nils A... more On the multiplicity of sign changing solutions to nonlinear periodic Schrodinger equations Nils Ackermann Mathematisch. es Institut, Universitat Giessen Amdtstr. 2, 35392 Giessen Germany Abstract. Under weak non degeneracy assumptions we show that for certain superlinear, ...
12 smooth functions the result follows. 9 Alternatively, one uses directly in the variations-of-c... more 12 smooth functions the result follows. 9 Alternatively, one uses directly in the variations-of-constants formula thatA is resolvent positive. 84 Strong monotonicity has striking consequences for the qualitative be-havior of the flow. It is used, for example, in the study of convergence of ...
Communications in Contemporary Mathematics, 2005
ABSTRACT We prove the existence of infinitely many geometrically distinct two bump solutions of p... more ABSTRACT We prove the existence of infinitely many geometrically distinct two bump solutions of periodic superlinear Schrödinger equations of the type -Δu+V(x)u=f(x,u), where x∈ℝ N and lim |x|→∞ u(x)=0. The solutions we construct change sign and have exactly two nodal domains. The usual multibump constructions for these equations rely on strong non-degeneracy assumptions. We present a new approach that only requires a weak splitting condition. In the second part of the paper we exhibit classes of potentials V for which this splitting condition holds.
Calculus of Variations and Partial Differential Equations, 1998
Abstract. For a smooth domain Ω with compact boundary we investigate the problem −d2∆u + u = f (u... more Abstract. For a smooth domain Ω with compact boundary we investigate the problem −d2∆u + u = f (u) with Neumann boundary conditions, where f has superlinear but subcritical growth. Provided that d > 0 is sufficiently small we show the existence of at least cat(∂Ω) positive solutions ...
We are concerned with the properties of weak solutions of the stationary Schrödinger equation... more We are concerned with the properties of weak solutions of the
stationary Schrödinger equation −Deltau+Vu=f(u)-\Delta u + Vu = f(u)−Deltau+Vu=f(u), $u\in
H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where VVV is Hölder
continuous and infV>0\inf V>0infV>0. Assuming fff to be continuous and
bounded near 000 by a power function with exponent larger than 111
we provide precise decay estimates at infinity for solutions in
terms of Green's function of the Schrödinger operator. In some
cases this improves known theorems on the decay of solutions. If
fff is also real analytic on (0,infty)(0,\infty)(0,infty) we obtain that the set of
positive solutions is locally path connected. For a periodic
potential VVV this implies that the standard variational functional
has discrete critical values in the low energy range and that a
compact isolated set of positive solutions exists, under additional
assumptions.
Journal of Differential Equations, 2013
We consider the supercritical problem
Milan Journal of Mathematics, 2011
Let M be a smooth k-dimensional closed submanifold of R N , N ≥ 2, and let Ω R be the open tubula... more Let M be a smooth k-dimensional closed submanifold of R N , N ≥ 2, and let Ω R be the open tubular neighborhood of radius 1 of the expanded manifold M R := {Rx : x ∈ M }. For R sufficiently large we show the existence of positive multibump solutions to the problem
is Borel measurable, define for σ -finite positive Borel measures µ, ν on R n the bilinear integr... more is Borel measurable, define for σ -finite positive Borel measures µ, ν on R n the bilinear integral expression
In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vect... more In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vector fields that are asymptotically additive in a well-defined sense. This result is used to obtain multibump solutions for two basic types of periodic stationary Schrödinger equations with superlinear nonlinearity. The nonlinear term may be of convolution type. If the superquadratic term in the energy functional is convex, our results also apply in certain cases if 0 is in a gap of the spectrum of the Schrödinger operator.
For a domain Ω ⊂ R N we consider the equation
We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems... more We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain, posed with homogeneous Dirichlet boundary conditions. The distinguishing feature of this class is the indefinite superlinear (but subcritical) growth of the nonlinearity at infinity. We present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria. In addition, we can prove the existence of connecting orbits in many cases.
Let Γ denote a smooth simple curve in R N , N ≥ 2, possibly with boundary. Let Ω R be the open no... more Let Γ denote a smooth simple curve in R N , N ≥ 2, possibly with boundary. Let Ω R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ := {Rx | x ∈ Γ ∂Γ}. Consider the superlinear problem −∆u + λu = f (u) on the domains Ω R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd.
We consider the singularly perturbed semilinear parabolic problem u t − d 2 ∆u + u = f (u) with h... more We consider the singularly perturbed semilinear parabolic problem u t − d 2 ∆u + u = f (u) with homogeneous Neumann boundary conditions on a smoothly bounded domain Ω ⊆ R N . Here f is superlinear at 0 and ±∞ and has subcritical growth. For small d > 0 we construct a compact connected invariant set X d in the boundary of the domain of attraction of the asymptotically stable equilibrium 0. The main features of X d are that it consists of positive functions that are pairwise non-comparable, and that its topology is at least as rich as the topology of ∂Ω in a certain sense. If the number of equilibria in X d is finite this implies the existence of connecting orbits within X d that are not a consequence of a well known result by Matano. * Supported by DFG Grants BA 1009/15-1, BA 1009/15-2. † Supported by the research project MSM 0021620839, financed by MSMT, and partly supported by the GACR project 201/06/0352.
We consider the supercritical problem
We consider the Choquard-Pekar equation
The time-independent superlinear Schrödinger equation with spatially periodic and positive potent... more The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials.
We investigate the dynamics of the semiflow ϕ induced on H 1 0 ( ) by the Cauchy problem of the s... more We investigate the dynamics of the semiflow ϕ induced on H 1 0 ( ) by the Cauchy problem of the semilinear parabolic equation
We prove a variant of the Brézis-Lieb Lemma that applies to more general nonlinear superposition ... more We prove a variant of the Brézis-Lieb Lemma that applies to more general nonlinear superposition operators within a certain range of growth exponents, at the expense of stronger conditions on the admissible sequences of functions. This new set of conditions is well adapted to second order semilinear elliptic partial differential equations on R N . The proof rests on the uniform continuity of superposition operators on bounded subsets of Sobolev space, which we obtain from an application of the concentration compactness method.