Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|^p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.     

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

On nonlinear conservation laws with a nonlocal diffusion term

Scalar one-dimensional conservation laws with a nonlocal diffusion term corresponding to a Riesz-Feller differential operator are considered. Solvability results for the Cauchy problem in L^∞ are adapted from the case of a fractional derivative with homogeneous symbol. The main interest of this work is the investigation of smooth shock profiles. In the case of a genuinely nonlinear smooth flux function we prove the existence of such travelling waves, which are monotone and satisfy the standard entropy condition. Moreover, the dynamic nonlinear stability of the travelling waves under small perturbations is proven, similarly to the case of the standard diffusive regularisation, by constructing a Lyapunov functional.     

Lagrangian dynamics on an infinite-dimensional torus; a Weak KAM theorem

The space L²(0,1) has a natural Riemannian structure on the basis of which we introduce an L²(0,1)-infinite-dimensional torus T. For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute action-minimizing solutions of prescribed rotation number for the one-dimensional nonlinear Vlasov system with periodic potential.     

KKM-Fan’s lemma for solving nonlinear vector evolution equations with nonmonotonic perturbations

In this paper we are interested in the existence of solutions of the following initial value problem: on (0,T) with u(0)=u₀ where A:V→V^′ is a monotone operator, G:V→V^′ is a nonlinear nonmonotone operator and f:(0,T)→V^′ is a measurable function, by means of a recent generalization of the famous KKM-Fan’s lemma.     

Elliptic problems in generalized Orlicz-Musielak spaces

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a ??₂ nor ?₂-condition for an inhomogeneous and anisotropic N-function but assume it to be log-H?lder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ^??-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

We consider the problem −Δu+a(x)u=f(x)|u|^2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|^2*−2u in Ω, u=0 on ∂Ω.     

Multiple solutions for a Hénon-like equation on the annulus

For the equation −Δu=||xα|−2|up−1, 1<|x|<3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent ^2∗=2N/(N−2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions.     

On a backward Cauchy problem associated with continuous spectrum operator

The nonlinear backward Cauchy problem

u_t+Au(t)=f(u(t)),u(T)=φ,     

Boundary value problem for first order impulsive functional integro-differential equations

This paper investigates extremal solutions of the boundary value problem for impulsive functional integro-differential equations with nonlinear boundary conditions and deviating arguments. In the presence of a lower solution u and an upper solution v with u≥v, existence of extremal solutions is proved by establishing a new comparison principle and using the monotone iterative technique.     

Equivalence of communication and projective boundedness properties for monotone and homogeneous functions

This work concerns the eigenvalue problem for a monotone and homogeneous self-mapping f of a finite-dimensional positive cone. A communication criterion is formulated such that it is equivalent to the projective boundedness of the upper eigenspaces associated with f, a property that yields the existence of a nonlinear eigenvalue. Using the idea of dual function, a similar result is obtained for lower eigenspaces.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

Existence of solutions to nonlinear Hammerstein integral equations and applications

In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L²(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.     

Biharmonic nonlinear Schrödinger equation and the profile decomposition

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L²-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H²(R^N), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given.     

On ground state solutions for singular and semi-linear problems including super-linear terms at infinity

We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation −Δu=λf(x,u),x∈R^N, where f:R^N×(0,∞)→[0,∞) is a suitable function and λ>0 is a real parameter. This result completes the principal theorem of A. Mohammed [A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.11.080] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure will be used.     

An implicit iterative scheme for monotone variational inequalities and fixed point problems

In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of N$N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of N+1$N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other N$N$ mappings are nonexpansive.     

Eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance

The abstract boundary value problem Lu + Gu = f, u ϵ dom(L) ⊂ H, is considered. Here H is used to denote a real separable Hilbert space, L a closed symmetric linear operator, and G a nonlinear operator assumed to be Lipschitz continuous and strongly monotone. In addition L is assumed to have a complete set of eigenfunctions in H, and is allowed to have an infinite dimensional null space. The existence of unique solutions, depending continuously on f, is established by a constructive approach. Galerkin approximations are considered and error estimates are given. As an application of the main result, the existence of time periodic weak solutions of the n-dimensional wave equation is shown.     

A note on the implicit function theorem for quasi-linear eigenvalue problems

We consider the quasi-linear eigenvalue problem −Δ_pu=λg(u) subject to Dirichlet boundary conditions on a bounded open set Ω, where g is a locally Lipschitz continuous function. Imposing no further conditions on Ω or g, we show that for λ near zero the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions parameterized by λ depends continuously on the parameter.