Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side

We deal with existence, non-existence and multiplicity of solutions to the model problem

(P)     

Infinitely many solutions to perturbed elliptic equations

A new version of perturbation theory is developed which produces infinitely many sign-changing critical points for uneven functionals. The abstract result is applied to the following elliptic equations with a Hardy potential and a perturbation from symmetry:

     

Perturbation semi-linéaire des résolvantes et des semi-groupes

Résumé La perturbation semi-linéaire des résolvantes et des semi-groupes linéaires, nous donne des résolvantes et des semi-groupes non linéaires. Nous étudions alors les propriétés de ces opérateurs non linéaires et en particulier les fonctions surmédianes et excessives associées.

---

We are concerned with nonlinear resolvents and semi-groups. They are obtained by perturbing linear ones. Properties of these nonlinear operators are investigated, particularly supermedian and excessive functions.

Ce travail est soutenu par la fondation nationale pour la recherche scientifique. Projet MA4-89-FST.     

On (r, 2)-capacities for a class of elliptic pseudo differential operators

Support by DFG contract Ja 511/1-1 is gratefully acknowledged     

Degree and global bifurcation for elliptic equations with multivalued unilateral conditions

A bifurcation problem for an elliptic multivalued boundary value problem with a real parameter is considered. The existence of global bifurcation between two eigenvalues of a certain type of the Laplacian is proved. For a class of abstract inclusions with compact multivalued mappings in a Hilbert space, it is shown how the degree can be determined near the eigenvalues of a particular type of an associated linear single-valued problem, and the jump of the degree is proved. As a consequence, global bifurcation for such abstract inclusions is obtained. The weak formulation of the boundary value problem mentioned is a particular case.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

A Hausdorff measure classification ofG-polar sets for the superdiffusions

Summary We establish relations betweenG-polar sets of superdiffusions and the restricted Hausdorff dimension. As an application, we give new proofs of Dynkin's criteria for theS-polarity andH-polarity (established earlier by Dawson, Iscoe, Perkins, and Le Gall under more restrictive assumptions.)     

A density result for Sobolev spaces in dimension two, and applications to stability of nonlinear Neumann problems

We prove that if Ω⊆R² is bounded and R²?Ω satisfies suitable structural assumptions (for example it has a countable number of connected components), then W1,2(Ω) is dense in W1,p(Ω) for every 1?p<2. The main application of this density result is the study of stability under boundary variations for nonlinear Neumann problems of the form

     

Nonlinear elliptic equations with measure data

In this paper we prove the existence of solutions of nonlinear equations of the type-div(a(x, u, Du)+H(x, u, Du)=f, wherea andH are Caratheodory functions andf is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results.     

Fully nonlinear elliptic equations and the dini condition

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.     

Multiple solutions of nonlinear elliptic systems

We proved a multiplicity result for a nonlinear elliptic system in R^N. The functional related to the system is strongly indefinite. We investigated the relation between the number of solutions and the topology of the set of the global maxima of the coefficients.     

Markov processes associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Nonlinear potential theory and PDEs

We consider equations like -div(|u| ^p–2u)=, where is a nonnegative Radon measure and 1u and the measure are reviewed. A link between potential estimates and the boundary regularity of the Dirichlet problem is established.     

Optimal boundary regularity for nonlinear singular elliptic equations

In this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary. By a careful construction of super- and sub-solutions, we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument.     

Comparison principle for some classes of nonlinear elliptic equations

We prove a comparison principle for second order quasilinear elliptic operators in divergence form when a first order term appears. We deduce uniqueness results for weak solutions to Dirichlet problems when data belong to the natural dual space.     

Eigenvalue problems for nonlinear elliptic equations with unilateral constraints

In this paper we study eigenvalue problems for hemivariational and variational inequalities driven by the p$p$-Laplacian differential operator. Using topological methods (based on multivalued versions of the Leray–Schauder alternative principle) and variational methods (based on the nonsmooth critical point theory), we prove existence and multiplicity results for the eigenvalue problems that we examine.     

Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials

We study the equation Δu+u|u|^p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|⁻² where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.     

A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations

We consider nonlinear elliptic equations driven by the p-Laplacian differential operator. Using degree theoretic arguments based on the degree map for operators of type (S)₊ , we prove theorems on the existence of multiple nontrivial solutions of constant sign.