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.     

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.     

Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups

In this paper, we consider a class of resonant cooperative elliptic systems. Based on some new results concerning the computations of the critical groups and the Morse theory, we establish some new results about the existence and multiplicity of solutions under new classes of conditions. It turns out that our main results sharply improve some known results in the literature.     

Nonlinear self-adjointness of a 2D generalized second order evolution equation

We study the nonlinear self-adjointness of a general class of quasilinear 2D second order evolution equations which do not possess variational structure. For this purpose, we use the method of Ibragimov, devised and developed recently. This approach enables one to establish the conservation laws for any differential equation. We first obtain conditions determining the self-adjoint sub-class in the general case. Then, we establish the conservation laws for important particular cases: the Ricci Flow equation, the modified Ricci Flow equation and the nonlinear heat equation.     

Multiple solutions for a class of elliptic equations with jumping nonlinearities

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.     

Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian

In this paper we consider a nonlinear eigenvalue problem driven by the p$p$-Laplacian differential operator and with a nonsmooth potential. Using degree theoretic arguments based on the degree map for certain operators of monotone type, we show that the problem has at least two nontrivial positive solutions as the parameter λ>0$\lambda >0$ varies in a half-line.     

On sign-changing and multiple solutions of the p-Laplacian

In this paper, we construct the pseudo-gradient vector field in , by which we obtain the positive and negative cones of are both invariant sets of the descent flow of the corresponding functional. Then we use differential equations theory in Banach spaces and dynamics theory to study p-Laplacian boundary value problems with “jumping” nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems of p-Laplacian.     

Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential

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.     

An infinite-dimensional linking theorem and applications

In this paper, we establish a new infinite-dimensional linking theorem without (PS)-type assumptions. The new theorem needs a weaker linking geometry and produces bounded (PS) sequences. The abstract result will be applied to the study of the existence of solutions of the strongly indefinite partial differential systems. For the first application, we consider the system

     

Nonexistence of solutions for singular elliptic equations with a quadratic gradient term

This paper concerns the nonexistence of solutions for singular elliptic equations with a quadratic gradient term. The main results complement and partly extend some works by Arcoya et al. (2009) [1]. As a by-product of the main results, we fill in a gap in one of their works.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

Interior estimates for some semilinear elliptic problem with critical nonlinearity

We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary.     

An Orlicz-space approach to superlinear elliptic systems

In this paper we study superlinear elliptic systems in Hamiltonian form. Using an Orlicz-space setting, we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth. Existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense.     

Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems

In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Hardy–Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients

We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations.     

Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index

In this paper we study a quasilinear elliptic problem with the Dirichlet boundary condition in a bounded domain involving the operator Au=−Δ_pu−Δ_qu. Assuming that the nonlinearity has a concave-convex behavior we obtain some multiplicity results. More precisely, we obtain five nontrivial solutions; two by a minimization argument, two by the mountain pass theorem, and the other by a cohomological linking theorem.     

Blow-up and symmetry of sign-changing solutions to some critical elliptic equations

In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis-Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball.