Existence of multiple solutions with precise sign information for superlinear Neumann problems

We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator and having a p-superlinear nonlinearity. Using truncation techniques combined with the method of upper–lower solutions and variational arguments based on critical point theory, we prove the existence of five nontrivial smooth solutions, two positive, two negative and one nodal. For the semilinear (i.e., p = 2) problem, using critical groups we produce a second nodal solution. This paper was completed while N.S. Papageorgiou was visiting the University of Aveiro as an invited scientist. The hospitality and financial support of the host institution are gratefully acknowledged. V. Staicu acknowledges partial financial support from the Portuguese Foundation for Sciences and Technology (FCT) under the project POCI/MAT/55524/2004.     

Sobolev type embedding and quasilinear elliptic equations with radial potentials

We study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials. The existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions.     

The existence of two nontrivial solutions via homological local linking for the non-coercive p-Laplacian Neumann problem

In this paper we consider a nonlinear Neumann problem driven by the p-Laplacian and with a Carathéodory right hand side nonlinearity f(z,x). The hypothesis on f(z,x) does not imply the coercivity of the corresponding Euler functional. Using variational arguments and critical groups we show that the problem has at least two nontrivial smooth solutions.     

Bifurcation direction and exchange of stability for variational inequalities on nonconvex sets

This paper concerns Crandall–Rabinowitz type bifurcation for abstract variational inequalities on nonconvex sets and with multidimensional bifurcation parameter. We derive formulae which determine the bifurcation direction and, in the case of potential operators, the stability of all solutions close to the bifurcation point. In particular, it follows that in some cases an exchange of stability appears similar to the case of equations, but in some other cases stable nontrivial solutions bifurcate at points where there is no loss of stability of the trivial solution. As an application we consider a system of two second order ODEs with nonconvex unilateral boundary conditions.     

Two solutions for inhomogeneous nonlinear elliptic equations at critical growth

The existence of two nontrivial solutions for a class of fully nonlinear problems at critical growth with perturbations of lower order is proved. The first solution is obtained via a local minimization argument while the second solution follows by a non-smooth mountain pass theorem.     

Solutions to a gradient system with resonance at both zero and infinity

In this paper, we study the existence and multiplicity of nontrivial solutions for a gradient system with resonance at both zero and infinity via Morse theory.     

Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

In this paper, we study the Dirichlet problem for a class of infinitely degenerate nonlinear elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy's inequality, the existence and regularity of multiple nontrivial solutions have been proved.     

Multiplicity results for quasi-linear problems

Applying the minimax arguments and Morse theory, we establish some results on the existence of multiple nontrivial solutions for a class of p$p$-Laplacian elliptic equations.     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem

We consider a general nonlinear elliptic problem of the second order whose associated functional presents two linking structures and we prove the existence of three nontrivial solutions to the problem.     

Periodic solutions for an asymptotically linear wave equation with resonance

In this paper, we study the existence and multiplicity of nontrivial periodic solutions for an asymptotically linear wave equation with resonance, both at infinity and at zero. The main features are using Morse theory for the strongly indefinite functional and the precise computation of critical groups under conditions which are more general.     

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.     

On resonant Neumann problems: Multiplicity of solutions

We consider a semilinear Neumann problem with an asymptotically linear reaction term. We assume that resonance occurs at infinity. Using variational methods based on the critical point theory, together with the reduction technique and Morse theory, we show that the problem has at least four nontrivial smooth solutions.     

Radial entire solutions for supercritical biharmonic equations

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13. Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.     

Soliton solutions for quasilinear Schrödinger equations with critical growth

In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9].     

Multiplicity of solutions for a class of fourth elliptic equations

We show that there exist at least three nontrivial solutions for a class of fourth elliptic equations under Navier boundary conditions by linking approaches.     

Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC ²-function at both isolated critical point and infinity.     

On some nonlinear elliptic problems for p–Laplacian in \mathbb {R}^N

In this paper, we consider a nonlinear elliptic problem involving the p-Laplacian with perturbation terms in the whole . Via variational arguments, we obtain existence and regularity of nontrivial solutions. The research of the first and second authors is supported by grant num, #28/12 from the Al-Imam University, Riyadh, KSA.