Existence and multiplicity results for classes of elliptic resonant problems

In this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems. We also establish several multiplicity results. The methods used here are based on combining the minimax methods and the Morse theory especially some new observations on the critical groups of a local linking-type degenerate critical point.     

Existence and multiplicity results for some elliptic systems with discontinuous nonlinearities

In this paper we complete two tasks. We first extend the critical point theorem obtained by Li and Willem [S.J. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), 6-32] to the nonsmooth case in which the energy functional is locally Lipschitz and satisfies the weaker nonsmooth Cerami condition. Then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results.     

Existence and multiplicity results for some quasilinear elliptic equation with weights

Using variational methods, we show the existence and multiplicity of solutions of singular boundary value problems of the type

     

Existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions

In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.     

Existence and multiplicity results to Neumann problems for elliptic equations involving the p-Laplacian

In this paper, existence results of positive solutions to a Neumann problem involving the p-Laplacian are established. Multiplicity results are also pointed out. The approach is based on variational methods.     

Existence and multiplicity results for quasilinear elliptic differential systems

We use a nonsmooth critical point theory to prove existence results for a variational system of quasilinear elliptic equations in both the sublinear and superlinear cases. We extend a technique of Bartsch to obtain multiplicity results when the system is invariant under the action of a compact Lie group. The problem is rather different from its scalar version, because a suitable condition on the coefficients of the system seems to be necessary in order to prove the convergence of the Palais-Smale sequences. Such condition is in some sense a restriction to the "distance" between the quasilinear operator and a semilinear one.     

Existence and multiplicity results for partially superquadratic elliptic systems

In this paper we establish the existence and multiplicity of solutions for a class of partially superquadratic elliptic systems by using the Morse theory.     

Multiplicity results for a class of superlinear elliptic problems

We study a class of superlinear elliptic problems under the Dirichlet boundary condition on a bounded smooth domain in . Assuming that the nonlinearity is superlinear in a neighborhood of , we study the dependence of the number of signed and sign-changing solutions on the parameter .

     

Existence results for nonlinear elliptic problems

The existence of non-trivial solutions for nonlinear Dirichlet problems involving the p-Laplacian is investigated. In particular, an existence result of at least one non-trivial solution, without requiring any asymptotic condition on the nonlinear term either at zero or at infinity, is presented. As a consequence, also a multiplicity result is pointed out. The approach is based on a local minimum theorem for differentiable functionals.     

Existence of positive radial solutions for some nonvariational superlinear elliptic systems

The system under consideration is

     

Existence and multiplicity results for critical growth polyharmonic elliptic systems

In the present paper, we deal with the existence and multiplicity of nontrivial solutions for a class of polyharmonic elliptic systems with Sobolev critical exponent in a bounded domain. Some new existence and multiplicity results are obtained. Our proofs are based on the Nehari manifold and Ljusternik–Schnirelmann theory. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity results for singular quasilinear elliptic equation on unbounded domain

In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities (0.1) where is an unbounded exterior domain with smooth boundary ?Ω, 1 < p < N,0 ≤ a < (N ? p) ∕ p,λ > 0,1 < s < p < r < q = pN ∕ (N ? pd),d = a + 1 ? b,a ≤ b < a + 1. By the variational methods, we prove that problem 0.1 admits a sequence of solutions u_k under the appropriate assumptions on the weight functions H(x) and H(x). For the critical case, s = q,h(x) = | x | ^{? bq}, we obtain that problem 0.1 has at least a nonnegative solution with p < r < q and a sequence of solutions u_k with 1 < r < p < q and J(u_k) → 0 as k → ∞ , where J(u) is the energy functional associated to problem 0.1 . Copyright © 2013 John Wiley & Sons, Ltd.     

Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity

In this paper we study the existence of positive solutions for the problem

(0.1)     

Multiplicity theorems for superlinear elliptic problems

In this paper we study second order elliptic equations driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti–Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. Finally we indicate how in the semilinear case, Morse theory can be used to produce six nontrivial solutions. This paper was completed while the first author was visiting the University of Aveiro as an Invited Scientist. The hospitality and financial support of the host institution are gratefully acknowledged. The second and third authors acknowledge the partial financial support of the Portuguese Foundation for Science and Technology (FCT) under the research project POCI/MAT/55524/2004.