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.     

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.     

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.     

Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains

By using the fibering method, we study the existence of non-negative solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing subcritical and supercritical lower order nonlinearities.     

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.     

Nontrivial solutions of some quasilinear problems via a cohomological local splitting

This paper deals with a generalization of the p-Laplacian type boundary value problem

     

Critical point theorems in cones and multiple positive solutions of elliptic problems

We obtain critical point variants of the compression fixed point theorem in cones of Krasnoselskii. Critical points are localized in a set defined by means of two norms. In applications to semilinear elliptic boundary value problems this makes possible the use of local Moser-Harnack inequalities for the estimations from below. Multiple solutions are found for problems with oscillating nonlinearity.     

Multiple solutions for a class of semilinear elliptic equations with general potentials

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear elliptic equations −△u+a(x)u=g(x,u)$-△u+a\left(x\right)u=g(x,u)$ in a bounded smooth domain of R^N(N≥3)${\mathbb{R}}^N(N\ge 3)$ with the Dirichlet boundary value, where the primitive of the nonlinearity g$g$ is of superquadratic growth near infinity in u$u$ and the potential a$a$ is allowed to be sign-changing. Recent results in the literature are generalized and significantly improved.     

Gradient estimates for Dirichlet parabolic problems in unbounded domains

We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂R^N involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.     

Multiplicity results for problems involving the Hardy-Sobolev operator via Morse theory

We establish some multiplicity results for a class of boundary value problems involving the Hardy-Sobolev operator using Morse theory.     

Existence of solutions for a class of singular quasilinear elliptic resonance problems

The paper concerns a resonance problem for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The equation set studied is one of the most useful sets of Navier-Stokes equations; these describe the motion of viscous fluid substances such as liquids, gases and so on. By using Galerkin-type techniques, the Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro, we show the existence of a nontrivial solution.     

Monotonicity constraints and supercritical Neumann problems

We prove the existence of a positive and radially increasing solution for a semilinear Neumann problem on a ball. No growth conditions are imposed on the nonlinearity. The method introduces monotonicity constraints which simplify the existence of a minimizer for the associated functional. Special care must be employed to establish the validity of the Euler equation.     

On an eigenvalue problem involving variable exponent growth conditions

We study the problem in Ω, u=0 on ∂Ω, where Ω is a bounded domain in R^N, is a continuous function and λ and ε are two positive constants. We prove that for any ε>0 each λ∈(0,λ₁) is an eigenvalue of the above problem, where λ₁ is the principal eigenvalue of the Laplace operator on Ω. Moreover, for each eigenvalue λ∈(0,λ₁) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques.     

On a class of quasilinear eigenvalue problems on unbounded domains

We prove several existence and non-existence results for a quasilinear eigenvalue problem with nonlinear boundary condition on unbounded domain. Our paper extends previous results obtained in Chabrowski [1] and Pflüger [4].     

On superlinear problems without the Ambrosetti and Rabinowitz condition

Existence and multiplicity results are obtained for superlinear p-Laplacian equations without the Ambrosetti and Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. Our results extend the recent results of Miyagaki and Souto [O. Miyagaki, M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008) 3628-3638].     

Multiple solutions for a class of quasilinear Schrödinger equations

In this paper, we study the following quasilinear Schrödinger equations of the form where , , . Some existence results for positive solutions, negative solutions and sequence of high energy solutions are obtained via a perturbation method.