Quasilinear elliptic equations in \RN via variational methods and Orlicz-Sobolev embeddings

In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.Comment: 18 pages, 1 figur

Existence and multiplicity of solutions to equations of N−N-Laplacian type with critical exponential growth in RN\mathbb{R}^{N}

In this paper, we deal with the existence and multiplicity of solutions to the nonuniformly elliptic equation of the N-Lapalcian type with a potential and a nonlinear term of critical exponential growth and satisfying the Ambrosetti-Rabinowitz condition. In spite of a possible failure of the Palais-Smale compactness condition, in this article we apply minimax method to obtain the weak solution to such an equation. In particular, in the case of $N-$Laplacian, using the minimization and the Ekeland variational principle, we obtain multiplicity of weak solutions. Finally, we will prove the above results when our nonlinearity doesn't satisfy the well-known Ambrosetti-Rabinowitz condition and thus derive the existence and multiplicity of solutions for a much wider class of nonlinear terms $f$.Comment: 30 pages. First draft in November, 201

On local compactness in quasilinear elliptic problems

One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work \cite{BN}, Br\'ezis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relatively compact