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

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

2011-07-02

arxiv.org/abs/1107.0375v1

Mathematics

Analysis of PDEs

30 pages. First draft in November, 2010

Scientific paper

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 $f$.

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