Sign changing solutions for elliptic equations with critical growth in cylinder type domains

Pedro Girão; Miguel Ramos

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The aim of this paper is to study the existence of variational solutions to a nonhomogeneous elliptic equation involving the $N$ -Laplacian $- Δ_{N} u \equiv - {div (| \nabla u |}^{N - 2} \nabla u) = e (x, u) + h (x) in Ω$ where $u \in W_{0}^{1, N} (ℝ^{N})$ , $Ω$ is a bounded smooth domain in $ℝ^{N}$ , $N \geq 2$ , $e (x, u)$ is a critical nonlinearity in the sense of the Trudinger-Moser inequality and $h (x) \in {(W_{0}^{1, N})}^{*}$ is a small perturbation.

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Weak linking theorems and Schrödinger equations with critical Sobolev exponent

Martin Schechter, Wenming Zou (2003)

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In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $- Δ u + V (x) u = {K (x) | u |}^{2^{*} - 2} u + g (x, u), u \in W^{1, 2} (𝐑^{N})$ , where $N \geq 4; V, K, g$ are periodic in $x_{j}$ for $1 \leq j \leq N$ and 0 is in a gap of the spectrum of $- Δ + V$ ; $K > 0$ . If $0 < g (x, u) u \leq {c | u |}^{2^{*}}$ for an appropriate constant $c$ , we show that this equation has a nontrivial solution.