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In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem: Here $$1

We consider the following singularly perturbed nonlinear elliptic problem: where Ω is a bounded domain in $${\mathbb{R}^N (N \ge 3)}$$ with a boundary $${\partial \Omega \in C^2}$$ and the nonlinearity f is of critical growth. In this paper, we construct a solution $${u_\varepsilon}$$ of the above problem which exhibits one spike near a maximum point of the distance function from the boundary ∂Ω under a critical growth condition on f. Our result complements the study made in [] in the sense that, in that paper, only the subcritical growth was considered. [ABSTRACT FROM AUTHOR]

Abstract: Let Ω be a bounded domain in with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω, where ν is the exterior normal to ∂Ω and the nonlinearity f is of subcritical growth. It has been known that under Berestycki and Lions conditions for and , there exists a solution of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂Ω for small . In this paper, we extend the previous result for and . [Copyright &y& Elsevier]

For elliptic equations ε2Δ u − V( x) u + f( u) = 0, x ∈ R N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. [ABSTRACT FROM AUTHOR]

We consider the problem: Δu+up=0 in ΩR,u=0 on ∂ΩR,u>0 in ΩR, where ΩR≡{x∈RN|R-1<|x|0. [ABSTRACT FROM AUTHOR]