Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

Positive Solutions of Critical Quasilinear Elliptic Problems in General Domains

We consider a certain class of quasilinear elliptic equations with a term in the critical growth range. We prove the existence of positive solutions in bounded and unbounded domains. The proofs involve several generalizations of standard variational arguments.

Positive Solutions of Elliptic Equations with Singular Nonlinearity

In this paper, a nonlinear elliptic boundary value problem with singular nonlinearity Lu(x) = f(x, u(x)), x ∈ Ω, u(x) = φ(x), x ∈ ∂Ω, is studied, where L is a uniformly elliptic operator, Ω is a bounded domain in R , N ≥ 2, φ ≥ 0 may take the value 0 on ∂Ω, and f(x, s) is possibly singular near s = 0. Some results regarding the existence of positive solutions for the problem are given under a s...

Nodal Solutions of Elliptic Equations with Critical Sobolev Exponents

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

We show that the critical nonlinear elliptic Neumann problem ∆u− μu+ u = 0 in Ω, u > 0 in Ω and ∂u ∂ν = 0 on ∂Ω where Ω is a bounded and smooth domain in R, has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K, there exists μK > 0 such that for 0 < μ < μK , the above problem has a nontrivial solution which blows up at K interior points ...

Positive Solutions for Elliptic Equations with Singular Nonlinearity

We study an elliptic boundary-value problem with singular nonlinearity via the method of monotone iteration scheme: −∆u(x) = f(x, u(x)), x ∈ Ω, u(x) = φ(x), x ∈ ∂Ω, where ∆ is the Laplacian operator, Ω is a bounded domain in RN , N ≥ 2, φ ≥ 0 may take the value 0 on ∂Ω, and f(x, s) is possibly singular near s = 0. We prove the existence and the uniqueness of positive solutions under a set of hy...