The Dirichlet problem for superlinear elliptic equations

Boundary value problems for nonlinear elliptic partial differential equations have been a major focus of research in nonlinear analysis for decades. This chapter focuses on some basic ideas in a simple setting and presents survey selected results on the Dirichlet problem for the equation given in this chapter with superlinear nonlinearity. The chapter consists of three sections: (1) discusses the positive solutions of the equation, (2) focuses on sign-changing solutions on bounded domains, and (3) treats the unbounded domain Ω =ℝ N . No effort is being made to be as general as possible. This chapter does not present results on the bifurcation of solutions nor for the p-Laplace operator, nor does it treat singularly perturbed equations in detail.

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