Note on spectral theory of nonlinear operators: Extensions of some surjectivity theorems of Fučík and Nečas
We present some monotonicity and symmetry results for positive solutions of the equation satisfying an homogeneous Dirichlet boundary condition in a bounded domain . We assume 1 < p < 2 and locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if is a ball then the solutions are radially symmetric and strictly radially decreasing.
In this Note we consider the following problem where is a bounded smooth starshaped domain in , , , , and . We prove that if is a solution of Morse index than cannot have more than maximum points in for sufficiently small. Moreover if is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for sufficiently small.
We consider the semilinear Lane–Emden problem where and is a smooth bounded domain of . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of , as . Among other results we show, under some symmetry assumptions on , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...
We consider the Yamabe type family of problems , in , on , where is an annulus-shaped domain of , , which becomes thinner as . We show that for every solution , the energy as well as the Morse index tend to infinity as . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...
Page 1