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In the present paper, we study the p-Laplace equation in a hollow symmetric bounded domain. Let H and G be closed subgroups of the orthogonal group such that H⊊G{H\varsubsetneq G}. Then we prove the existence of a positive solution which is H-invariant and G-non-invariant. Furthermore, we give several examples of H, G and Ω, and find symmetric and asymmetric solutions.

In this paper we study the p-Laplace Emden-Fowler equation with a radial and sign-changing weight in the unit ball under the Dirichlet boundary condition. We show that if the weight function is negative in the unit ball except for a small neighborhood of the boundary and positive at somewhere in this neighborhood, then no least energy solution is radially symmetric. Therefore the equation has both a positive radial solution and a positive nonradial solution. Moreover, we prove in the one dimensional case that if the neighborhood is large, then a positive solution is unique. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the Emden–Fowler equation is studied in a hollow thin domain which is invariant under the action of a closed subgroup of the orthogonal group. Then it is proved that if the domain is thin enough, a least energy solution is not group invariant. [Copyright &y& Elsevier]

Abstract: We study the one-dimensional -Laplace Emden–Fowler equation under the Dirichlet boundary condition, where the coefficient function is even in , small for , and large for with close to 1. Then we prove that a least energy solution is not even. [Copyright &y& Elsevier]

Abstract: We study the generalized Hénon equation in reflectionally symmetric or point symmetric domains and prove that a least energy solution is neither reflectionally symmetric nor even. Moreover, we prove the existence of a positive solution with prescribed exact symmetry. [Copyright &y& Elsevier]

In this paper, we study the Emden-Fowler equation whose coefficient is even in the interval (-1, 1) under the Dirichlet boundary condition. We prove that if the density of the coefficient function is thin in the interior of (-1, 1) and thick on the boundary, then a least energy solution is not even. Therefore the equation has at least three positive solutions: the first one is even, the second one is a non-even least energy solution u(t) and the third one is the reflection u(-t). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the generalized Hénon equation with a radial coefficient function in the unit ball and show the existence of a positive non-radial solution. Our result is applicable to a wide class of coefficient functions. Our theorem ensures that if the ratio of the density of the coefficient function in to that in is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric. [Copyright &y& Elsevier]