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In this paper, we study the p -Laplace elliptic equations under the Dirichlet boundary condition. We give a general and weak sufficient condition for the existence of a sequence of solutions converging to zero. This result is proved by applying the symmetric mountain pass lemma obtained in our earlier paper. For some elliptic equations with parameters, we decide whether the zero solution is an accumulation point or an isolated point in the set of all solutions. [ABSTRACT FROM AUTHOR]

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]

We study the sublinear Emden-Fowler equation in small domains. As the domain becomes smaller, so does any solution. We investigate the convergence rate of the Sobolev norm of solutions as the volume of the domain converges to zero. The result is obtained by estimating the first eigenvalue of the Laplacian with the help of the variational method. [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]

In this paper, we study the semilinear elliptic equation with a small perturbation. We assume the main term in the equation to have a mountain pass structure but do not suppose any condition for the perturbation term. Then we prove the existence of a positive solution. Moreover, we prove the existence of at least two positive solutions if the perturbation term is nonnegative. [ABSTRACT FROM AUTHOR]

Abstract: We study the sublinear elliptic equation having two nonlinear terms, where the main term is sublinear and odd with respect to and the perturbation term is any continuous function with a small coefficient. Then we prove the existence of multiple small solutions. [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]

Abstract: In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to . [Copyright &y& Elsevier]

Abstract: For an even functional on a Banach space, the symmetric mountain pass lemma gives a sequence of critical values which converges to zero. Under the same assumptions on the functional, this paper establishes a new critical point theorem which provides a sequence of critical points converging to zero. The theorem is applied to sublinear elliptic equations. Then a sequence of solutions converging to zero is obtained. [Copyright &y& Elsevier]

We deal with sublinear elliptic equations in a ball and prove the existence of infinitely many solutions which are not radially symmetric but G invariant. Here G is any closed subgroup of the orthogonal group and is not transitive on the unit sphere. [Copyright &y& Elsevier]