Existence Of Sign-changing Solutions Research Articles

On the Existence and the Profile of Nodal solutions of Elliptic Equations Involving Critical Growth

We study the existence of sign changing solutions to the slightly subcritical problem $$-\Delta u=|u|^{p-1-\epsilon} u\ {\rm in}\ \Omega,\quad u=0\ {\rm on}\ \partial \Omega,$$ where ω is a smooth bounded domain in ℝ N , N ≥ 3, p = (N + 2)/(N − 2) and ɛ > 0. We prove that, for ɛ small enough, there exist N pairs of solutions which change sign exactly once. Moreover, the nodal surface of these solutions intersects the boundary of ω, provided some suitable conditions are satisfied.

Sign-changing solutions of critical growth nonlinear elliptic systems

In this paper, we are concerned with the existence of sign-changing solutions of a class of nonlinear elliptic systems with critical growth.

Sign Changing Solutions of Superlinear Schrödinger Equations

We prove the existence of sign changing solutions in H 1(ℝ N ) for a stationary Schrödinger equation −Δu + a(x)u = f(x, u) with superlinear and subcritical nonlinearity f, and control the number of nodal domains. If f is odd we obtain an unbounded sequence of sign changing solutions u k , k ≥ 1, so that u k has at most k + 1 nodal domains. The bound on the number of nodal domains follows from a nonlinear version of Courant's nodal domain theorem which we also prove.

New Linking Theorem and Sign-Changing Solutions

In this paper, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A general liking type theorem is given with the location of the critical point produced specified in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities, and may or may not be resonant with respect to Fučík spectrum.

Existence of sign changing solutions for some critical problems on $\mathbb R^N$

The main purpose of this paper is to construct families of positive and changing-sign solutions for both the slightly subcritical and slightly supercritical equations <p align="center"> $-\Delta u+V(x)u=N(N-2)|u|^{\frac{4}{N-2}\pm\varepsilon}u$ in $\mathbb R^N,$ <p align="left" class="times"> which blow-up and concentrate at different points of $\mathbb R^N$ as $\varepsilon$ goes to 0, under certain conditions on the potential $V.$

On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth**The authors are supported by M.U.R.S.T. project ‘Metodi variazionali e topologici nello studio di fenomeni non lineari’.

This paper deals with the existence of sign changing solutions of the problemwhere Ω is a bounded regular domain in , N ⩾ 4, ε > 0, p = (N + 2)/(N − 2), q ⩾ 1, q ≠ p and .

Multiple existence of sign changing solutions for semilinear elliptic problems on RN

Multiple existence of sign changing solutions for semilinear elliptic problems on RN

A numerical investigation of sign-changing solutions to superlinear elliptic equations on symmetric domains

In this paper, we investigate numerically sign-changing solutions of superlinear elliptic equations on symmetric domains. Based upon the symmetric criticality principle of Palais, the existence of sign-changing solutions which reflect the symmetry of Ω is studied first. A simple numerical algorithm, the modified mountain pass algorithm, is then proposed to compute the sign-changing solutions. This algorithm is discussed and compared with the high-linking algorithm for sign-changing solutions developed by Ding et al. [Nonlinear Anal. 37 (1999) 151–172]. By implementing both algorithms on several numerical examples, the sign-changing solutions and their nodal curves are displayed and discussed.

Multiple and sign changing solutions of an elliptic eigenvalue problem with constraint

We prove the existence of sign changing solutions of a semilinear elliptic eigenvalue problem with constraint by using variational methods. Among those three solutions we obtained, one is positive, one negative and one sign changing. We also prove the existence of multiple sign changing solutions under some additional condition.

Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli

We study the existence of many nonradial sign-changing solutions of a superlinear Dirichlet boundary value problem in an annulus in $\mathbb R^N$. We use Nehari-type variational method and group invariance techniques to prove that the critical points of an action functional on some spaces of invariant functions in $H_{0}^{1,2}(\Omega_{\varepsilon})$, where $\Omega_{\varepsilon}$ is an annulus in $\mathbb R^N$ of width $\varepsilon$, are weak solutions (which in our case are also classical solutions) to our problem. Our result generalizes an earlier result of Castro et al. (See [A. Castro, J. Cossio and J. M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems , Electron. J. Differential Equations 2 (1998), 1–18]).

On the existence of sign changing solutions for semilinear Dirichlet problems

On the existence of sign changing solutions for semilinear Dirichlet problems

On the existence of sign changing solutions for semilinear Dirichlet problems

where Ω ⊂ R is a bounded domain with Lipschitz boundary and f : R → R is of class C with f(0) = 0. Thus u0 ≡ 0 is a trivial solution of (D) and we are interested in finding and studying nontrivial solutions. One way of obtaining these is to compare the behavior of f near the origin and near infinity. We shall always assume that f grows subcritically at infinity so that variational methods can be applied and the associated functional satisfies the Palais–Smale condition. Suppose f ′(0) < λ1 where 0 < λ1 < λ2 ≤ λ3 ≤ . . . are the eigenvalues (counted with multiplicities) of −∆ on Ω with homogeneous Dirichlet boundary conditions. If f grows superlinearly at infinity then the mountain pass theorem of Ambrosetti and Rabinowitz [AR], [R] together with the maximum principle guarantees the existence of a positive solution u+ and a negative solution u− of (D). Using linking or Morse type arguments Wang [Wa] obtained a third nontrivial solution u1. In this paper we shall refine Wang’s result and obtain more information on u1 and on other solutions whose existence is proved via Morse theory. Let us illustrate this with the following two theorems. More general results will be stated and proved later.