Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

Sign-changing solutions for elliptic problems with critical Sobolev-Hardy exponents

Let be a smooth bounded domain such that 0∈Ω, N?7, 0?s<2, 2∗(s)=2(N−s)/(N−2). We prove the existence of sign-changing solutions for the singular critical problem −Δu−μ(u/|x|2)=(|u|2∗(s)−2/|x|s)u+λu with Dirichlet boundary condition on Ω for suitable positive parameters λ and μ.     

A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms

Let be a bounded domain such that . We obtain existence of sign-changing solutions for the Dirichlet problem on Ω,u=0 on ∂Ω for suitable positive numbers μ and λ.     

Multiplicity of nodal solutions for elliptic problems involving non-odd nonlinearities

In this paper, we study the effect of domain shape on the number of 2-nodal solutions for the semilinear elliptic equation involving non-odd nonlinearities. We prove that a semilinear elliptic equation in an m$m$-bump domain (possibly unbounded) has m²$m^2$ 2-nodal solutions and we can find a least energy nodal solution in those solutions. Furthermore, we can describe the bump location of these solutions.     

Three weak solutions for elliptic Dirichlet problems

An existence result of three non-zero solutions for non-autonomous elliptic Dirichlet problems, under suitable assumptions on the nonlinear term, is presented. The approach is based on a recent three critical points theorem for differentiable functionals.     

Multiplicity of positive solutions for semilinear elliptic problems in unbounded domains

In this paper, we study the effect of domain shape on the multiplicity of positive solutions for the semilinear elliptic equations. We prove a Palais-Smale condition in unbounded domains and assert that the semilinear elliptic equation in unbounded domains has multiple positive solutions.     

Very weak solutions with boundary singularities for semilinear elliptic Dirichlet problems in domains with conical corners

Let Ω⊂Rn be a bounded Lipschitz domain with a cone-like corner at 0∈∂Ω. We prove existence of at least two positive unbounded very weak solutions of the problem −Δu=up in Ω, u=0 on ∂Ω, which have a singularity at 0, for any p slightly bigger that the generalized Brezis-Turner exponent p^*. On an example of a planar polygonal domain the actual size of the p-interval on which the existence result holds is computed. The solutions are found variationally as perturbations of explicitly constructed singular solutions in cones. This approach also makes it possible to find numerical approximations of the two very weak solutions on Ω following a gradient flow of a suitable functional and using the mountain pass algorithm. Two-dimensional examples are presented.     

Remarks on multiple solutions for elliptic resonant problems

We obtain four nontrivial solutions for an elliptic resonant problem via Morse theory and Lyapunov-Schmidt reduction method. Our result improves some recent works.     

Infinitely many solutions for polyharmonic elliptic problems with broken symmetries

By means of a perturbation argument devised by P. Bolle, we prove the existence of infinitely many solutions for perturbed symmetric polyharmonic problems with non–homogeneous Dirichlet boundary conditions. An extension to the higher order case of the estimate from below for the critical values due to K. Tanaka is obtained.     

Fucik spectrum, sign-changing and multiple solutions for semilinear elliptic boundary value problems with jumping nonlinearities at zero and infinity

In this paper, Fucik spectrum, ordinary differential equation theory of Banach spaces and Morse theory are used to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and some new results on the existence of nontrivial solutions, multiple solutions and sign-changing solutions are obtained. In one case seven nontrivial solutions are got. The techniques have independent interest.     

Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian

In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35].     

Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms

We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problem

(P_ε)     

Sign-changing solutions for supercritical elliptic problems in domains with small holes

Let \(\mathcal {D}\) be a bounded, smooth domain in \(\mathbb {R}^N\) , N ≥ 3, \(P\in \mathcal {D}\) . We consider the boundary value problem in \(\Omega = \mathcal {D} \setminus B_\delta(P)\) ,

$\begin{aligned}\Delta u + |u|^{p-1} u = 0\, \quad in\, \Omega,\\u = 0\quad on\, \partial\Omega,\end{aligned}$

with p supercritical, namely \(p > \frac{N+2}{N-2}\) . Given any positive integer m, we find a sequence \(p_1 < p_2 < p_3 < \cdots , \quad with \lim_{k\to+\infty} p_k = +\infty \), such that if p is given, with p ≠ p _j for all j, then for all δ > 0 sufficiently small, this problem has a sign-changing solution which has exactly m + 1 nodal domains.

     

Global behaviour of the components of nodal solutions for Lidstone boundary value problems

By employing bifurcation techniques, we investigate the global behaviour of the components of nodal solutions for Lidstone boundary value problems depending on higher order derivatives. Our results improve on those in the literature.