Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains (original) (raw)

Existence of signed solutions for a semilinear elliptic boundary value problem

Differential and Integral Equations

We consider the problem of existence of signed solutions for the semilinear elliptic boundary value problem Δu+u + q -u - p =0inΩ,u=0on∂Ω, where Ω is an open bounded subset of ℝ N with smooth boundary, 0<q<1 and 2<p+1<2N/(N-2) (if N≥3) and <∞ (if N=2). Using well known results we see that this problem has a positive and a negative solution. We show that for certain large domains there exists a solution that changes sign. The proof of our result is based on the Mountain Pass Theorem.

Sign Changing Solutions for Quasilinear Superlinear Elliptic Problems

The Quarterly Journal of Mathematics

Results on existence and multiplicity of solutions for a nonlinear elliptic problem driven by the Φ-Laplace operator are established. We employ minimization arguments on suitable Nehari manifolds to build a negative and a positive ground state solutions. In order to find a nodal solution we employ additionally the well known Deformation Lemma and Topological Degree Theory.

Existence of Strictly Positive Solutions for Sublinear Elliptic Problems in Bounded Domains

Advanced Nonlinear Studies, 2014

Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k1ξp ≤ f (ξ) ≤ k2ξp for all ξ ≥ 0 and some k1, k2 > 0 and p ∈ (0, 1). We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form −Δu = m(x) f (u) in Ω, u = 0 on ∂Ω.

Multiple sign changing solutions of nonlinear elliptic problems in exterior domains

Advanced Nonlinear Studies

We consider the problem -Δu+(V_{∞}+V(x))u=|u|^{p-2}u, u∈H₀¹(Ω), where Ω is an exterior domain in R^{N}, V_{∞}>0, V∈C(R^{N},R), inf_{R^{N}}V>-V_{∞} and V(x)→0 as |x|→∞. Under symmetry conditions on Ω and V, and some assumptions on the the decay of V at infinity, we show that there is an effect of the topology of the orbit space of certain subsets of the domain on the number of low energy sign changing solutions to this problem, having some specific symmetries.

Superlinear Elliptic Problems with Sign Changing Coefficients

Communications in Contemporary Mathematics, 2012

Via variational methods, we study multiplicity of solutions for the problem [Formula: see text] where a simple example for g(x, u) is |u|p-2u; here a, λ are real parameters, 1 < q < 2 < p ≤ 2* and b(x) is a function in a suitable space Lσ. We obtain a class of sign changing coefficients b(x) for which two non-negative solutions exist for any λ > 0, and a total of five nontrivial solutions are obtained when λ is small and a ≥ λ1. Note that this type of results are valid even in the critical case.

Sign changing solutions of nonlinear elliptic equations

Advances in Differential Equations, 1996

This paper is concerned with a class of nonlinear elliptic Dirichlet problems approximating degenerate equations. If the degeneration set consists of k connected components, by using variational methods, it is proved the existence of k 2 distinct nodal solutions, having exactly two nodal regions, whose positive and negative parts concentrate near subsets of the degeneration set.