Superlinear Elliptic Problems Research Articles

A sharp oscillation property involving the critical Sobolev exponent for a class of superlinear elliptic problems

This paper studies the asymptotic behaviour as α := u(0) ↑∞ of the first zero R(α) of the radially symmetric solution of the semilinear equationin ℝn, n ≥ 1, where h > 0 and β > 1. We establish that R(α) = O(α−(β−1)/2) if n = 1, 2 or n ≥ 3 and β < (n + 2)/(n − 2), and conjecture that lim inf α→∞R(α) > 0 if n ≥ 3 and β > (n + 2)/(n − 2).

Existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli

We consider the nonlinear elliptic problem−Δu=upin ΩR,u>0in ΩR,u=0in ΩR where p>1 and ΩR={x∈RN:R<|x|<R+1} with N⩾3. It is known that as R→∞, the number of nonequivalent solutions of the above problem goes to ∞ when p∈(1,(N+2)/(N−2)), N⩾3. Here we prove the same phenomenon for any p>1 by finding O(N−1)-symmetric clustering bump solutions which concentrate near the set {(x1,…,xN)∈ΩR:xN=0} for large R>0.

On the superlinear problem involving thep(x)-Laplacian

This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: ( − div(|∇u| p(x)−2 ∇u) = �f(x,u) in , u = 0 on@, where ⊂ R N (N ≥ 2) is a bounded domain with smooth boundary @, � > 0 is a parameter. Existence of nontrivial solution is established for arbitrary � > 0. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter � > 0. Then, it is considered the continuation of the solutions. Our results are a generalization of Miyagaki and Souto.

Multiple positive solutions for a class of [formula omitted]-superlinear elliptic problems

We study nonlinear elliptic Dirichlet problems driven by the p -Laplacian and with a nonlinearity which consists of a p -superlinear term perturbed by a p -linear one. Using critical point theory, coupled with suitable truncation techniques, we show that the problem has two positive smooth solutions. When the conditions are symmetric in the two semiaxes, we have four nontrivial smooth solutions of constant sign. Finally for the semilinear problem, using Morse theory, we produce a fifth nontrivial smooth solution.

Infinitely Many Solutions for a Robin Boundary Value Problem

By combining the embedding arguments and the variational methods, we obtain infinitely many solutions for a class of superlinear elliptic problems with the Robin boundary value under weaker conditions.

Multibump nodal solutions for an indefinite superlinear elliptic problem

We define some Nehari-type constraints using an orthogonal decomposition of the Sobolev space H 0 1 and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem.

Multiplicity theorems for superlinear elliptic problems

In this paper we study second order elliptic equations driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti–Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. Finally we indicate how in the semilinear case, Morse theory can be used to produce six nontrivial solutions.

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

In this article, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise, a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck [16]. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. The method enables us to obtain new results on universal estimates of spatial singularities for elliptic equations and systems, under optimal growth assumptions that, unlike previous results, involve only the behavior of the nonlinearity at infinity. For semilinear systems of Lane-Emden type, these seem to be the first results on singularity estimates to cover the full subcritical range. In addition, we give an affirmative answer to the so-called Lane-Emden conjecture in three dimensions

Boundary singularities for weak solutions of semilinear elliptic problems

Let Ω be a bounded domain in RN, N⩾2, with smooth boundary ∂Ω. We construct positive weak solutions of the problem Δu+up=0 in Ω, which vanish in a suitable trace sense on ∂Ω, but which are singular at prescribed isolated points if p is equal or slightly above N+1N−1. Similar constructions are carried out for solutions which are singular at any given embedded submanifold of ∂Ω of dimension k∈[0,N−2], if p equals or it is slightly above N−k+1N−k−1, and even on countable families of these objects, dense on a given closed set. The role of the exponent N+1N−1 (first discovered by Brezis and Turner [H. Brezis, R. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601–614]) for boundary regularity, parallels that of NN−2 for interior singularities.

A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains

A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems − Δ u = f ( x , u ) on a bounded domain Ω ⊂ R n with u = 0 on ∂ Ω are studied, where the nonlinearity 0 ⩽ f ( x , s ) grows at most like s p . If Ω is a Lipschitz domain we exhibit two exponents p * and p * , which depend on the boundary behavior of the Green function and on the smallest interior opening angle of ∂ Ω. We prove that for 1 < p < p * all positive very weak solutions are a priori bounded in L ∞ . For p > p * we construct a nonlinearity f ( x , s ) = a ( x ) s p together with a positive very weak solution which does not belong to L ∞ . Finally we exhibit a class of domains for which p * = p * . For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains p * = p * = n + 1 n − 1 is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601–614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49–81].

On the superlinear Lazer-McKenna conjecture: the non-homogeneous case

We prove the Lazer-McKenna conjecture for the superlinear elliptic problem of the Ambrosetti-Prodi type with a non-homogeneous non-linearity by constructing solutions with sharp peaks. We also compute the critical groups provided the critical points are isolated.

Multiple positive solutions for a superlinear elliptic problem in [formula omitted] with a sublinear Neumann boundary condition

Let Ω ⊂ R 2 be a bounded domain with C 2 boundary. Let u ∈ H 1 ( Ω ) be a weak solution of the following problem: ( P λ) { − Δ u + u = p ( u ) e u α u > 0 } in Ω , ∂ u ∂ ν = λ ψ u q on ∂ Ω , where α ∈ ( 0 , 2 ] , λ > 0 , q ∈ [ 0 , 1 ) and ψ ≥ 0 , a Hölder continuous function on Ω ¯ . Here p ( u ) is a polynomial perturbation of e u α as u → ∞ . Using variational methods we show that there exists 0 < Λ < ∞ such that ( P λ ) admits at least two solutions if λ ∈ ( 0 , Λ ) , no solution if λ > Λ and at least one solution when λ = Λ .

Radial solutions of semilinear elliptic equations with broken symmetry

The aim of this paper is to prove the existence of infinitely many radial solutions of a superlinear elliptic problem with rotational symmetry and non-homogeneous boundary data.

On numerical solutions of a superlinear elliptic problem

In this work we consider numerical positive solutions of the equation −Δu = λf (u) with Dirichlet boundary condition in a bounded domain Ω, where λ> 0 and f (u) is a superlinear function of u. We study the behavior of the branches of numerical positive solutions for varying λ. Mathematics Subject Classification: 35J60, 35B30, 65N99

Partial symmetry of least energy nodal solutions to some variational problems

We investigate the symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are nodal solutions of superlinear elliptic problems, or eigenfunctions of weighted asymmetric eigenvalue problems, or they lie on the first curve in the Fucik spectrum. In all instances, we prove that the minimizers are foliated Schwarz symmetric. We give examples showing that the minimizers are in general not radially symmetric. The basic tool which we use is polarization, a concept going back to Ahlfors. We develop this method of symmetrization for sign changing functions.

Concentration phenomena for solutions of superlinear elliptic problems

In this paper we look for positive solutions of the problem −Δu+λu=up−1 in Ω, u=0 on ∂Ω, where Ω is a bounded domain in Rn, n⩾3, p>2 and λ is a positive parameter. We describe new concentration phenomena, which occur as λ→+∞, and exploit them to construct (for λ large enough) positive solutions that concentrate near spheres of codimension 2 as λ→+∞; these spheres approach the boundary of Ω as λ→+∞. Notice that the existence and multiplicity results we obtain hold also in contractible domains arbitrarily close to starshaped domains (no solution can exist if p⩾2nn−2 and Ω is starshaped, because of Pohožaev's identity). The method we use is completely variational and based on a blow up analysis in the equivariant setting. In order to avoid concentration phenomena near points and to overcome some difficulties related to the lack of compactness, we first modify the nonlinear term in a suitable region, then we solve the modified problem by minimizing the related energy functional on a suitable infinite dimensional manifold and, finally, we show that the solutions of the modified problem solve also our problem, for λ large enough, because they are localized in the prescribed region where the nonlinear term has not been modified.

Multiple positive solutions of superlinear elliptic problems with sign-changing weight

We study the existence of multiple positive solutions for a superlinear elliptic PDE with a sign-changing weight. Our approach is variational and relies on classical critical point theory on smooth manifolds. A special care is paid to the localization of minimax critical points.

On the superlinear Lazer–McKenna conjecture

We prove the Lazer–McKenna conjecture for a superlinear elliptic problem of Ambrosetti–Prodi type by constructing solutions with sharp peaks.

Multiplicity results for a class of superlinear elliptic problems

We study a class of superlinear elliptic problems -Δu = λf(u) under the Dirichlet boundary condition on a bounded smooth domain in R N . Assuming that the nonlinearity f(u) is superlinear in a neighborhood of u = 0, we study the dependence of the number of signed and sign-changing solutions on the parameter λ.

Multiplicity of positive solutions for an indefinite superlinear elliptic problem on RN

We consider the elliptic problem −Δu−λu=a(x)up, with p>1 and a(x) sign-changing. Under suitable conditions on p and a(x), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space RN. More precisely, we show that there exists Λ>0 such that this equation on RN has no positive solution for λ>Λ, at least two positive solutions for λ∈(0,Λ), and at least one positive solution for λ∈(−∞,0]∪{Λ}.