Solutions Of Semilinear Elliptic Problems Research Articles

Corrigendum to “Existence of solutions for semilinear elliptic problems without (𝑃𝑆) condition”

In this corrigendum we give a correct proof for equation (3.17), as well as the assertion on line 15, page 1364, of “Existence of solutions for semilinear elliptic problems without ( P S ) (PS) condition”, Proc. Amer. Math. Soc. 132(5) (2004), 1355-1366.

Existence of solutions for semilinear elliptic problems in exterior of ball

We prove the existence of solutions for the semilinear elliptic problem in Ω = B ( 0 , R ) c , N ⩾ 3 . − Δ u = G ′ ( u ) , under suitable general assumptions on the nonlinear term G.

New classes of entire solutions for semilinear elliptic problems in $ℝ^n$

The goal of this paper is to describe some new results concerning entire solutions of semilinear elliptic equations in ℝ^n with non trivial asymptotic behavior at infinity. We describe in particular the (focusing, subcritical) nonlinear Schrödinger equation and the Allen–Cahn equation, which enjoy some common features but also present rather different aspects.

Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem Δu = λp(x)f(u) in Ω subject to some Dirichlet conditions, where Ω is a regular domain in ℝn (n ≥ 3) with compact boundary. The nonlinearity f is nonnegative continuous and the potential p belongs to some Kato class K(Ω). So we prove the existence of a positive continuous solution depending on λ by the use of a potential theory approach.

Multiple Positive Solutions of Semilinear Elliptic Problems in Exterior Domains

Assume that is a positive continuous function in and satisfies the suitable conditions. We prove that the Dirichlet problem admits at least three positive solutions in an exterior domain.

Existence of multiple nontrivial solutions for semilinear elliptic problems

The aim of this paper is to prove the existence of multiple nontrivial solutions to a semilinear elliptic problem at resonance. The proofs used here are based on combining the Morse theory and the minimax methods.

Symmetry of solutions of semilinear elliptic problems

We study symmetry properties of least energy positive or nodal solutions of semilinear elliptic problems with Dirichlet or Neumann boundary conditions. The proof is based on symmetrizations in the spirit of Bartsch, Weth and Willem (J. Anal. Math., 2005) together with a strong maximum principle for quasi-continuous functions of Ancona and an intermediate-value property for such functions.

Multiplicity of 2-nodal solutions for semilinear elliptic problems in [formula omitted

In this paper, we study the effect of the graph of weight functions on the number of 2-nodal solutions for semilinear elliptic problems in R N . Our results generalize and improve on the results of [D. Cao, E.S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in R N , Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (5) (1996) 567–588]. In the present paper, we give a new analytical method to study this issue.

ON AXISYMMETRIC TRAVELING WAVES AND RADIAL SOLUTIONS OF SEMI-LINEAR ELLIPTIC EQUATIONS

ABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowlyvarying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisym-metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed.

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems on S n

We consider the following superlinear elliptic equation on S n where D is a geodesic ball on S n with geodesic radius θ1, and Δ S n is the Laplace–Beltrami operator on S n . We prove that for any and for any positive integer N ≥ 1, there exist at least 2N + 1 solutions to the above problem for ϵ sufficiently small. Moreover, the asymptotic behavior of such solutions is also characterized. We then apply this result to the Brezis–Nirenberg problem and establish the existence of solutions which are not minimizers of the associated energy.

Existence and multiplicity for a resonance problem for the [formula omitted]-Laplacian on bounded domains in [formula omitted

The existence of at least two solutions for a resonance problem involving the p -Laplacian is shown for the case of bounded domains in R N . This work constitutes an extension of a previous result of Landesman, Robinson and Rumbos for the case p = 2 [E. Landesman, S. Robinson, A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Analysis TMA 24 (7) (1995) 1049–1059].

Non-radial clustered spike solutions for semilinear elliptic problems on sN

We consider the superlinear elliptic equation on Sn $$\varepsilon ^2 \Delta _{S^n } u - u + u^p = 0 in S^n , u > 0 in S^n ,$$ where ΔSn is the Laplace-Beltrami operator on S n. We prove that for any k = 1,..., n − 1, there exists p k > 1 such that for 1 < p < p k and e sufficiently small, there exist at least n−k positive solutions concentrating on a k-dimensional subset of the equator. We also discuss the problem on geodesic balls of S n and establish the existence of positive non-radial solutions. The method extends to Dirichlet problems with more general non-linearities. The proofs are based on the finite-dimensional reduction procedure which was successfully used by the second author in singular perturbation problems.

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.

Sign changing solutions of semilinear elliptic problems in exterior domains

We prove the existence of a sign changing solution to the semilinear elliptic problem \(\ -\Delta u+u=\mid u\mid^{p-2}u, \ u\in H_{0}^{1}(\Omega)\), in an exterior domain Ω having finite symmetries.

The exact boundary blow-up rate of large solutions for semilinear elliptic problems

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem − △ u = λ u − a ( x ) u p , u | ∂ Ω = + ∞ , where Ω is a bounded smooth domain in R N . The weight function a ( x ) in front of the nonlinearity can vanish on the boundary of the domain Ω at different rates according to the point x 0 of the boundary. The decay rate of the weight function a ( x ) may not be approximated by a power function of distance near the boundary ∂ Ω . We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution u ( x ) must satisfy lim x → x 0 u ( x ) K ( b x 0 ∗ ( dist ( x , ∂ Ω ) ) ) − β = 1 for each x 0 ∈ ∂ Ω , where b x 0 ∗ ( r ) = ∫ 0 r ∫ 0 s b x 0 ( t ) d t d s , K = [ β ( ( β + 1 ) C 0 − 1 ) ] 1 p − 1 , β = 1 p − 1 , C 0 = lim r → 0 ( ∫ 0 r b x 0 ( t ) d t ) 2 b x 0 ∗ ( r ) b x 0 ( r ) and b x 0 ( r ) is the boundary normal section of a ( x ) at x 0 ∈ ∂ Ω , i.e., b x 0 ( r ) = a ( x 0 − r n x 0 ) , r > 0 , r ∼ 0 , and n x 0 stands for the outward unit normal vector at x 0 ∈ ∂ Ω .

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.

Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary

This paper is concerned with the existence and multiplicity of positive solutions of the equation −Δu+u=up−1, 2<p<2∗=2NN−2, with Dirichlet zero data, in an unbounded smooth domain Ω⊂RN having unbounded boundary. Under the assumptions:(h1)∃τ1,τ2,…,τk∈R+∖{0}, 1⩽k⩽N−2, such that(x1,x2,…,xN)∈Ω⟺(x1,…,xi−1,xi+τi,…,xN)∈Ω,∀i=1,2,…,k,(h2)∃R∈R+∖{0} such that RN∖Ω⊂{(x1,x2,…,xN)∈RN:∑j=k+1Nxj2⩽R2} the existence of at least k+1 solutions is proved.

Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ≥ 5 , 0 ≤ s < 2 , 2 ∗ ( s ) = 2 ( N − s ) N − 2 . We prove the existence of nontrivial solutions for the singular critical problem − Δ u − μ u | x | 2 = | u | 2 ∗ ( s ) − 2 | x | s u + λ u with Dirichlet boundary condition on Ω for all λ > 0 and 0 ≤ μ < ( N − 2 2 ) 2 − ( N + 2 N ) 2 .

Multiplicity of positive solutions for semilnear elliptic problems with antipodal symmetry

In this paper, we show the multiple existence of positive solutions of semilinear elliptic problems of the form $$ -\Delta u=\vert u\vert ^{2^{*}-2}u+f, \quad u\in H_{0}^{1}(\Omega), $$ where $\Omega\subset{\mathbb R}^{N}$ is a bounded domain, $2^{*}$ is the Sobolev critical exponent and $f\in L^{2}(\Omega)$.

Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities

In this paper we develop a critical point theory for nonsmooth locally Lipschitz functionals defined on a closed, convex set extending this way the work of Struwe (Variational Methods, Springer, Berlin, 1990). Through a deformation result, we obtain minimax principles producing critical points. Then we use the theory to obtain positive and negative solutions of nonlinear and semilinear hemivariational inequalities. In this context we improve a result on positive solutions for semilinear elliptic problems due to Nirenberg (Variational methods in nonlinear problems, in: Topics in Calculus of Variations, Lecture Notes in Mathematics, vol. 1365, Springer, Berlin, 1987).