Solutions Of Semilinear Elliptic Equations Research Articles

Multiple positive solutions of a class of non autonomous Schrödinger–Poisson systems

For the nonlinear Schrödinger equation coupled with Poisson equation of the version −Δu+u+ϕu=a(x)∣u∣p−2u+λk(x)u in R3 and −Δϕ=u2 in R3, we prove the existence of two positive solutions in H1(R3) when a(x) is sign changing and the linear part is not coercive. We show that the coupled term ϕu is helpful to find multiple positive solutions when a(x) is sign changing, which gives striking contrast to the known result where ϕu is proven to be an obstacle to get the existence of nontrivial solutions. Surprisingly we show that the term ϕu can play the role similar to a sign condition ∫a(x)e1pdx<0, which has turned out to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Alama et al. (1993)).

Existence and separation of positive radial solutions for semilinear elliptic equations

We consider the semilinear elliptic equation Δu+K(|x|)up=0 in RN for N>2 and p>1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r−ℓK(r) with ℓ>−2 around a positive constant is small near r=∞ and p is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent pc which is determined by N and the order of the behavior of K(r) as r=|x|→0 and ∞. In order to understand how subtle the structure is on K at p=pc, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2).

Convergence Analysis of a Minimax Method for Finding Multiple Solutions of Semilinear Elliptic Equation: Part I—On Polyhedral Domain

In 2001 and 2002, Li---Zhou designed minimax methods for finding saddle critical points in Hilbert space. These methods were applied to numerically solve semilinear elliptic equation for multiple solutions. Subsequence and sequence convergence results for the methods in functional analysis were established in 2002. But, since these convergence results do not consider discretization, they are not convergence results in numerical analysis. In this paper, we point out what approximation problem is, when Li---Zhou's methods are used to solve semilinear elliptic equation and the finite element method is used in discretization. Global sequence convergence result on computation of the approximation problem is verified. Finally, as diameter of element domain goes to zero, convergence of solutions of the approximation problem to solutions of semilinear elliptic equation is proved.

Multiple Positive Solutions for Semilinear Elliptic Equations Involving Subcritical Nonlinearities in &lt;em&gt;R&lt;/em&gt;&lt;sup&gt;&lt;em&gt;N&lt;/em&gt;&lt;/sup&gt;

Multiple Positive Solutions for Semilinear Elliptic Equations Involving Subcritical Nonlinearities in &lt;em&gt;R&lt;/em&gt;&lt;sup&gt;&lt;em&gt;N&lt;/em&gt;&lt;/sup&gt;

Removable singularities and singular solutions of semilinear elliptic equations

This note proves that a closed set with appropriate properties is removable for solutions of semilinear elliptic equations satisfying a certain growth condition near that set. Also, we give the existence theorem of positive solutions with singularities on a prescribed compact set, which shows that a growth condition in our removability theorem is optimal.

Uniqueness of radial solutions of semilinear elliptic equations on hyperbolic space

We consider the uniqueness of radial solutions for the semilinear elliptic equation −ΔHdu+λu−up+1=0 on hyperbolic space Hd. The proof is based on suitable transformations, energy functions and an idea of Yanagida. A similar approach was used by Kwong and Li.

Radial symmetry of large solutions of semilinear elliptic equations with convection

We study the radial symmetry of large solutions of the semilinear elliptic problem Δu + ∇h · ∇u = f (∣x∣, u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.

Multiple positive solutions to a class of modified nonlinear Schrödinger equations

This paper is concerned with the existence and multiplicity of positive solutions of the following equation−u″+u−2((|u|2)″)u=a(x)|u|p−2u+λk(x)u where u∈H1(R). We will prove that the modified term 2((|u|2)″)u is helpful to find multiple positive solutions in the case of the nonlinearity with sign changing coefficient a(x). Surprisingly, we do not need a sign condition ∫RNa(x)e1pdx<0, which has been proven to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with sign changing nonlinearity (Alama and Tarantello (1993) [1]).

Estimates for Solutions of Semilinear Elliptic Equation in Two Dimensions

In this paper we study a family of nonlinear Elliptic problems in two dimensions, we give some estimates for the solutions of this problem, and we decompose it on two problems, the first is the Poisson's equation and the second is the Liouville equation.

Infinitely many Solutions of Semi-Linear Elliptic Equation with a Logarithmic Nonlinear Term

The semi-linear elliptic equation is an important model in Mathematic, Physics. In this paper, we study the Dirichlet problem of semi-linear elliptic equation with a logarithmic nonlinear term. By using the logarithmic Sobolev inequality, mountain pass theorem and perturbation theorem, we obtain infinitely many nontrivial weak solutions, and also the energy of the solution is positive.

Multiple positive solutions to a class of Kirchhoff equation on [formula omitted] with indefinite nonlinearity

We study the existence and multiplicity of positive solutions to a class of Kirchhoff equation −(1+b∫R3|∇u(x)|2dx)Δu(x)+u(x)=V(x)|u(x)|p−2u(x)+λk(x)u(x),u(x):=u∈H1(R3). When V(x) is sign changing in R3, we prove that there is δ̄>0 such that the Kirchhoff equation possesses at least one positive solution for 0<λ<λ1+δ̄ and at least two positive solutions for λ1<λ<λ1+δ̄. Here λ1 is the first eigenvalue of −Δu+u=λk(x)u, u∈H1(R3). A positive eigenfunction corresponding to λ1 is denoted by e1. An interesting phenomena is that the term b(∫R3|∇u|2dx)Δu can be used to compete with the indefinite nonlinearity and we do not assume the condition ∫R3V(x)e1pdx<0, which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (Alama and Tarantello, 1993).

Positive bounded solutions for semilinear elliptic equations in smooth domains

We are concerned with the following semilinear elliptic equation $\Delta u=\lambda f\left( x,u\right) $ in $D,$ subject to some Dirichlet conditions, where $\lambda \geq 0$ is a parameter and $D$ is a smooth domain in $\mathbb{R}^{n}\left( n\geq 3\right) $. Under some appropriate assumptions on the nonnegative nonlinearity term $f\left( x,u\right) ,$ we show the existence of a positive bounded solution for the above semilinear elliptic equation. Our approach is based on Schauder's fixed point Theorem.

Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity

We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system:{−Δu+u+l(x)ϕu=k(x)|u|p−2u+μh(x)uin R3,−Δϕ=l(x)u2in R3, where 4<p<6, k∈C(R3), k changes sign in R3 and lim|x|→∞k(x)=k∞<0. We mainly prove the existence of at least two positive solutions in the case that μ>μ1 and near μ1, where μ1 is the first eigenvalue of −Δ+id in H1(R3) with weight function h, whose corresponding eigenfunction is denoted by e1. An interesting phenomenon is that we do not need the condition ∫R3k(x)e1pdx<0, which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. S. Alama, G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993) 439–475).

Existence of solutions for semilinear elliptic equations on R N

In this paper, the existence of at least one nontrivial solution for a class of semilinear elliptic equations on R N is established by using the linking methods.

Dirichlet forms and semilinear elliptic equations with measure data

We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right-hand side of the equation is monotone and satisfies mild integrability assumption, and the measure is smooth. We also study regularity of solutions under the assumption that the measure is smooth and has finite total variation. Some applications of our general results are given.

Positive solutions of semilinear elliptic equations: a dynamical approach

This paper is devoted to the study of the structure of positive radial solutions for the following semi-linear equation: $$\Delta u + f(u,|x|)=0 .$$ We require $f$ to be nonnegative and to exhibit both subcritical and supercritical behavior with respect to the Sobolev critical exponent. More precisely we assume that $f$ is subcritical for $u$ small and $|x|$ large and supercritical for $u$ large and $|x|$ small, and we give existence and non-existence results for ground states regular and singular, with either fast or slow decay. We find a surprisingly rich structure, which is characterized by two different patterns of bifurcations. We perform a Fowler transformation and we use a dynamical approach, exploiting some ideas borrowed from Bamon, Del Pino, and Flores, combining them with the use of the translation of the Pohozaev function for this dynamical context.

Eigenfunction Expansion Method for Multiple Solutions of Semilinear Elliptic Equations with Polynomial Nonlinearity

Eigenfunction expansion discretization is considered for finding multiple solutions of semilinear elliptic equations with polynomial nonlinearity. Error estimates of the discretization are derived. An efficient polynomial homotopy is constructed for computing all solutions of the resulting polynomial system on coarse level. A filter strategy on successively finer levels is designed to remove spurious solutions and to refine nonspurious solutions, simultaneously. The filtered solutions are further refined by a finite element Newton method. Numerical results are included to verify the error estimates derived, the efficiency of the homotopy, and the effectiveness of the filter strategy. A specific case of the Lazer--McKenna conjecture is numerically verified.

Multiple positive solutions of semilinear elliptic equations with weight function in exterior domains

In this paper, we are concerned with the multiplicity of solutions for semilinear elliptic problems with weight functions in exterior domains. We prove that, if the decay of the weight function at spatial infinity is sufficiently slow, then the equation admits at least three solutions and two of them escape away to the spatial infinity as the decay rate of the weight function tends to $0$. The result is proved via the variational method combined with the Ljusternik--Schnirelman-type multiplicity theorem.

Solutions of Semilinear Elliptic Equations in Tubes

Given a smooth compact k-dimensional manifold Λ embedded in ℝ m , with m≥2 and 1≤k≤m−1, and given ϵ>0, we define B ϵ (Λ) to be the geodesic tubular neighborhood of radius ϵ about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation $$\begin{cases} \Delta u + u^p = 0 &\mbox{in } B_{\epsilon}(\varLambda) \\ u = 0 & \mbox{on } \partial B_{\epsilon}(\varLambda) , \end{cases} $$ when the parameter ϵ is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m−k≤2 or $p\in(1, \frac{n+2}{n-2})$ when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ϵ tends to 0, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for $p>\frac{n+2}{n-2}$ , n≥3, if ϵ is sufficiently small.

Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior

In this paper, we study the existence and boundary behavior ofsolutions to boundary blow-up elliptic problems\begin{eqnarray*}\triangle u =b(x)f(u)(1+|\nabla u|^q), u\geq 0, \ x\in \Omega,\ u|_{\partial \Omega}=\infty,\end{eqnarray*} where $\Omega$ is a bounded domain with smooth boundaryin $\mathbb R^N$, $q\in (0, 2]$, $b \in C^{\alpha}(\bar{\Omega})$ which ispositive in $\Omega$, may be vanishing on the boundary, and $f$ isnormalised regularly varying at infinity with positive index $p$ and$p+q>1$.