Solutions Of Semilinear Elliptic Equations Research Articles

Dependence of blowup rate of large solutions of semilinear elliptic equations, on the curvature of the boundary*

Let D be a smooth bounded domain in . Let f be a positive monotone increasing function on which satisfies the Keller–Osserman condition. It is well-known that the solutions of Δ u=f(u), which blow up at the boundary behave, to a first order approximation, like a function of dist(x,∂ D). In this paper we show that the second order approximation depends on the mean curvature of ∂ D. This paper is an extension of results in [4] which dealt with radially symmetric solutions. It extends also the results in [5] for f = tp .

Some nonexistence results for positive solutions of elliptic equations in unbounded domains

We prove some Liouville type theorems for positive solutions of semilinear elliptic equations in the whole space \mathbb{R}^N , N\geq 3 , and in the half space \mathbb{R}^N_{+} with different boundary conditions, using the technique based on the Kelvin transform and the Alexandrov-Serrin method of moving hyperplanes. In particular we get new nonexistence results for elliptic problems in half spaces satisfying mixed (Dirichlet-Neumann) boundary conditions.

Solutions for semilinear elliptic equations with critical exponents and Hardy potential

In this paper, we answer affirmatively an open problem (cf. Theorem 4′ in Ferrero and Gazzola (J. Differential Equations 177 (2001) 494): Let Ω∋0 be an open-bounded domain, Ω⊂R N(N⩾5) and assume that 0⩽μ<( N−2 2 ) 2−( N+2 N ) 2 , then, for all λ>0 there exists a nontrivial solution with critical level in the range (0, 1 N S μ N 2 ) for the problem −Δu−μ u |x| 2 =λu+|u| 2 ∗−2 u in Ω; u=0 on ∂Ω.

On the stability of radial solutions of semilinear elliptic equations in all of [formula omitted

We establish that every nonconstant bounded radial solution u of −Δ u= f( u) in all of R n is unstable if n⩽10. The result applies to every C 1 nonlinearity f satisfying a generic nondegeneracy condition. In particular, it applies to every analytic and every power-like nonlinearity. We also give an example of a nonconstant bounded radial solution u which is stable for every n⩾11, and where f is a polynomial. To cite this article: X. Cabré, A. Capella, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Partial regularity for weak solutions of semilinear elliptic equations with supercritical exponents

Let Ω be an open subset in R n (n > 3). In this paper, we study the partial regularity for stationary positive weak solutions of the equation (1.1) Δu + h 1 (x)u + h 2 (x)u α = 0 in Ω. We prove that if α > n+2/n-2, and u ∈ H 1 (Ω) n L α+1 (Ω) is a stationary positive weak solution of (1.1), then the Hausdorff dimension of the singular set of u is less than n - 2α+1/α-1, which generalizes the main results in Pacard 1993 and Pacard 1994.

Classification of positive solutions of semilinear elliptic equations

We give a classification of all solutions of a general semilinear PDE in the positive quadrant of R 2 . To cite this article: J. Busca et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Moving planes, moving spheres, and a priori estimates

In this paper, we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi-linear elliptic equations. Using the ‘method of moving planes’, we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of Lin. Applying ‘the method of moving spheres’, we give a different proof for a well-known sup+inf inequality established by Brezis, Li and Shafrir.

On Uniqueness of Positive Solutions of Semilinear Elliptic Equations with Singular Potential

Abstract We prove the uniqueness of positive W0 1,2 solutions for the following problem, which involves the singular potential |x|-2 : where N ≥ 3, 1 &lt; p &lt; , and 0 &lt; λ &lt; . We also give a complete description of the set of positive singular solutions of the problem.

Positive solution for semilinear elliptic equation on general domain

We give the comparison principle, the uniqueness theorem, and the necessary and sufficient conditions for the solvability with positive solutions in W loc 2, p ∩ L ∞ of the Dirichlet problems for semilinear elliptic equations on general bounded domains. The domains may be irregular while the equations are in general form. The resultant theorem includes previous work either in sublinear and some superlinear cases on the smooth domains or in linear case on the general domains. Our methods are the refined a priori estimates and the degree theory.

Existence of a global branch of solutions of semilinear elliptic equations in [formula omitted

Existence of a global branch of solutions of semilinear elliptic equations in [formula omitted

The symmetry of least-energy solutions for semilinear elliptic equations

In this paper we will apply the method of rotating planes (MRP) to investigate the radial and axial symmetry of the least-energy solutions for semilinear elliptic equations on the Dirichlet and Neumann problems, respectively. MRP is a variant of the famous method of moving planes. One of our main results is to consider the least-energy solutions of the following equation: (∗) Δu+K(x)u p=0, x∈B 1, u>0 in B 1, u| ∂B 1 =0, where 1<p< n+2 n−2 and B 1 is the unit ball of R n with n⩾3. Here K( x)= K(| x|) is not assumed to be decreasing in | x|. In this paper, we prove that any least-energy solution of (∗) is axially symmetric with respect to some direction. Furthermore, when p is close to n+2 n−2 , under some reasonable condition of K, radial symmetry is shown for least-energy solutions. This is the example of the general phenomenon of the symmetry induced by point-condensation. A fine estimate for least-energy solution is required for the proof of symmetry of solutions. This estimate generalizes the result of Han (Ann. Inst. H. Poincaré Anal. Nonlinéaire 8 (1991) 159) to the case when K( x) is nonconstant. In contrast to previous works for this kinds of estimates, we only assume that K( x) is continuous.

Large positive solutions of semilinear elliptic equations with critical and supercritical growth

Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic equations with critical and supercritical growth on general bounded smooth domains. It is shown that the large positive solution develops a boundary layer. The boundary derivative estimate of the large solution is also established.

Symmetry Results for Solutions of Semilinear Elliptic Equations with Convex Nonlinearities

In this paper, we study the symmetry properties of the solutions of the semilinear elliptic problem {−Δu=f(x,u)in Ωu=g(x)on ∂Ω, where Ω is a bounded symmetric domain in RN, N⩾2, and f:Ω×R→R is a continuous function of class C1 in the second variable, g is continuous and f and g are somehow symmetric in x. Our main result is to show that all solutions of the above problem of index one are axially symmetric when Ω is an annulus or a ball, g≡0 and f is strictly convex in the second variable. To do this, we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of Ω is a sufficient condition for the symmetry of the solution, when f is a convex function.

Solutions of semilinear elliptic equations with superlinear sign changing nonlinearities

Solutions of semilinear elliptic equations with superlinear sign changing nonlinearities

Solutions of semilinear elliptic equations with asymptotic linear nonlinearity

, and ; and bare positive real numbers.The solutions structure of (1) is quite dierent when is equal to zero or not. When is equal to zero or a nonzero real number but small enough, there are some resultsof solutions structure of (1) in [1,9]. Here we extend these results to the case when is an arbitrary positive real number.Forconvenience,wedenotef =(u

Existence and Asymptotic Behaviour of Large Solutions of Semilinear Elliptic Equations

In this paper we are interested in the semilinear elliptic equations of the type Δu=u⋅ϕ(⋅,u), on bounded smooth domain Ω of Rn. We also treat existence of positive solution of Δu=p(x)f(u), which explodes near the boundary of Ω (called large solutions). Our approach is based on potential theory.

Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems

We propose a method to investigate the structure of positive radial solutions to semilinear elliptic problems with various boundary conditions. It is already shown that the boundary value problems can be reduced to a canonical form by a suitable change of variables. We show structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.

CANONICAL FORM RELATED WITH RADIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS AND ITS APPLICATIONS

CANONICAL FORM RELATED WITH RADIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS AND ITS APPLICATIONS

The existence of a symmetric global minimizer on [formula omitted] implies the existence of a counter-example to a conjecture of De Giorgi in [formula omitted

For solutions of semilinear elliptic equations, we show the link that exists between global minimizers on R n−1 and the existence of nontrivial monotonic solutions on R n . We prove that the existence of a symmetric global minimizer on R n−1 implies the existence of a nonplanar monotonic solution on R n which is a global minimizer. Moreover, we prove that every monotonic solution on R n is a global minimizer if and only if its up and down limits are global minimizers.

A unified approach to the structure of radial solutions for semilinear elliptic problems

Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is shown that the boundary value problems can be reduced to a canonical form after suitable change of variables. Through the canonical form, we can study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations. We also clarify the implication of the Kelvin transformation and the Rellich-Pohozaev identity, and give their generalized forms.