Bounded positive solutions of semilinear elliptic equations

In this paper, by using the fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result of bounded positive solutions of semilinear elliptic equations in exterior domain of Rn, n?3.     

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 $\text{R}{}^2$. To cite this article: J. Busca et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Three positive solutions of nonhomogeneous semilinear elliptic equations

In this paper, we use new analyses to assert that there are three positive solutions of Eq. (1.1) in infinite cylinder domain with hole .     

Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ .     

Removable sets for continuous solutions of semilinear elliptic equations

We discuss the possible removability of sets for continuous solutions of semilinear elliptic equations of the form ???u =?F(x, u). In particular, we show that a set E in ${\mathbb{R}^{n}}$ is removable for ??-H?lder continuous solutions of such equations if and only if n ? 2?+???-dimensional Hausdorff measure of E is zero.     

Asymptotic symmetry of singular solutions of semilinear elliptic equations

For dimensions 3?n?6, we derive lower bound for positive solution of

     

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:

(∗)     

On positive entire solutions of indefinite semilinear elliptic equations

We establish that the elliptic equation Δu+K(x)up+μf(x)=0 in Rn has a continuum of positive entire solutions for small μ?0 under suitable conditions on K, p and f. In particular, K behaves like l|x| at ∞ for some l?−2, but may change sign in a compact region. For given l>−2, there is a critical exponent pc=pc(n,l)>1 in the sense that the result holds for p?pc and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l=−2, the result holds for any p>1 while pc is decreasing to 1 as l decreases to −2.     

Positive entire stable solutions of inhomogeneous semilinear elliptic equations

For n≥3 and p>1, the elliptic equation Δu+K(x)u^p+μf(x)=0 in possesses a continuum of positive entire solutions, provided that (i) locally Hölder continuous functions K and f vanish rapidly, for instance, K(x),f(x)=O(|x|^l) near ∞ for some l<−2 and (ii) μ≥0 is sufficiently small. Especially, in the radial case with K(x)=k(|x|) and f(x)=g(|x|) for some appropriate functions k,g on [0,∞), there exist two intervals I_μ,1, I_μ,2 such that for each α∈I_μ,1 the equation has a positive entire solution u_α with u_α(0)=α which converges to l∈I_μ,2 at ∞, and u_α1<u_α2 for any α₁<α₂ in I_μ,1. Moreover, the map α to l is one-to-one and onto from I_μ,1 to I_μ,2. If K≥0, each solution regarded as a steady state for the corresponding parabolic equation is stable in the uniform norm; moreover, in the radial case the solutions are also weakly asymptotically stable in the weighted uniform norm with weight function |x|ⁿ⁻².     

Symmetry of solutions to semilinear elliptic equations via Morse index

In this paper we prove symmetry results for solutions of semilinear elliptic equations in a ball or in an annulus in , , in the case where the nonlinearity has a convex first derivative. More precisely we prove that solutions having Morse index are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce, under some additional hypotheses on the nonlinearity, that the nodal set of sign changing solutions with Morse index intersects the boundary of the domain.

     

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.