Positive solutions for semilinear elliptic equations with singular forcing terms

We consider the existence of solutions to the semilinear elliptic problem

(∗)_κ     

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.     

Two-gap elliptic solutions to integrable nonlinear equations

We study spectral surfaces associated with elliptic two-gap solutions to the nonlinear Schrödinger equation (NLS), the Korteweg-de Vries equation (KdV), and the sine-Gordon equation (SG). It is shown that elliptic solutions to the NLS and SG equations, as well as solutions to the KdV equation elliptic with respect tot, can be assigned to any hyperelliptic surface of genus 2 that forms a covering over an elliptic surface.     

Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Clocα(Ω) is positive in but may be vanishing or appropriately singular on the boundary,f∈C[0,∞),f(0)=0,and f is strictly increasing on [0,∞)(or f∈C(R),f(s)> 0,■s∈R,f is strictly increasing on R).We show unified boundary behavior of such solutions to the problem under a new structure condition on f.     

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).     

Multiple solutions of semilinear elliptic equations with one-sided growth conditions

Multiplicity results for semilinear elliptic equations are obtained under one-sided growth conditions on the nonlinearity. Techniques of nonsmooth critical point theory are employed.     

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 ${(x, y) \in \mathbb {R}^{2}}$ .     

Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term

In this paper, we introduce weighted p-Sobolev spaces on manifolds with edge singularities. We give the proof for the corresponding edge type Sobolev inequality, Poincaré inequality and Hardy inequality. As an application of these inequalities, we prove the existence of nontrivial weak solutions for the Dirichlet problem of semilinear elliptic equations with singular potentials on manifolds with edge singularities.     

Positive solutions to a semilinear elliptic equation with a Sobolev-Hardy term

In this paper, we study the existence of positive solutions to the following semilinear elliptic equation with a Sobolev-Hardy term

(0.1)     

Asymptotic symmetry of singular solutions of semilinear elliptic equations

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