Solutions Of Semilinear Elliptic Equations Research Articles

The accelerated search-extension method for computing multiple solutions of semilinear PDEs

In this paper, we propose an accelerated search-extension method (ASEM) based on the interpolated coefficient finite element method, the search-extension method (SEM) and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in 1-D and 2-D cases will show the efficiency of our approach.

Some new entire solutions of semilinear elliptic equations on [formula omitted

We prove existence of a new type of positive solutions of the semilinear equation − Δ u + u = u p on R n , where 1 < p < n + 2 n − 2 . These solutions are bounded, but do not tend to zero at infinity. Indeed, they decay to zero away from three half-lines with a common origin, and their asymptotic profile is periodic along these half-lines.

The behavior of solutions of semilinear elliptic equations of second order of the form Lu = e u in the infinite cylinder

We consider a semilinear elliptic equation of second order with variable coefficients of the form Lu = e u in the semi-infinite cylinder whose solution satisfies a homogeneous Neumann condition on the lateral surface of the cylinder.

Generalized boundary values of the solutions of semilinear elliptic equations from weighted functional spaces

In weighted C-spaces, we establish the solvability of a boundary-value problem for a semilinear elliptic equation of order 2m in a bounded domain with generalized functions given on its boundary, strong power singularities at some points of the boundary, and finite orders of singularities on the entire boundary. The behavior of the solution near the boundary of the domain is analyzed.

Multiplicity of positive and nodal solutions for semilinear elliptic equations in infinite strips

In this paper, we study the multiplicity of positive and nodal solutions of the following semilinear elliptic equation: { − Δ u + u = g μ ( x ) | u | p − 2 u in A , u ∈ H 0 1 ( A ) , where 2 < p < 2 N N − 2 ( N ≥ 3 ) , the parameter μ ≥ 0 , A = Θ × R is an infinite strip in R N and Θ is a bounded domain in R N − 1 . We assume that g μ ( x ) = a ( x ) + μ b ( x ) is a positive function, where the functions a and b are satisfying some suitable conditions.

Existence and multiplicity of solutions for semilinear elliptic equations with critical weighted Hardy–Sobolev exponents

Some existence results are obtained for solutions of semilinear elliptic equations with critical weighted Hardy–Sobolev exponents and superlinear nonlinearity by the variational methods and some analysis techniques.

Multiplicity of 2-nodal solutions for a semilinear elliptic equation

In this paper, we consider the multiplicity of 2-nodal solutions of semilinear elliptic equations. Using the generalized barycenter map, we prove that existence of multiple 2-nodal solutions for semilinear elliptic equations in some domains with hole.

Entire large solutions of semilinear elliptic equations of mixed type

We prove existence and nonexistence of nonnegative entire largesolutions for the semilinear elliptic equation $\Delta u = p(x)f(u) +q(x)g(u)$ in which $f$ and $g$ are nondecreasing and vanish at theorigin. The locally Hölder continuous functions $p$ and $q$ arenonnegative. We extend results previously obtained for special cases of$f$ and $g$.

Existence of global positive solutions of semilinear elliptic equations

Using a very simple approach, we prove the existence of asymptotically vanishing global positive solutions of the semilinear elliptic equation Δ u + f ( x , u ) + g ( | x | ) x ⋅ ∇ u = 0 in all of R n , n ≥ 3 . In particular, earlier conditions on the negative part of the function f are dropped.

Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities

We extend our recent results on the classification of stable solutions of the equation − Δ u = f ( u ) in R N , where f ≥ 0 is a general convex, non-decreasing function.

Limits at Infinity of Superharmonic Functions and Solutions of Semilinear Elliptic Equations of Matukuma Type

In an unbounded domain Ω in ℝn (n ≥ 2) with a compact boundary or Ω = ℝn, we investigate the existence of limits at infinity of positive superharmonic functions u on Ω satisfying a nonlinear inequality like as $$ -\Delta u(x) \le \frac{c}{1+|x|^2}u(x)^p \quad \text{for } x\in{\Omega}, $$ where Δ is the Laplacian and c > 0 and p > 0 are constants. The result is applicable to positive solutions of semilinear elliptic equations of Matukuma type.

Ground state solutions for singular semi-linear elliptic equations

We prove the existence of a ground state solution for the semi-linear elliptic equation − Δ u = f ( x , u ) on R N under suitable conditions on a locally Hölder continuous non-linearity f ( x , t ) . The non-linearity may exhibit a singularity as t → 0 + .

Existence of positive entire solutions of semilinear elliptic equations

For n ≥ 3 and p > 1 , the elliptic equation Δ u + K ( x ) u p + f ( x ) = 0 in R n possesses a positive entire solution under proper conditions on locally Hölder continuous functions K and f . After obtaining a priori estimates on decay of solutions near ∞ , we study the existence of positive solutions, and present necessary conditions of f for the existence.

A [formula omitted] inequality near [formula omitted

Let Ω be a domain in R 2 , not necessarily bounded. Consider the semi-linear elliptic equation − Δ u = R ( x ) e u , x ∈ Ω . We prove that, for any compact subset K of Ω, there is a constant C, such that the inequality (1) sup K u + inf Ω u ⩽ C holds for all solutions u. This type of inequality was first established by Brezis, Li, and Shafrir [H. Brezis, Y.Y. Li, I. Shafrir, A sup + inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal. 115 (1993) 344–358] under the assumption that R ( x ) is positive and bounded away from zero. It has become a useful tool in estimating the solutions of semi-linear elliptic equations either in Euclidean spaces or on Riemannian manifolds (see [H. Brezis, Y.Y. Li, I. Shafrir, A sup + inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal. 115 (1993) 344–358; C.-C. Chen, C.-S. Lin, A sharp sup + inf inequality for a nonlinear elliptic equation in R 2 , Comm. Anal. Geom. 6 (1998) 1–19; W. Chen, C. Li, Gaussian curvature in the negative case, Proc. Amer. Math. Soc. 131 (2003) 741–744; W. Chen, C. Li, Indefinite elliptic problems with critical exponent, in: Advances in Non-linear PDE and Related Areas, World Scientific, 1998, pp. 67–79; Y.Y. Li, I. Shafrir, Blow up analysis for solutions of − Δ u = V e u in dimension two, Indiana Univ. Math. J. 43 (1994) 1255–1270]). In Brezis, Li, and Shafrir's result, the constant C depends on the lower bound of the function R ( x ) . In this paper, we remove this restriction and extend the inequality to the case where R ( x ) is allowed to have zeros, so that it can be applied to obtain a priori estimates for a broader class of equations, as we will illustrate in the last section. The key to prove this inequality is the analysis of asymptotic decay near infinity of solutions for a corresponding limiting equation in R 2 , which is interesting and useful in its own right.

Multiple solutions of semilinear elliptic equations in exterior domains

In this paper, assume that in an exterior domain admits at least two positive solutions and a solution which changes sign.

Multiplicity of positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$

In this paper, we study the multiplicity of positive solutions for the following semilinear elliptic equation:\begin{alignat*}{2} -\Delta u+\lambda u&amp;=f(x)u^{p-1}+h(x)u^{q-1} &amp;\quad&amp;\text{in }\mathbb{R}^{N}, \\ u&amp;&gt;0 &amp;&amp;\text{in }\mathbb{R}^{N}, \\ u&amp;\in H^{1}(\mathbb{R}^{N}), \end{alignat*}where affects the number of positive solutions.

Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems

We apply the “monotone separation of graphs” technique of L.A. Peletier and J. Serrin [L.A. Peletier, J. Serrin, Uniqueness of positive solutions of semilinear equations in R n , Arch. Ration. Mech. Anal. 81 (2) (1983) 181–197; L.A. Peletier, J. Serrin, Uniqueness of nonnegative solutions of semilinear equations in R n , J. Differential Equations 61 (3) (1986) 380–397], as developed further by L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202], to the question of exact multiplicity of positive solutions for a class of semilinear equations on a unit ball in R n . We also observe that using P. Pucci and J. Serrin [P. Pucci, J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (2) (1998) 501–528] improvement of a certain identity of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] produces a short proof of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] result on the uniqueness of positive solution of ( 1 < p , q < n + 2 n − 2 ) Δ u + u p + u q = 0 for | x | < 1 , u = 0 when | x | = 1 .

Asymptotic symmetry of singular solutions of semilinear elliptic equations

For dimensions 3 ⩽ n ⩽ 6 , we derive lower bound for positive solution of Δ u − μ u + K ( x ) u n + 2 n − 2 = 0 in B 2 ∖ { 0 } , u ∈ C 2 ( B 2 ∖ { 0 } ) in the neighbourhood of singularity. Here μ > 0 . We prove that there exists a constant c 1 > 0 depending on n, μ, | K | , | ∇ K | such that u ( x ) > c 1 | x | − n − 2 2 in B 1 ∖ { 0 } . For dimension 3, we assume that K is Hölder continuous with exponent θ with 1 2 < θ ⩽ 1 while for dimensions n = 4 , 5 , 6 , assume that K ∈ C 1 is bounded between two positive constants c | x | θ − 1 ⩽ | ∇ K ( x ) | ⩽ C | x | θ − 1 for c, C > 0 and n − 2 2 ⩽ θ ⩽ n − 2 . As an application, we derive the asymptotic symmetry and a priori estimates for the solutions.

Pseudo-radial solutions of semi-linear elliptic equations on symmetric domains

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $\Delta u = \pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains.

Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations

In this paper, we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation. We also obtain a precise global bifurcation diagram of the solution set. As a result, an open problem presented by C.-H. Hsu and Y.-W. Shih [C.-H. Hsu, Y.-W. Shih, Solutions of semilinear elliptic equations with asymptotic linear nonlinearity, Nonlinear Anal. 50 (2002) 275–283] is completely solved. Our argument is mainly based on bifurcation theory and continuation method.