Class Of Semilinear Elliptic Equations Research Articles

Local bifurcation from the second eigenvalue of the Laplacian in a square

In this work we study local bifurcation from the branch of trivial solutions for a class of semilinear elliptic equations, at the second eigenvalue λ 2 \lambda _2 of a square. We find that the bifurcation set can be locally described as the union of exactly four bifurcation branches of nontrivial solutions which cross the bifurcation point ( λ 2 , 0 ) (\lambda _2,0) . We also compute the Morse index of the solutions in the four branches.

On the existence of positive solutions for a class of semilinear elliptic equations

We prove some existence and nonexistence results for the semilinear elliptic equation Δ u= q( x) f( u) on Ω⊆ R n (n⩾2) where u is required to blow up on the boundary of Ω and f is a nonnegative function which is assumed to be Lipschitz continuous and bounded away from zero on each interval [ ε,∞) and have at worst linear growth. In particular, we extend some results already obtained in the case where f( u)= u γ , 0< γ⩽1.

On a Class of Semilinear Elliptic Equations in Rn

We establish that for n⩾3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses separated positive entire solutions of infinite multiplicity, provided that a locally Hölder continuous function K⩾0 in Rn\\{0}, satisfies K(x)=O(∣x∣σ) at x=0 for some σ>−2, and K(x)=c∣x∣−2+O(∣x∣−n[log∣x∣]q) near ∞ for some constants c>0 and q>0. In the radial case K(x)=∣x∣l1+∣x∣τ with l>−2 and τ⩾0, or K(x)=∣x∣λ−2(1+∣x∣2)λ/2 with λ>0, we investigate separation phenomena of positive radial solutions, and show that if n and p are large enough, the equation possesses a positive radial solution with initial value α at 0 for each α>0 and a unique positive radial singular solution among which any two solutions do not intersect.

On a double resonant problem in $\Bbb R^N$

It is established, via variational methods, the existence and multiplicity of solutions for a class of semilinear elliptic equations in ${{\mathbb R}^N}$. The condition on the potential implies that the associated linear problem possesses a sequence of positive eigenvalues. This fact is used to study double resonant problems under a local nonquadraticity condition at infinity and pointwise limits. For the existence of solution the nonlinearity may satisfy a critical growth condition.

Remarks on a Class of Semilinear Elliptic Equations on ℝn, via Perturbat ion Met hods

Remarks on a Class of Semilinear Elliptic Equations on ℝⁿ, via Perturbat ion Met hods

Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations

The existence and multiplicity results are obtained for solutions of a class of the Dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods, respectively.

Existence Results for the Equation $– \Delta u = f(x, u) in \mathbb R^n$

Strong solutions to the class of semilinear elliptic equations – \Delta u = f(x, u) on the entire space and with possibly supercritical growth for f(x, \cdot)$ are obtained by mainly using fixed points arguments. The case of discontinuous non-linearities is then examined.

A remark on the existence of explosive solutions for a class of semilinear elliptic equations

A remark on the existence of explosive solutions for a class of semilinear elliptic equations

ON SEMILINEAR ELLIPTIC EQUATIONS WITH CONCAVE AND CONVEX NONLINEARITIES IN RN

In this paper, a class of semilinear elliptic equations with sublinear and superlinear nonlinearities in RN is studied. By making use of variational method and L∞ estimation, the authors obtain some results about existence of multiple positive solutions and asymptotic behavior of the solutions.

Positive solutions for a class of nonlinear elliptic problems in RN

We are concerned with positive solutions decaying at infinity for a class of semilinear elliptic equations in all of RN having superlinear subcritical nonlinearity. The corresponding variational problem lacks compactness because of the unboundedness of the domain and, in particular, it cannot be solved by minimization methods. However, we prove the existence of a positive solution, corresponding to a higher critical value of the related functional, under a suitable fast decay condition on the coefficient of the linear term. Moreover, we analyse the behaviour of the solution as this coefficient goes to infinity and show that the solution tends to split as the sum of two positive functions sliding to infinity in opposite directions. Finally, we use this property to prove the existence of at least 2k − 1 distinct positive solutions, when this coefficient splits as the sum of k bumps sufficiently far apart.

On a class of semilinear elliptic problems near critical growth

We use Minimax Methods and explore compact embedddings in the context of Orlicz and Orlicz‐Sobolev spaces to get existence of weak solutions on a class of semilinear elliptic equations with nonlinearities near critical growth. We consider both biharmonic equations with Navier boundary conditions and Laplacian equations with Dirichlet boundary conditions.

Multiple solutions for a class of semilinear elliptic equations

Multiple solutions for a class of semilinear elliptic equations

Asymptotic Behavior of Radial Solutions for a Class of Semilinear Elliptic Equations

Equation (1) has its origin from, e.g., the prescribed curvature problems in Riemannian geometry, and astrophysics (i.e., the Lane Emden Fowler equation and the Matukuma equation as special cases). The asymptotic behavior of the positive solutions to (1) has recently received much attention, see e.g., [L1, LN, Na]. However, it is well-known that the above equation (1) does not always have positive solutions or positive radial solutions. In other words, under suitable conditions, radial solutions to (1) must oscillate about the zero at infinite times (see e.g., [DCC, NY, N]). Thus it becomes very interesting to know as precisely as possible the asympototic behaviors of the oscillating periods, the amplitudes of the oscillatory solutions. In this paper, we restrict our attention to the study of radial solutions to (1). More precisely, we shall discuss the asymptotic behavior of oscillatory article no. DE963208

The existence and uniqueness of the decaying positive entire solutions for a class of semilinear elliptic equations

This paper first proves the following equations $$\Delta u - m^2 u + f(x,u) = 0, x\varepsilon R^n , n \geqslant 3, m > 0$$

Numerical approximation of bounded solutions for semilinear elliptic equations in an unbounded cylindrical domain

AbstractWe consider the approximation of bounded solutions for a class of semilinear elliptic equations in an unbounded cylindrical domain. On certain conditions, the existence of approximate solutions for the semidiscretization of the problem is proved. Error estimates, applications, and a numerical example are also given. © 1993 John Wiley &amp; Sons, Inc.

Singularly perturbed semilinear elliptic equation with boundary-interior layer interaction

In this paper, we consider the phenomenon of the boundary and interior layer interactions for a class of semilinear elliptic equation. Under some appropriate conditions, we get the existence of the exact solution for the problem and its high order uniformly valid expansion.

The positive radial solutions of a class of semilinear elliptic equations.

Article The positive radial solutions of a class of semilinear elliptic equations. was published on January 1, 1989 in the journal Journal für die reine und angewandte Mathematik (volume 1989, issue 401).

Existence of Decaying Entire Solutions of a Class of Semilinear Elliptic Equations

The main result establishes the existence of a nontrivial nonnegative radial solution $u \in C_{\operatorname {loc} }^2({{\mathbf {R}}^N})$ of a semilinear elliptic eigenvalue problem in ${{\mathbf {R}}^N},N \geq 3$, such that $u(|x|)$ has uniform limit zero as $|x| \to \infty$. Asymptotic decay estimates and necessary conditions are obtained. Since such solutions do not exist in the space $W_0^{1,2}({{\mathbf {R}}^N})$, a considerable departure from standard procedures is required.

Existence of ground states with exponential decay for semi-linear elliptic equations in ofR n

In this paper, we prove the existence of a positive solution, tending exponentially to zero at infinity, for a class of semi-linear elliptic equations with nonconstant coefficients in ofR n .

Existence of decaying entire solutions of a class of semilinear elliptic equations

The main result establishes the existence of a nontrivial nonnegative radial solution u ∈ C loc 2 ( R N ) u \in C_{\operatorname {loc} }^2({{\mathbf {R}}^N}) of a semilinear elliptic eigenvalue problem in R N , N ≥ 3 {{\mathbf {R}}^N},N \geq 3 , such that u ( | x | ) u(|x|) has uniform limit zero as | x | → ∞ |x| \to \infty . Asymptotic decay estimates and necessary conditions are obtained. Since such solutions do not exist in the space W 0 1 , 2 ( R N ) W_0^{1,2}({{\mathbf {R}}^N}) , a considerable departure from standard procedures is required.