Solutions Of Semilinear Elliptic Problems Research Articles

Multiple solutions for resonant semilinear elliptic problems in [formula omitted

We prove the existence of multiple nontrivial solutions for the semilinear elliptic problem − Δ u = h ( λ u + g ( u ) ) in R N , u ∈ D 1 , 2 , where h ∈ L 1 ∩ L α for α > N / 2 , N ⩾ 3 , g is a C 1 ( R , R ) function that has at most linear growth at infinity, g ( 0 ) = 0 , and λ is an eigenvalue of the corresponding linear problem − Δ u = λ h u in R N , u ∈ D 1 , 2 . Existence of multiple solutions, for certain values of g ′ ( 0 ) , is obtained by imposing a generalized Landesman–Lazer type condition. We use the saddle point theorem of Ambrosetti and Rabinowitz and the mountain pass theorem, as well as a Morse-index result of Ambrosetti [A. Ambrosetti, Differential Equations with Multiple Solutions and Nonlinear Functional Analysis, Equadiff 82, Lecture Notes in Math., vol. 1017, Springer-Verlag, Berlin, 1983] and a Leray–Schauder index theorem for mountain pass type critical points due to Hofer [H. Hofer, A note on the Topological Degree at a critical Point of Mountain Pass Type, Proc. Amer. Math. Soc. 90 (1984) 309–315]. The results of this paper are based upon multiplicity results for resonant problems on bounded domains in [E. Landesman, S. Robinson, A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Anal. 24 (1995) 1049–1059] and [S. Robinson, Multiple solutions for semilinear elliptic boundary value problems at resonance, Electron. J. Differential Equations 1995 (1995) 1–14], and complement a previous existence result by the authors in [G. López Garza, A. Rumbos, Resonance and strong resonance for semilinear elliptic equations in R N , Electron. J. Differential Equations 2003 (2003) 1–22] for resonant problems in R N in which g was assumed to be bounded.

Constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems

A new numerical algorithm for solving semilinear elliptic problems is presented. A variational formulation is used and critical points of a C1-functional subject to a constraint given by a level set of another C1-functional (or an intersection of such level sets of finitely many functionals) are sought. First, constrained local minima are looked for, then constrained mountain pass points. The approach is based on the deformation lemma and the mountain pass theorem in a constrained setting. Several examples are given showing new numerical solutions in various applications.

Existence of solutions for semilinear elliptic problems without (PS) condition

We establish an existence result for semilinear elliptic problems with the associated functional not satisfying the Palais-Smale condition. The nonlinearity of our problem does not satisfy the Ambrosetti-Rabinowitz condition.

Mountain pass solutions and an indefinite superlinear elliptic problem on $\mathbb R^{\mathbb N}$

We consider the elliptic problem $$ -\Delta u-\lambda u=a(x) g(u), $$ with $a(x)$ sign-changing and $g(u)$ behaving like $u^p$, $p> 1$. Under suitable conditions on $g(u)$ and $a(x)$, we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space $\mathbb R^N$. More precisely, we show that there exists $\Lambda> 0$ such that this equation on $\mathbb R^N$ has no positive solution for $\lambda> \Lambda$, at least two positive solutions for $\lambda\in (0,\Lambda)$, and at least one positive solution for $\lambda\in (-\infty,0]\cup\{\Lambda\}$. Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.

Multiple solutions for semilinear elliptic systems with non-homogeneous boundary conditions

Since the early seventies, many authors have widely investigated existence and multiplicity of solutions for semilinear elliptic problems with Dirichlet boundary conditions, especially by means of variational methods (see [22] and references therein). In particular, if ’ is a real L2-function on a bounded domain ⊂ Rn, p? 2 and pi 2∗ if n? 3 (here, 2∗ = 2n n−2 ), the following model problem   −6u= |u|p−2u+ ’ in ;

On the existence of explosive solutions for semilinear elliptic problems

On the existence of explosive solutions for semilinear elliptic problems

Blow-up boundary solutions of semilinear elliptic problems

We consider the following semilinear elliptic equation: :u= p(x)f(u) in <; u ≥ 0; u ≡ 0 in <; (1) where <⊂RN (N ≥ 3) is a smooth domain (bounded or possibly unbounded) with compact (possibly empty) boundary. We assume throughout this paper that p is a non-negative function such that p∈C0; ( = <) if < is bounded, and p∈C loc (<), otherwise. The nonlinearity f is assumed to ful?ll (f 1) f ∈ C1[0;∞); f′ ≥ 0; f(0) = 0 and f? 0 on (0;∞) and the Keller–Osserman condition (see [7,13])

Multiple existence of sign changing solutions for semilinear elliptic problems on RN

Multiple existence of sign changing solutions for semilinear elliptic problems on RN

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.

Existence and exact multiplicity of positive radial solutions of semilinear elliptic problems in annuli

We study the existence, nonexistence and exact multiplicity of positive radial solutions of the problem: $- \Delta u = f(u) $ in $ B(R_1, R)$, where $ R_1 > 0 $ is a fixed real number and $ B(R_1,R) \subset \mathbb R^n (n \geq 3) $, $ R > R_1 $, is an annulus, for the following boundary conditions: (1) $ u = 0 $ on $ \partial B(R_1,R) $. For $ f(t) = t^p - t^q $, $ 1 < p < q $ and $ p \leq (n+2)/(n-2) $, there exist $ R_0 \geq \tilde{R} > R_1 $ such that the problem admits exactly two solutions for $ R > R_0 $ and no solution for $ R < \tilde{R}$. (2) $ u = 0 $ on $ |x| = R_1 $ and $\partial u/ \partial \nu = 0 $ on $ |x| = R $. (a) For $ f(t) = t^p - t^q $, $ 1 < p < q $ and $ p \leq (n+2)/(n-2)$, there exists $ R_0 > R_1 $ such that the problem admits exactly two solutions for $ R > R_0 $, exactly one solution for $ R = R_0 $ and no solution for $ R < R_0 $. (b) Let $ f^{\prime}(t)t > f(t) $. If $ f(t) > 0$ for $ t > 0 $, then the problem admits at most one solution for any $R > R_1 $. For a large class of nonlinearities $ f $ changing sign (for example $ f(t) = t^p -t $), there exists $ R_0 > R_1 $ such that the problem admits a unique solution for all $ R < R_0 $. These results have been proved by analyzing the behaviour of the first and the second variations of $ u(\gamma,r) $, the unique solution of the corresponding initial value problem with $ u^{\prime}(R_1) = \gamma > 0 $, with respect to $ \gamma$ together with an identity (2.8) which may be regarded as variations of the generalized Pohozaev's identity with respect to $ \gamma $.

Multiple solutions for semilinear elliptic problems

A method of finding critical points in a half space is developed. It is then applied to the study of semilinear boundary value problems, and used to determine conditions which lead to multiple non-trivial solutions.

Classification of the Solutions of Semilinear Elliptic Problems in a Ball

In this paper we fully describe the set of the positive and nodal (regular and singular) radial solutions of the superlinear elliptic PDE. −Δu=λu+|u|p−1uinB1,u=0on∂B1,p>1, (1)without restriction on the range of λ∈R. Here, B1 is the unit ball in RN. More precisely, in all subcritical, critical and supercritical cases, we analyze the possible singularities of radial solutions at the origin and the number of bounded and unbounded solutions. The solutions will be of three different types: bounded with a finite number of zeroes in (0, 1), singular at the origin, still with a finite number of zeroes and singular with sign changing oscillations at the origin.

One-dimensionality of Solutions of Semilinear Elliptic Problems in Cylindrical Domains

One-dimensionality of Solutions of Semilinear Elliptic Problems in Cylindrical Domains

One-dimensionality of Solutions of Semilinear Elliptic Problems in Cylindrical Domains

One-dimensionality of Solutions of Semilinear Elliptic Problems in Cylindrical Domains

Eigenvalues of singular boundary value problems and existence results for positive radial solutions of semilinear elliptic problems in exterior domains

Eigenvalues of singular boundary value problems and existence results for positive radial solutions of semilinear elliptic problems in exterior domains

On the extremal solutions of semilinear elliptic problems

We investigate here the properties of extremal solutions for semilinear elliptic equation −Δu = λf(u) posed on a bounded smooth domain of ℝn with Dirichlet boundary condition and with f exploding at a finite positive value a.

Large solutions of semilinear elliptic problems

Large solutions of semilinear elliptic problems

Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli

We study the existence of many nonradial sign-changing solutions of a superlinear Dirichlet boundary value problem in an annulus in $\mathbb R^N$. We use Nehari-type variational method and group invariance techniques to prove that the critical points of an action functional on some spaces of invariant functions in $H_{0}^{1,2}(\Omega_{\varepsilon})$, where $\Omega_{\varepsilon}$ is an annulus in $\mathbb R^N$ of width $\varepsilon$, are weak solutions (which in our case are also classical solutions) to our problem. Our result generalizes an earlier result of Castro et al. (See [A. Castro, J. Cossio and J. M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems , Electron. J. Differential Equations 2 (1998), 1–18]).

Oscillating global continua of positive solutions of semilinear elliptic problems

Oscillating global continua of positive solutions of semilinear elliptic problems

Existence of entire solutions for semilinear elliptic problems on ${\mathbb R}^{N}$

In this paper, we consider the existence of positive and negative entire solutions of semilinear elliptic problem $$ -\Delta u + u = g(x,u), \quad u \in H^{1}({\mathbb R}^{N})\tag{P} $$ where $N \geq 2$ and $g:{\mathbb R}^{N} \times {\mathbb R }\to {\mathbb R}$ is a continuous function with superlinear growth and $g(x,0) = 0$ on ${\mathbb R}^{N} $.