Solutions Of Semilinear Elliptic Problems Research Articles

Bifurcation and one-sign solutions for semilinear elliptic problems in $ \mathbb{R}^{N} $

<abstract><p>In this work, we study the existence of one-sign solutions without signum condition for the following problem:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray} \left\{ \begin{array}{ll} -\Delta u = \lambda a(x)f(u), \, \, x\in\mathbb{R}^{N}, & {\rm{}}\ u(x)\rightarrow0, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\mathrm{as}}\, \, |x|\rightarrow +\infty, & {\rm{}} \end{array} \right. \end{eqnarray} $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ \lambda $ is a real parameter and $ a\in C^{\alpha}_{loc}(\mathbb{R}^{N}, \mathbb{R}) $ for some $ \alpha\in(0, 1) $ is a weighted function, $ f\in C^{\alpha}(\mathbb{R}, \mathbb{R}) $, and there exist two constants $ s_{2} < 0 < s_{1}, $ such that $ f(s_{1}) = f(s_{2}) = f(0) = 0 $ and $ sf(s) > 0 $ for $ s\in\mathbb{R}\backslash\{s_{1}, 0, s_{2}\}. $ Furthermore, we consider the exact multiplicity of one-sign solutions for above problem under more strict hypotheses. We use bifurcation techniques and the approximation of connected components to prove our main results.</p></abstract>

Computer-assisted proofs for some nonlinear diffusion problems

In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form Δu+f(u,∇u)=0. By studying a well chosen fixed point problem defined around the numerical solution, these techniques make it possible to prove the existence of a solution in an explicit (and usually small) neighborhood of the numerical solution. In this work, we develop a similar approach for a broader class of systems, including nonlinear diffusion terms of the form ΔΦ(u). In particular, this enables us to obtain new results about steady states of a cross-diffusion system from population dynamics: the (non-triangular) SKT model. We also revisit the idea of automatic differentiation in the context of computer-assisted proof, and propose an alternative approach based on differential–algebraic equations.

Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \] where $L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviours.

A Landesman–Lazer Local Condition for Semilinear Elliptic Problems

The objective of this paper is to study the existence, multiplicity and non existence of solutions for semilinear elliptic problems under a local Landesman–Lazer condition. There is no growth restriction at infinity on the nonlinear term and it may change sign. In order to establish the existence of solution we combine the Lyapunov–Schmidt reduction method with truncation and approximation arguments via bootstrap methods. In our applications we also consider the existence of a bifurcation point which may have multiple positive solutions for a fixed value of the parameters.

Positive solutions of semilinear elliptic problems with a Hardy potential

AbstractLet Ω ⊂ ℝNbe a bounded domain and δ(x) be the distance of a pointx∈ Ω to the boundary. We study the positive solutions of the problem Δu+ (μ/(δ(x)2))u=upinΩ, wherep> 0,p≠ 1 and μ ∈ ℝ, μ ≠ 0 is smaller than the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper, we first give the complete picture of the radial solutions in balls. In particular, we establish forp> 1 the existence of a unique large solution behaving like δ−(2/(p−1))at the boundary. In general domains, we extend the results of Bandle and Pozio and show that there exists a unique singular solutionsusuch that$u/\delta ^{\beta _-}\to c$on the boundary for an arbitrary positive function$c \in C^{2+\gamma }(\partial \Omega ) \, (\gamma \in (0,1)), c \ges 0$. Here β−is the smaller root of β(β − 1) + μ = 0.

Uncountable sets of finite energy solutions for semilinear elliptic problems in exterior domains

In this article we discuss conditions suitable for the existence of asymptotically vanishing positive solutions for the following semilinear elliptic problem △ u ( x ) + f ( x , u ( x ) ) + g ( x ) x ⋅ ∇ u ( x ) = 0 , where x ∈ R n and ‖ x ‖ > R . The main result of our investigation is the construction of uncountable sets of minimal solutions which have finite energy in a neighbourhood of infinity. We apply an iteration scheme based on the subsolution and supersolution method. Our approach allows us to consider sublinear as well as superlinear problems without radial symmetry.

A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schrödinger equations.

Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains

We consider the semilinear elliptic equation $ -\Delta u =\lambda f(u) $ in a smooth bounded domain $ \Omega $ of $ \Bbb{R}^{n} $ with Dirichlet boundary condition, where $ f $ is a $ C^{1} $ positive and nondeccreasing function in $ [0, \infty) $ such that $ \frac{f(t)}{t} \rightarrow \infty $ as $ t \rightarrow \infty $. When $ \Omega $ is an arbitrary domain and $ f $ is not necessarily convex, the boundedness of the extremal solution $ u^{*} $ is known only for $ n = 2 $, established by X. Cabré[5]. In this paper, we prove this for higher dimensions depending on the nonlinearity $ f $. In particular, we prove that if \begin{document}$ \frac{1}{2} < \beta_{-}:=\liminf\limits_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}\leq \beta_{+}:=\limsup\limits_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}} < \infty, $ \end{document} where $ F(t)=\int_{0}^{t}f(s)ds $, then $ u^{*} \in L^{\infty}(\Omega) $, for $ n \leq 6 $. Also, if $\beta_{-}=\beta_{+}>\frac{1}{2} $ or $ \frac{1}{2} < \beta_{-}\leq \beta_{+} < \frac{7}{10} $, then $ u^{*} \in L^{\infty}(\Omega) $, for $ n \leq 9 $. Moreover, under the sole condition that $ \beta_{-} > \frac{1}{2} $ we have $ u^{*} \in H^{1}_{0}(\Omega) $ for $ n \geq 1 $. The same is true if for some $ \epsilon > 0 $ we have \begin{document}$$$ \frac{tf'(t)}{f(t)} \geq 1+\frac{1}{(\ln t)^{2-\epsilon}} ~~ \text{for large} ~ t, $$$ \end{document} which improves a similar result by Brezis and Vázquez [4].

Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Let BR be a ball of radius R in RN. We analyze the positive solutions to the problem {−Δu+u=|u|p−2u,in BR,∂νu=0,on ∂BR, that branch out from the constant solution u=1 as p grows from 2 to +∞. The nonzero constant positive solution is the unique positive solution for p close to 2. We show that there exist arbitrarily many positive solutions as p→∞ (in particular, for supercritical exponents) or as R→∞ for any fixed value of p>2, partially answering a conjecture in Bonheure et al. (2012). We give explicit lower bounds for p and R so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.

Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space

In this article, we will study the nondegeneracy properties of positive finite energy solutions of the equation -Δ𝔹u - λu = |u|p-1u in the hyperbolic space. We will show that the degeneracy occurs only in an N-dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls.

Infinitely many solutions for semilinear elliptic problems with sign-changing weight functions

In this paper, we study the elliptic problem where is the critical Sobolev exponent, and or is a sign-changing function. Under different assumptions on and we prove the existence of infinitely many solutions to the above problem. We also show that one of these solutions is positive.

Existence of Blow-up Solutions of Semilinear Elliptic Problems

We consider the boundary blow-up nonlinear elliptic problems \(\Delta u \pm \lambda |\nabla u|^\beta = k(x) f(u)\) in a smooth bounded domain \( D\subset \mathbb R ^N\), with \(u|_{\partial D}=\infty \), where \(\beta \in [0,2)\) and \(\lambda \ge 0.\) Under suitable assumptions on \(k\) and \(f\), we show the existence of at least one solution \(u\).

Multiple positive solutions of semilinear elliptic equations with weight function in exterior domains

In this paper, we are concerned with the multiplicity of solutions for semilinear elliptic problems with weight functions in exterior domains. We prove that, if the decay of the weight function at spatial infinity is sufficiently slow, then the equation admits at least three solutions and two of them escape away to the spatial infinity as the decay rate of the weight function tends to $0$. The result is proved via the variational method combined with the Ljusternik--Schnirelman-type multiplicity theorem.

A geometric inequality for stable solutions of semilinear elliptic problems in the Engel group

We prove that, if E is the Engel group and u is a stable solution of ∆Eu = f(u), then ˆ {∇Eu 6=0}  |∇Eu|2 {( p + 〈 (Hu) ν, v 〉 |∇Eu| )2 + h } − J   η ≤ ˆ E |∇Eη||∇Eu| for any test function η ∈ C∞ 0 (E). Here above, h is the horizontal mean curvature, p is the imaginary curvature and J := 2(X3X2uX1u−X3X1uX2u) + (X4u)(X1u−X2u) This can be interpreted as a geometric Poincare inequality, extending the work of [21, 22, 13] to stratified groups of step 3. As an application, we provide a non-existence result.

Existence and multiplicity of nonnegative solutions for semilinear elliptic problems involving nonlinearities indefinite in sign

Let r,s∈]1,2[ and λ,μ∈]0,+∞[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0 boundary data for the semilinear elliptic equation −Δu=λus−1−ur−1 in Ω⊂RN, where N≥2. We prove that there exists a positive constant Λ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λ, λ=Λ or λ<Λ. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given.

On strictly positive solutions for some semilinear elliptic problems

Let B be a ball in \({\mathbb{R}^{N}}\), N ≥ 1, let m be a possibly discontinuous and unbounded function that changes sign in B and let 0 < p < 1. We study existence and nonexistence of strictly positive solutions for semilinear elliptic problems of the form \({-\Delta u=m(x) u^{p}}\) in B, u = 0 on ∂B.

Multiple Solutions of Semilinear Elliptic Problems with Degenerate Boundary Conditions

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet and Robin problems. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the Leray–Schauder degree, we prove very exact results on the number of solutions of our problem. The results here extend earlier theorems due to Berger–Podolak, Castro–Lazer and Ambrosetti–Mancini to the degenerate case.

Three positive solutions for semilinear elliptic problems involving concave and convex nonlinearities

We study the existence and multiplicity of positive solutions for the Dirichlet problemwhere λ &gt; 0, 1 &lt; q &lt; 2, p = 2* = 2N/(N − 2), 0 ε Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂Ω and f is a non-negative continuous function on $\bar{\varOmega}$. Assuming that f satisfies some hypothesis, we prove that the equation admits at least three positive solutions for sufficiently small λ.

Second order expansion for blowup solutions of semilinear elliptic problems

In this paper, we mainly study the second order expansion of solutions to the elliptic problems △ u = b ( x ) f ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = ∞ , where Ω is a bounded domain with a smooth boundary in R N ( N ≥ 2 ) , b ∈ C α ( Ω ̄ ) which is positive in Ω and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p > 1 . Our analysis is based on the sub-supersolution method and Karamata regular variation theory.

Corrigendum to “Existence of solutions for semilinear elliptic problems without (𝑃𝑆) condition”

In this corrigendum we give a correct proof for equation (3.17), as well as the assertion on line 15, page 1364, of “Existence of solutions for semilinear elliptic problems without ( P S ) (PS) condition”, Proc. Amer. Math. Soc. 132(5) (2004), 1355-1366.