Elliptic Problems In RN Research Articles

Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

Blow-up solutions for a general class of the second-order differential equations on the half line

In this paper, we study the existence of positive blow-up solutions for a general class of the second-order differential equations and systems, which are positive radially symmetric solutions to many elliptic problems in RN. We explore fixed point arguments applied to suitable integral equations to get solutions.

Doubly Critical Problems Involving Fractional Laplacians in ℝ N

Abstract In this paper, we show the existence of nontrivial solutions for doubly critical nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians. Our approach is elementary without the use of neither the concentration-compactness principle nor the extension argument developed in [4].

Formula omitted] versus [formula omitted] local minimizer and a Hopf-type maximum principle

We consider functionals of the form Φ(u)=12∫RN|∇u|2−∫RNb(x)G(u) on D1,2(RN), N≥3, whose critical points are the weak solutions of a corresponding elliptic equation in the whole RN. We present a Brezis–Nirenberg type result and a Hopf-type maximum principle in the context of the space D1,2(RN). More precisely, we prove that a local minimizer of Φ in the topology of the subspace V must be a local minimizer of Φ in the D1,2(RN)-topology, where V is given by V:={v∈D1,2(RN):v∈C(RN)withsupx∈RN⁡(1+|x|N−2)|v(x)|<∞}. It is well-known that the Brezis–Nirenberg result has been proved a strong tool in the study of multiple solutions for elliptic boundary value problems in bounded domains. We believe that the result obtained in this paper may play a similar role for elliptic problems in RN.

Infinitely many weak solutions for some elliptic problems in RN

We investigate the existence of infinitely many weak solutions to some elliptic problems involving the p-Laplacian in RN by using variational method and critical point theory.

Multiple positive solutions for a class of concave–convex elliptic problems in ℝNinvolving sign-changing weight, II

In this paper, we study the following concave–convex elliptic problems: [Formula: see text] where N ≥ 3, 1 &lt; q &lt; 2 &lt; p &lt; 2* = 2N/(N - 2), λ &gt; 0 and μ &lt; 0 are two parameters. By using several variational methods and a perturbation argument, we obtain three positive solutions to this problem under the predefined conditions of fλ(x) and gμ(x), which simultaneously extends the result of [T. Hsu, Multiple positive solutions for a class of concave–convex semilinear elliptic equations in unbounded domains with sign-changing weights, Bound. Value Probl. 2010 (2010), Article ID 856932, 18pp.; T. Wu, Multiple positive solutions for a class of concave–convex elliptic problems in ℝNinvolving sign-changing weight, J. Funct. Anal. 258 (2010) 99–131]. We also study the concentration behavior of these three solutions both for λ → 0 and μ → -∞.

The existence of multiple positive solutions for a class of non-local elliptic problem in RN

This paper is concerned with the existence of at least two positive solutions for a class of non-local elliptic equations in . Using variational methods and the decomposition of the Nehari manifold, we show multiple existence of positive solutions.

Existence and concentration of solutions with compact support for elliptic problems with critical growth in [formula omitted

For a continuous and sign-changing function a(x), we study the existence and asymptotic behavior of compactly supported solutions of the following elliptic problem in RN with N⩾3:−Δu=aγ(x)uq+up,u⩾0 and u∈D1,2(RN), where γ>0, 0<q<1, p=N+2N−2 and aγ=γa or aγ=γa+−a−. When γ is small, both cases admit two solutions vγ and Vγ, where vγ is a local minimizer and Vγ is a minimizing sequence for the Sobolev embedding from D1,2(RN) to Lp+1(RN) as γ→0. In the case aγ=γa+−a−, Vγ blows up in the L∞-norm and concentrates at one point as γ→0, consequently we show that Vγ cannot be radially symmetric when a(0)<0 and γ is small enough.

Multiple solutions with compact support for a semilinear elliptic problem with critical growth

For a sign-changing function a(x), we consider solutions of the following semilinear elliptic problem in RN with N⩾3:−Δu=(γa+−a−)uq+up,u⩾0 and u∈D(RN), where γ>0, 0<q<1 and p=N+2N−2. We show that all solutions are compactly supported if liminf|x|→∞a−(x)>0. When Ω+={x∈RN|a(x)>0} has several connected components, we prove that there exists an interval on γ, in which two solutions exist and are positive in Ω+, moreover one solution blows up as γ→0. A uniqueness result for solution with small L∞-norm is also given.

Compactly supported solutions for a semilinear elliptic problem in ℝn with sign-changing function and non-Lipschitz nonlinearity

For a sign-changing function a(x) we consider the solutions of the following semilinear elliptic problem in ℝn with n ≥ 3:where γ &gt; 0 and 0 &lt; q &lt; 1 &lt; p &lt; (n + 2)/(n − 2). Under an appropriate growth assumption on a− at infinity, we show that all solutions are compactly supported. When Ω+ = {x ∈ ℝn | a(x) &gt; 0} has several connected components, we prove that there exists an interval on γ in which the solutions exist. In particular, if a(x) = a(|x|), by applying the mountain-pass theorem there are at least two solutions with radial symmetry that are positive in Ω+.

On the existence of signed solutions for a quasilinear elliptic problem in RN

We study the existence of positive and negative weak solutions for the equation −¢pu + V (x)|u| p−2 u = ¸f(u) in R N , where −¢pu = div(|∇u| p−2 ∇u) is the p-Laplacian operator, 1 < p < N, ¸ is a positive real parameter and the potential V : R N → R is bounded from below for a positive constant and ”large” at infinity. It is assumed that the nonlinearity f : R → R is continuous and just superlinear in a neighborhood of the origin.

Bifurcation results for semilinear elliptic problems in RN

In this paper we obtain, for a semilinear elliptic problem in RN, families of solutions bifurcating from the bottom of the spectrum of −Δ. The problem is variational in nature and we apply a nonlinear reduction method that allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

Nonlinear Perturbations of a Periodic Elliptic Problem with Critical Growth

Using the well-known Mountain Pass Theorem due to Ambrosetti and Rabinowitz, we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in RN with critical growth.

ON MULTIPLE POSITIVE SOLUTIONS FOR A NONLINEAR ELLIPTIC PROBLEM IN RN

By making use of variational method, the authors obtain some results about existence of multiple positive solutions and their asymptotic behavior as the parameter λ → + ∞ for a semilinear elliptic problem in RN.

On the Liouville property for fully nonlinear equations

In this paper we prove some Liouville theorems for nonnegative viscosity supersolutions of a class of fully nonlinear uniformly elliptic problems in ℝN.

On a class of semilinear elliptic problems in ℝN with critical growth

On a class of semilinear elliptic problems in ℝN with critical growth

Multiplicity of positive and nodal solutions for nonlinear elliptic problems in ℝN

We are concerned with the multiplicity of positive and nodal solutions of{−Δu+μu=Q(x)|u|p−2uinℝNu∈H1(ℝN) where 2<p<2NN−2, N ≥ 3, μ > 0, Q ∈ C(ℝN) and Q(x) ≥ 0 for x ∈ ℝN. We show how the “shape” of the graph of Q(x) affects the number of both positive and nodal solutions.

Critical semilinear biharmonic equations in RN

SynopsisAn existence theorem is obtained for a fourth-order semilinear elliptic problem in RNinvolving the critical Sobolev exponent (N + 4)/(N − 4), N&gt;4. A preliminary result is that the best constant in the Sobolev embeddingL2N/(N–4)(RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be