The aim of this paper is to study the critical elliptic equations with Stein–Weiss type convolution parts $$\begin{aligned} \displaystyle -\Delta u =\frac{1}{|x|^{\alpha }}\left( \int _{\mathbb {R}^{N}}\frac{|u(y)|^{2_{\alpha , \mu }^{*}}}{|x-y|^{\mu }|y|^{\alpha }}dy\right) |u|^{2_{\alpha }^{*}-2}u,\quad x\in \mathbb {R}^{N}, \end{aligned}$$ where exponent due weighted Hardy–Littlewood–Sobolev...

This article concerns with the problem ?div(jruj m?2 ru) = hu q + u m ?1 ; in R N : Using the Ekeland Variational Principle and the Mountain Pass Theorem, we show the existence of > 0 such that there are at least two non-negative solutions for each in (0;).

In the present paper we study the weak lower semicontinuity of the functional

We consider the following nonlinear singular elliptic equation −div (|x| −2a ∇u) = K(x)|x| −bp |u| p−2 u + λg(x) in R N , where g belongs to an appropriate weighted Sobolev space, and p denotes the Caffarelli–Kohn– Nirenberg critical exponent associated to a, b, and N. Under some natural assumptions on the positive potential K(x) we establish the existence of some λ 0 > 0 such that the above pr...

Given (M, g) a smooth, compact Riemannian n-manifold, we consider equations like ∆gu + hu = u −1−ε, where h is a C-function on M , the exponent 2∗ = 2n/ (n− 2) is critical from the Sobolev viewpoint, and ε is a small real parameter such that ε→ 0. We prove the existence of blowing-up families of positive solutions in the subcritical and supercritical case when the graph of h is distinct at some...

where Ω ⊂ R is a smooth domain with smooth boundary ∂Ω such that 0 Î Ω, Δpu = div(|∇u|∇u), 1 < p < N, μ < μ̄ = ( N−p p ), l >0, 1 < q < p, sign-changing weight functions f and g are continuous functions on ̄, μ̄ = ( N−p p ) p is the best Hardy constant and p∗ = Np N−p is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the multiplicity of positive solu...

In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent −∆u = λu− αu + u −1, u > 0, in Ω, u = 0, on ∂Ω. where Ω ⊂ Rn, n ≥ 3 is a bounded C2-domain λ > λ1, 1 < p < 2∗ − 1 = n+2 n−2 and α > 0 is a bifurcation parameter. Brezis and Nirenberg [2] showed that a lower order (non-negative) perturbation can contribute t...

v > 0 in R \ {a1, . . . , ak}, where N ≥ 3, k ∈ N, hi ∈ C(S), (a1, a2, . . . , ak) ∈ R , ai 6= aj for i 6= j, and 2 = 2N N−2 is the critical Sobolev exponent. The interest in such a class of equations arises in nonrelativistic molecular physics. Inverse square potentials with anisotropic coupling terms turn out to describe the interaction between electric charges and dipole moments of molecules...

We establish the existence of an entire solution for a class of stationary Schrödinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows-up at infinity. The abstract framework is related to Lebesgue–Sobolev spaces with variable exponent. The proof is based on the critical point theory in the sense of Clarke and we apply Chang’s version of the Mountain ...

We study the asymptotic behavior of solutions to the nonlocal nonlinear equation (−∆p) u = |u|u in a bounded domain Ω ⊂ R as q approaches the critical Sobolev exponent p∗ = Np/(N − ps). We prove that ground state solutions concentrate at a single point x̄ ∈ Ω and analyze the asymptotic behavior for sequences of solutions at higher energy levels. In the semi-linear case p = 2, we prove that for s...