Sum of weighted Lebesgue spaces and nonlinear elliptic equations (original) (raw)
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Sum of wheighted Lebesgue spaces and nonlinear elliptic equations Marino Badiale — Lorenzo Pisani —
We study the sum of wheighted Lebesgue spaces, by considering an abstract measure space (Ω,A, μ) and investigating the main properties of both the Banach space L (Ω) = {u1 + u2 : u1 ∈ L1 (Ω) , u2 ∈ L2 (Ω)} , Li (Ω) := Li (Ω, dμ) , and the Nemytskĭı operator defined on it. Then we apply our general results to prove existence and multiplicity of solutions to a class of nonlinear p-laplacian equations of the form − pu+ V (|x|) |u|p−2 u = f (|x| , u) in R where V is a nonnegative measurable potential, possibly singular and vanishing at infinity, and f is a Carathéodory function satisfying a double-power growth condition in u.
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In this paper, we study the existence and uniqueness of weak solution to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic problems −div [ ω1A(x,∇u) + ν2B(x, u,∇u) ] + ν1C(x, u) + ω2|u|u = f − divF, where 1 < p < ∞, ω1, ν2, ν1 and ω2 are Ap-weight functions, and A : Ω × R −→ R, B : Ω×R×R −→ R, C : Ω×R −→ R are Caratéodory functions that satisfy some conditions and the right-hand side term f − divF belongs to Lp(Ω, ω ′ 2 ) + n ∏ j=1 L ′ (Ω, ω ′ 1 ). We will use the BrowderMinty Theorem and the weighted Sobolev spaces theory to prove the existence and uniqueness of weak solution in the weighted Sobolev space W 1,p 0 (Ω, ω1, ω2).
Existence of weak solutions for nonlinear elliptic systems involving the (p(x), q(x))-Laplacian
arXiv: Analysis of PDEs, 2009
In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -\\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{\\alpha(x)}|v|^{\\beta(x)} v + f(x) in \\Omega, \\Delta_{q(x)}v = c(x) |v|^{q(x)-2}v - d(x)|v|^{\\beta(x)}|u|^{\\alpha(x)} u + g(x) in \\Omega, u = v = 0 \\quad on \\partial\\Omega, where Omega\\OmegaOmega is an open bounded domains of mathbbRN\\mathbb{R}^NmathbbRN with a smooth boundary partialOmega\\partial\\OmegapartialOmega and Deltap(x)\\Delta_{p(x)}Deltap(x) denotes the p(x)p(x)p(x)-Laplacian.The existence of weak solutions is proved using the theory of monotone operators. Similar result will be proved when Omega=mathbbRN\\Omega=\\mathbb{R}^NOmega=mathbbRN.
Solutions in Lebesgue spaces to nonlinear elliptic equations with subnatural growth terms
St. Petersburg Mathematical Journal
We study the existence problem for positive solutions u ∈ L r (R n), 0 < r < ∞, to the quasilinear elliptic equation −∆ p u = σu q in R n in the sub-natural growth case 0 < q < p − 1, where ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian with 1 < p < ∞, and σ is a nonnegative measurable function (or measure) on R n. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators such as the A-Laplacian divA(x, ∇u), and the fractional Laplacian (−∆) α on R n , as well as linear uniformly elliptic operators with bounded measurable coefficients div(A∇u) on an arbitrary domain Ω ⊆ R n with a positive Green function.
arXiv (Cornell University), 2019
Given ≥ 3, 1 < p < N , two measurable functions V (r) ≥ 0, K (r) > 0, and a continuous function A(r) > 0 (r > 0), we study the quasilinear elliptic equation −div A(|x|)|∇u| p−2 ∇u u + V (|x|) |u| p−2 u = K(|x|)f (u) in R N. We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space X into the sum of Lebesgue spaces L q 1 K + L q 2 K , and thus into L q K (= L q K + L q K) as a particular case. Our results do not require any compatibility between how the potentials A, V and K behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of V and K, not of the potentials separately. The nonlinearity f has a double-power behavior, whose standard example is f (t) = min{t q 1 −1 , t q 2 −1 }, recovering the usual case of a single-power behavior when q1 = q2.
Compactness and existence results for the p -Laplace equation
Journal of Mathematical Analysis and Applications
Given 1 < p < N and two measurable functions V (r) ≥ 0 and K (r) > 0, r > 0, we define the weighted spaces W = u ∈ D 1,p (R N) : R N V (|x|) |u| p dx < ∞ , L q K = L q (R N , K (|x|) dx) and study the compact embeddings of the radial subspace of W into L q 1 K + L q 2 K , and thus into L q K (= L q K + L q K) as a particular case. We consider exponents q1, q2, q that can be greater or smaller than p. Our results do not require any compatibility between how the potentials V and K behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear p-Laplace equations of the form −△pu + V (|x|) |u| p−1 u = g (|x| , u) in R N , 1 < p < N, where V and g (|•| , u) with u fixed may be vanishing or unbounded at zero or at infinity. Both the cases of g super and sub p-linear in u are studied and, in the sub p-linear case, nonlinearities with g (|•| , 0) = 0 are also considered.