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It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation 
$\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $ in 
$\Omega _T\equiv \Omega \times (0,T)$ satisfies the uniform estimate
provided that 
$$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$
$q \gt 1+{N}/{2}$, where Ω is a bounded domain in 
${\open R}^N$ with Lipschitz boundary, T > 0, 
$\partial _p\Omega _T$ is the parabolic boundary of 
$\Omega _T$, 
$\omega \in L^1(\Omega _T)$ with 
$\omega \ges 0$, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if 
$q=1+{N}/{2}$. In this paper, we show that the linear growth of the upper bound in 
$\Vert f \Vert_{q,\Omega _T}$ can be improved. To be precise, we establish
$$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$
In this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation 
$-\Delta _p u = f(u)$ in bounded Steiner symmetric domains 
$ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.
The classical Alexandrov–Bakelman–Pucci estimate for the Laplacian states where 
$$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{s,n}\text{diam}(\unicode[STIX]{x03A9})^{2-\frac{n}{s}}\Vert \unicode[STIX]{x0394}u\Vert _{L^{s}(\unicode[STIX]{x03A9})},\end{eqnarray}$$
$\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$, 
$u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$ and 
$s>n/2$. The inequality fails for 
$s=n/2$. A Sobolev embedding result of Milman and Pustylnik, originally phrased in a slightly different context, implies an endpoint inequality: if 
$n\geqslant 3$ and 
$\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$ is bounded, then where 
$$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{n}\Vert \unicode[STIX]{x0394}u\Vert _{L^{\frac{n}{2},1}(\unicode[STIX]{x03A9})},\end{eqnarray}$$
$L^{p,q}$ is the Lorentz space refinement of 
$L^{p}$. This inequality fails for 
$n=2$, and we prove a sharp substitute result: there exists 
$c>0$ such that for all 
$\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$ with finite measure, This is somewhat dual to the classical Trudinger–Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces; the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.
$$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c\max _{x\in \unicode[STIX]{x03A9}}\int _{y\in \unicode[STIX]{x03A9}}\max \left\{1,\log \left(\frac{|\unicode[STIX]{x03A9}|}{\Vert x-y\Vert ^{2}}\right)\right\}|\unicode[STIX]{x0394}u(y)|dy.\end{eqnarray}$$
This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in (
$x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.
We use bifurcation and topological methods to investigate the existence/nonexistence and the multiplicity of positive solutions of the following quasilinear Schrödinger equation
involving sublinear/linear/superlinear nonlinearities at zero or infinity with/without signum condition. In particular, we study the changes in the structure of positive solution with κ as the varying parameter.
$$\left\{ {\matrix{ {-\Delta u-\kappa \Delta \left( {u^2} \right)u = \beta u-\lambda \Phi \left( {u^2} \right)u{\mkern 1mu} {\mkern 1mu} } \hfill & {{\rm in}\;\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm on}\;\partial \Omega } \hfill \cr } } \right.$$
In this paper, we analyse the semilinear fourth-order problem ( − Δ)2u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and
for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu > 0 in ℝN, and they do not exist when the previous condition fails.
$$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$
In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity
where 0 < γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.
$$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$
We consider an elliptic problem of the type
where Ω is a bounded Lipschitz domain in ℝN with a cylindrical symmetry, ν stands for the outer normal and 
$$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$
$\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $.
Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.
As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem
For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L2 (Ω) × L2(Γ2).
$$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$
Let n ⩾ 3, 0 ⩽ m < n − 2/n, ρ1 > 0, 
$\beta>\beta_{0}^{(m)}=(({m\rho_{1}})/({n-2-nm}))$, αm = ((2β + ρ1)/(1 − m)) and α = 2β+ρ1. For any λ > 0, we prove the uniqueness of radially symmetric solution υ(m) of Δ(υm/m) + αmυ + βx · ∇υ = 0, υ > 0, in ℝn∖{0} which satisfies 
$\lim\nolimits_{|x|\to 0|}|x|^{\alpha _m/\beta }v^{(m)}(x) = \lambda ^{-((\rho _1)/((1-m)\beta ))}$ and obtain higher order estimates of υ(m) near the blow-up point x = 0. We prove that as m → 0+, υ(m) converges uniformly in C2(K) for any compact subset K of ℝn∖{0} to the solution υ of Δlog υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn\{0}, which satisfies 
$\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$. We also prove that if the solution u(m) of ut = Δ (um/m), u > 0, in (ℝn∖{0}) × (0, T) which blows up near {0} × (0, T) at the rate 
$ \vert x \vert ^{-{\alpha_{m}}/{\beta}}$ satisfies some mild growth condition on (ℝn∖{0}) × (0, T), then as m → 0+, u(m) converges uniformly in C2 + θ, 1 + θ/2(K) for some constant θ ∈ (0, 1) and any compact subset K of (ℝn∖{0}) × (0, T) to the solution of ut = Δlog u, u > 0, in (ℝn∖{0}) × (0, T). As a consequence of the proof, we obtain existence of a unique radially symmetric solution υ(0) of Δ log υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn∖{0}, which satisfies 
$\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$.
In the present paper, we consider the nonlocal Kirchhoff problem
where a, b>0, 1<p<5 are constants, ϵ>0 is a parameter. Under some mild assumptions on the function V, we obtain multi-peak solutions for ϵ sufficiently small by Lyapunov–Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term 
$$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$
$\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of the elliptic singular perturbation problems.
