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The main result of this work is a new extension of the well-known inequality by Díaz and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian, 
$\Delta _{p(x)}u\equiv {\rm div}( \vert \nabla u \vert ^{p(x)-2}\nabla u)$. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.
We study radial symmetry of entire solutions of the equation
0.1It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called ‘maximal radial entire solution’, and infinitely many radial entire solutions with the latter behaviour, which are called ‘non-maximal radial entire solutions’. The necessary and sufficient conditions for an entire solution u of (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solution u of (0.1) to be a non-maximal radial entire solution.
$$\Delta ^2u = 8(N-2)(N-4)e^u\quad {\rm in}\;R^N\;\;(N \ges 5).$$
Let 1 < p < +∞ and let Ω ⊂ ℝN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type
We suppose that f(0) = f(1) = 0 and that f is negative between the two zeros and positive after. In case Ω is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focussing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behaviour (around 1) of non-constant radial solutions.
$-\Delta _pu = f(u),\quad u > 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega .$
$\mathbb{H}^{{\it n}}$
We investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball 
$B_1(0)\subset \mathbb{R}^n$ Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.
In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation
with radial structure, where n ⩾ 3, 1 < k < n/2 and p > 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research k-Hessian equations without radial structure.
$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$
We study the asymptotic behaviour, as p → ∞, of the least energy solutions of the problem
where xu is the (unique) maximum point of |u|, 
$$\left\{ {\matrix{ {-(\Delta _p + \Delta _{q(p)})u = \lambda _p \vert u(x_u) \vert ^{p-2}u(x_u)\delta _{x_u}} & {{\rm in}} & \Omega \cr {u = 0} \hfill \hfill \hfill & {{\rm on}} & {\partial \Omega ,} \cr } } \right.$$
$\delta _{x_{u}}$ is the Dirac delta distribution supported at xu,
and λp > 0 is such that
$$\mathop {\lim }\limits_{p\to \infty } \displaystyle{{q(p)} \over p} = Q\in \left\{ {\matrix{ {(0,1)} & {{\rm if}} & {N < q(p) < p} \cr {(1,\infty )} & {{\rm if}} & {N < p < q(p)} \cr } } \right.$$
$$\min \left\{ {\displaystyle{{{\rm \Vert }\nabla u{\rm \Vert }_\infty } \over {{\rm \Vert }u{\rm \Vert }_\infty }}:0 \ne u\in W^{1,\infty }(\Omega )\cap C_0(\bar{\Omega })} \right\} \les \mathop {\lim }\limits_{p\to \infty } (\lambda _p)^{1/p} < \infty .$$
We consider a class of Schrödinger operators on 
${\open R}^N$ with radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in 
$L^2({\open R}^N)$, we show that the following properties are generic with respect to the potential:
(P1) the eigenvalues below the essential spectrum are nonresonant (i.e., rationally independent) and so are the square roots of the moduli of these eigenvalues;
(P2) the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set.
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.
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 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.
We examine the elliptic system given by
where λ, γ are positive parameters, Ω is a smooth bounded domain in ℝN and f is a C2 positive, nondecreasing and convex function in [0, ∞) such that f(t)/t → ∞ as t → ∞. Assuming
$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill & {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$ we show that the extremal solution (u*, v*) associated with the above system is smooth provided that N < (2α*(2 − τ+) + 2τ+)/(τ+)max{1, τ+}, where α* > 1 denotes the largest root of the second-order polynomial
$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$ As a consequence, u*, v* ∈ L∞(Ω) for N < 5. Moreover, if τ− = τ+, then u*, v* ∈ L∞(Ω) for N < 10.
$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$
