Decreasing Regularly Varying Solutions of Sublinearly Perturbed Superlinear Thomas–Fermi Equation

We consider the perturbed Thomas–Fermi equation $$\begin{array}{ll} x^{\prime \prime}\, =\, p(t)|x|^{\gamma-1}x\, +\, q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (A) \end{array}$$ where δ and γ are positive constants with \({\delta < 1 < \gamma}\) and p(t) and q(t) are positive continuous functions on \({[a,\infty), a > 0}\) . Our aim here is to establish criteria for the existence of positive solutions of (A) decreasing to zero as \({t \to \infty}\) in the case where p(t) and q(t) are regularly varying functions (in the sense of Karamata). Generalization of the obtained results to equations of the form $$\begin{array}{ll} \left(r(t)x^{\prime}\right)^{\prime}\, =\, p(t)|x|^{\gamma-1}x \,+ \,q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (B) \end{array}$$ is also given.     

A convexity property and a new characterization of Euler’s gamma function

Our main results are:

1. Let α ≠ 0 be a real number. The function (Γ ? exp) ^α is convex on ${\mathbf{R}}$ if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x ₀ = 1.4616... denotes the only positive zero of ${\psi = \Gamma'/\Gamma}$ .

1. Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$

If there are a number b and a sequence of positive real numbers (a _n ) ${(n \in \mathbf{N})}$ with ${{\rm lim}_{n\to\infty} a_n =0}$ such that for every n the function ${(f \circ {\rm exp})^{a_n}}$ is Jensen convex on (b, ∞), then f is the gamma function.     

Strong Regularizing Effect of a Gradient Term in the Heat Equation with a Weight

We deal with the following parabolic problem, $$(P)\left\{\begin{array}{lll} u_t - \Delta{u} + |\nabla{u}|^q \quad=\quad \lambda{g}(x)u + f(x, t),\quad u > 0 \; {\rm in} \; \Omega \; \times \; (0, T),\\ \qquad\quad\quad\; u(x, t) \quad=\quad 0 \quad{\rm on}\; {\partial}{\Omega}\; \times ; (0, T),\\ \qquad\quad\quad\; u(x, 0) \quad=\quad u_{0}(x), \quad x \in {\Omega},\end{array}\right.$$ where is a bounded regular domain or ${\Omega = \mathbb{R}^N}$ , ${1 < q \leq 2, \lambda > 0\; {\rm and}\; f \geq 0, u_{0} \geq 0}$ are in a suitable class of functions. We give assumptions on g with respect to q for which for all λ >  0 and all ${f \in L^1(\Omega_T ), f \geq 0}$ , problem (P) has a positive solution. Under some additional conditions on the data, the Cauchy problem and the asymptotic behavior of the solution are also considered.     

On the solutions of a singular elliptic equation concentrating on two orthogonal spheres

Let \({A=\{x\in \mathbb{R}^{2m}: 0 < a < |x| < b\}}\) be an annulus. We consider the following singularly perturbed elliptic problem on A $$\left\{\begin{array}{lll}-\varepsilon ^2{\Delta u} + |x|^{\eta}u =|x|^{\eta}u^p, \quad {\rm in} A,\\ u > 0, \quad \quad \quad \quad \quad \quad \quad {\rm in} A, \\ u=0, \quad \quad \quad \quad \quad \quad \quad {\rm on}\partial A,\end{array}\right. $$ where \({1 < p < \frac{m+3}{m-1}}\) . We shall prove the existence of a positive solution \({u_\epsilon }\) which concentrates on two different orthogonal spheres of dimension (m?1) as \({\varepsilon \to 0}\) . We achieve this by studying a reduced problem on an annular domain in \({\mathbb{R}^{m+1}}\) and analysing the profile of a two point concentrating solution in this domain.     

H?lder estimates for parabolic p(x, t)-Laplacian systems

In this paper we establish the local H?lder continuity of the spatial gradient of weak solutions to the parabolic p(x, t)-Laplacian system $$\begin{array}{lll}\partial_{t}u - {\rm div} \left( a(x, t)|Du|^{p(x, t)-2}Du \right) = 0.\end{array}$$ More precisely, we prove that $$\begin{array}{lll}Du \in C_{\rm loc}^{0;\alpha,\alpha/2} \quad {\rm for\; some} \; \alpha \in (0, 1],\end{array}$$ provided p(·) and a(·) are H?lder-continuous.     

Uniform Estimates of Solutions of a Nonlinear Third-Order Differential Equation

One considers the differential equation $$y{\prime\prime\prime}(x) + p\bigl(x, y(x), y{\prime}(x), y{\prime\prime}(x)\bigr) |y(x)|^{k-1} y(x) = 0,$$ where k?>?1, the function p(x, y ₀ , y ₁ , y ₂) is continuous and satisfies the inequalities $$ 0 < p_* \le p(x, y_0, y_1, y_2) \le p^* < \infty,$$ as well as the Lipschitz condition with respect to the last three arguments. Uniform estimates are obtained for the moduli of the solutions with a common domain.     

Limits as p → ∞ of p-laplacian eigenvalue problems perturbed with a concave or convex term

We investigate the asymptotic behaviour as p → ∞ of sequences of positive weak solutions of the equation $$\left\{\begin{array}{l}-\Delta_p u = \lambda\,u^{p-1}+ u^{q(p)-1}\quad {\rm in}\quad \Omega,\\ u = 0 \quad {\rm on}\quad \partial\Omega,\end{array} \right.$$ where λ > 0 and either 1 < q(p) < p or p < q(p), with ${{\lim_{p\to\infty}{q(p)}/{p}=Q\neq1}}$ . Uniform limits are characterized as positive viscosity solutions of the problem $$\left\{\begin{array}{l}\min\left\{|\nabla u (x)| - \max\{\Lambda\,u (x),u ^Q(x)\}, -\Delta_{\infty}u (x)\right\} = 0 \quad {\rm in} \quad \Omega,\\ u = 0\quad {\rm on}\quad \partial\Omega.\end{array}\right.$$ for appropriate values of Λ > 0. Due to the decoupling of the nonlinearity under the limit process, the limit problem exhibits an intermediate behavior between an eigenvalue problem and a problem with a power-like right-hand side. Existence and non-existence results for both the original and the limit problems are obtained.     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Symmetry and regularity of extremals of an integral equation related to the Hardy�CSobolev inequality

Let ?? be a real number satisfying 0?<????<?n, ${0\leq t<\alpha, \alpha{^\ast}(t)=\frac{2(n-t)}{n-\alpha}}$ . We consider the integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{u^{{\alpha{^\ast}(t)}-1}(y)}{|y|^t|x-y|^{n-\alpha}}\,dy,\quad\quad\quad\quad\quad\quad\quad(1)$$ which is closely related to the Hardy?CSobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{|u(y)|^{p}u(y)}{|y|^t|x-y|^{n-\alpha}}\, dy\quad\quad\quad\quad\quad\quad\quad(2)$$ that corresponds to a large class of PDEs by regularity lifting method.     

Topological degree for solutions of fourth order mean field equations

We consider the following fourth order mean field equation with Navier boundary condition $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\,\,{\rm in}\, \Omega,{\quad}u = \Delta u = 0\,\,{\rm on}\,\partial \Omega,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ where h is a C ^2,?? positive function, ?? is a bounded and smooth domain in ${\mathbb{R}^4}$ . We prove that for ${\rho \in (32m\sigma_3, 32(m + 1)\sigma_3)}$ the degree-counting formula for (*) is given by $$d(\rho)=\left\{\begin{array}{ll}\frac{1}{m!} (-\chi (\Omega) +1) \cdot\cdot \cdot (-\chi(\Omega)+m) & {\rm for}\, m >0 ,\\ 1 & {\rm for}\, m=0\end{array}\right.$$ where ??(??) is the Euler characteristic of ??. Similar result is also proved for the corresponding Dirichlet problem $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$      

Principal Eigenvalue of p-Laplacian Operator in Exterior Domain

We consider an eigenvalue problem of the form $$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$ where \({\Omega \subset \mathrm{I\!R\!}^N}\) is a simply connected bounded domain, containing the origin, with C ² boundary \({\partial \Omega}\) and \({\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}\) is the exterior domain, \({1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}\) is the p-Laplacian operator and \({K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}\) is a positive function. Existence and properties of principal eigenvalue λ ₁ and its corresponding eigenfunction are established which are generally known in bounded domain or in \({\mathrm{I\!R\!}^N}\) . We also establish the decay rate of positive eigenfunction as \({|x| \to \infty}\) as well as near ?Ω.     

Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups

Let G be a homogeneous group, and let X ₁, X ₂, … , X _m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term: $$\left\{\begin{array}{ll}-\sum_{j=1}^{m}X_j^2u(x)-\frac{a}{\|x\|^\nu}u(x)=u^{\frac{Q+2}{Q-2}}(x), x\in\Omega,\\ u(x)=0, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\, x\in \partial\Omega,\end{array}\right.$$ where ${\Omega\subset G}$ is a bounded domain with smooth boundary and ${\mathbf{0}\in\Omega}$ , Q is the homogeneous dimension of G, ${a\in \mathbb{R},\ \nu <2 }$ . We boost u to ${L^p(\Omega)}$ for any ${1\leq p < \infty}$ if ${u\in S^{1,2}_0(\Omega)}$ is a weak solution of the problem above.     

Nonoscillation theory for second order half-linear differential equations in the framework of regular variation

Criteria are established for nonoscillation of all solutions of the second order half-linear differential equation A $(\mid y^\prime \mid^{\alpha-1}y^\prime)^\prime + q(t)\mid y \mid^{\alpha -1}y = 0,\ \ \ t \geq 0,$ where α > 0 is a constant and q: [0, ∞) → ? is continuous. The criteria are designed to exhibit the role played by the integral of q(t) in guaranteeing the existence of nonoscillatory solutions of (A) in specific classes of regularly varying functions in the sense of Karamata.     

A perturbation result for semi-linear stochastic differential equations in UMD Banach spaces

We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: $$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$ Here, A is the generator of an analytic C ₀-semigroup on a UMD Banach space X, H is a Hilbert space, W _H is an H-cylindrical Brownian motion, ${G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}$ , and ${F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}$ for some ${\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}$ , where ${\tau\in [1, 2]}$ denotes the type of the Banach space and ${X_{\theta_F}^{A}}$ denotes the fractional domain space or extrapolation space corresponding to A. We assume F and G to satisfy certain global Lipschitz and linear growth conditions. Let A ₀ denote the perturbed operator and U ₀ the solution to (SDE) with A substituted by A ₀. We provide estimates for ${\|U - U_0\|_{L^p(\Omega;C([0,T];X))}}$ in terms of ${D_{\delta}(A, A_0) := \|R(\lambda : A) - R(\lambda : A_0)\|_{\mathcal{L}(X^{A}_{\delta-1},X)}}$ . Here, ${\delta\in [0, 1]}$ is assumed to satisfy ${0\leq \delta < {\rm min}\{\frac{3}{2} - \frac{1}{\tau} + \theta_F,\, \frac{1}{2} - \frac{1}{p} + \theta_G \}}$ . The work is inspired by the desire to prove convergence of space approximations of (SDE). In this article, we prove convergence rates for the case that A is approximated by its Yosida approximation.     

{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms

In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, \Omega,\\ u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where ${0 < \alpha \leq a(x) \leq \beta, |b(x)| \leq \gamma, \gamma > 0, f \in L^2 (\Omega)}$ and Ω is a bounded subset of ${\mathbb{R}^N}$ with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ${W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}$ if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0.     

Global solutions of the heat equation with a nonlinear boundary condition

We consider the heat equation with a nonlinear boundary condition, $$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$ where ${\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}$ , p > 1 + 1/N, and (N ? 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P).     

Optimality estimations for approximately midconvex functions

Let V be a convex subset of a normed space and let a nondecreasing function α : [0, ∞) → [0, ∞) be given. A function ${f : V \rightarrow \mathbb{R}}$ is called α-midconvex if $$f\left(\frac{x+y}{2} \right)\leq \frac{f(x)+f(y)}{2}+\alpha(\|x-y\|) \quad \,{\rm for}\, x,y\in V.$$ It is known (Tabor in Control Cybern., 38/3:656–669, 2009) that if ${f : V \rightarrow \mathbb{R}}$ is α-midconvex, locally bounded above at every point of V then $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+P_\alpha(\|x-y\|) \quad \,{\rm for}\, x, y \in V,t \in [0,1],$$ where ${P_\alpha(r):=\sum_{k=0}^\infty \frac{1}{2^k} \alpha(2{\rm dist}(2^kr, \mathbb{Z}))}$ for ${r \in \mathbb{R}}$ . We show that under some additional assumptions the above estimation cannot be improved.     

Quasilinear elliptic equations in \mathbb{R }^{N} via variational methods and Orlicz-Sobolev embeddings

In this paper we prove the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\nabla \cdot \left[\phi ^{\prime }(|\nabla u|^2)\nabla u \right] +|u|^{\alpha -2}u =|u|^{s-2} u,&x\in \mathbb{R }^{N},\\ u(x) \rightarrow 0, \quad \text{ as} |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where $N\ge 2, \phi (t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t, 1< p<q<N, 1<\alpha \le p^* q^{\prime }/p^{\prime }$ and $\max \{q,\alpha \}< s<p^*,$ being $p^*=\frac{pN}{N-p}$ and $p^{\prime }$ and $q^{\prime }$ the conjugate exponents, respectively, of $p$ and $q$ . Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.     

A local mountain pass type result for a system of nonlinear Schr?dinger equations

We consider a singular perturbation problem for a system of nonlinear Schr?dinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$ where N?=?2, 3, ?? ₁, ?? ₂, ?? > 0 and V ₁(x), V ₂(x): R ^N ?? (0, ??) are positive continuous functions. We consider the case where the interaction ?? > 0 is relatively small and we define for ${P\in{\bf R}^N}$ the least energy level m(P) for non-trivial vector solutions of the rescaled ??limit?? problem: $$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$ We assume that there exists an open bounded set ${\Lambda\subset{\bf R}^N}$ satisfying $$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$ We show that (*) possesses a family of non-trivial vector positive solutions ${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$ which concentrates??after extracting a subsequence ?? _n ?? 0??to a point ${P_0\in\Lambda}$ with ${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$ . Moreover (v _1?? (x), v _2?? (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.