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.     

Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus

In this paper we consider the problem $$\left\{ \begin{array}{ll} -\Delta u=u^p+\lambda u & \quad\hbox{ in }A,\\ u > 0&\quad \hbox{ in }A,\\ u=0 &\quad \hbox{ on }\partial A, \end{array}\right. $$ where A is an annulus of ${\mathbb{R}^N,N\ge2}$ and p?>?1. We prove bifurcation of nonradial solutions from the radial solution in correspondence of a sequence of exponents {p _k } and for expanding annuli.     

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      4. Global solutions of the heat equation with a nonlinear boundary condition      Kazuhiro Ishige  Tatsuki Kawakami 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):429-457 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).      5. Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate      Chan-Gyun Kim  Junping Shi 《Results in Mathematics》2013,64(1-2):165-173 In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition $$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$ where Ω is a bounded smooth domain in ${\mathbb{R}^N}$ for ${N \ge 1, p > 1}$ , and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.      6. Topological degree for solutions of fourth order mean field equations      Changshou Lin  Juncheng Wei  Liping Wang 《Mathematische Zeitschrift》2011,268(3-4):675-705 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 $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$       7. Limits as p → ∞ of p-laplacian eigenvalue problems perturbed with a concave or convex term      Fernando Charro  Enea Parini 《Calculus of Variations and Partial Differential Equations》2013,46(1-2):403-425 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.      8. General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions      Shun-Tang Wu 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,63(1):65-106 In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain ??. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.      9. Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in {\mathbb{R}^{3}}      Jun Wang  Lixin Tian  Junxiang Xu  Fubao Zhang 《Calculus of Variations and Partial Differential Equations》2013,48(1-2):243-273 In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{array}\right.$$ where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a(x) has at least one minimum and b(x) has at least one maximum. We first prove the existence of least energy solution (u ε , φ ε ) for λ ≠ 0 and ε > 0 sufficiently small. Then we show that u ε converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.      10. Positive solutions of a Lidstone boundary value problem with variable coefficient function      Yun-Rui Yang  Sui Sun Cheng 《Journal of Applied Mathematics and Computing》2008,27(1-2):411-419 We establish the existence of positive solutions of the Lidstone boundary value problem $$\begin{array}{rcl}(-1)^{n}u^{(2n)}&=&\lambda a(t)f(u),\quad 0<t<1,\\[3pt]u^{(2i)}(0)&=&u^{(2i)}(1)=0,\quad 0\leq i\leq n-1\end{array}$$ for all sufficiently small positive real λ, where the function a may change sign in [0,1] and the function f:[0,∞)→R satisfies f(0)>0. We also show that our assumption is not vacuous.      11. Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities      K. Sreenadh  Sweta Tiwari 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(6):1831-1850 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.      12. Parabolic power concavity and parabolic boundary value problems      Kazuhiro Ishige  Paolo Salani 《Mathematische Annalen》2014,358(3-4):1091-1117 This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$ where $\varOmega $ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$ . We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$ .      13. Discontinuous superlinear elliptic equations of divergence form      Dian K. Palagachev 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(6):811-822 We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem $\left\{ {\begin{array}{lc} {{\rm div} \left( a^{ij} (x,u)D_{j} u \right) = b(x,u,Du) \quad {\rm in}\, \Omega \subset {\mathbb R}^{n}, \, n \ge 2,} \\ {u = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,{\rm on}\, \partial\Omega \in C^{1}. } \\ \end{array}} \right.$ The coefficients a ij (x, u) are supposed to be VMO functions with respect to x while the term b(x, u, Du) allows controlled growth with respect to the gradient Du and satisfies a sort of sign-condition with respect to u. Our results correct and generalize the announcements in Ragusa (Nonlinear Differ Equ Appl 13:605–617, 2007, Erratum in Nonlinear Differ Equ Appl 15:277–277, 2008).      14. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,63(2):395-400 Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d 1, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].      15. Local estimates for parabolic equations with nonlinear gradient terms      Tommaso Leonori  Francesco Petitta 《Calculus of Variations and Partial Differential Equations》2011,42(1-2):153-187 In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .      16. On the Fučik spectrum of non-local elliptic operators      Sarika Goyal  K. Sreenadh 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(4):567-588 In this article, we study the Fu?ik spectrum of the fractional Laplace operator which is defined as the set of all \({(\alpha, \beta)\in \mathbb{R}^2}\) such that $$\quad \left.\begin{array}{ll}\quad (-\Delta)^s u = \alpha u^{+} - \beta u^{-} \quad {\rm in}\;\Omega \\ \quad \quad \quad u = 0 \quad \quad \quad \qquad {\rm in}\; \mathbb{R}^n{\setminus}\Omega.\end{array}\right\}$$ has a non-trivial solution u, where \({\Omega}\) is a bounded domain in \({\mathbb{R}^n}\) with Lipschitz boundary, n > 2s, \({s \in (0, 1)}\) . The existence of a first nontrivial curve \({\mathcal{C}}\) of this spectrum, some properties of this curve \({\mathcal{C}}\) , e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fu?ik spectrum.      17. Bifurcation of positive solutions of a nonlinear discrete fourth-order boundary value problem      Ruyun Ma  Chenghua Gao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2013,64(3):493-506       18. Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions      Giuseppe Maria Coclite  Gisèle Ruiz Goldstein  Jerome A. Goldstein 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2013,64(3):733-753 Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions $$\left\{ \begin{array}{lll} u_{tt} ={\rm div} (\mathcal{A} \nabla u)-\gamma (x,u_t), && \quad {\rm in} \; (0, \infty) \times \Omega,\\ u(0, \cdot)=f, \, u_t(0,\cdot)=g, && \quad {\rm in}\; \Omega, \\ u_{tt} + \beta \partial^ \mathcal{A}_\nu u+c(x)u+ \delta (x,u_t)-q \beta \Lambda_{\rm LB} u=0,&& \quad {\rm on} \;(0, \infty ) \times \partial \Omega . \end{array}\right. $$ for t ≥  0 and ${x \in \Omega \subset \mathbb{R}^N}$ ; the last equation holds on the boundary ?Ω. Here ${\mathcal{A}= \{a_{ij}(x)\}_{ij}}$ is a real, hermitian, uniformly positive definite N × N matrix; ${\beta \in C(\partial \Omega)}$ , with β > 0; ${\gamma:\Omega \times \mathbb{R} \to \mathbb{R}; \delta:\partial \Omega \times \mathbb{R} \to \mathbb{R}; \,c:\partial \Omega \to \mathbb{R}; \, q \ge 0, \Lambda_{\rm LB}}$ is the Laplace–Beltrami operator on ?Ω, and ${\partial^\mathcal{A}_\nu u}$ is the conormal derivative of u with respect to ${\mathcal{A}}$ ; everything is sufficiently regular. We prove explicit stability estimates of the solution u with respect to the coefficients ${\mathcal{A},\,\beta,\,\gamma,\,\delta,\,c,\,q}$ , and the initial conditions f, g. Our arguments cover the singular case of a problem with q = 0 which is approximated by problems with positive q.      19. Dynamical system with boundary control associated with a symmetric semibounded operator      M. I. Belishev  M. N. Demchenko 《Journal of Mathematical Sciences》2013,194(1):8-20 Let L 0 be a closed densely defined symmetric semibounded operator with nonzero defect indices in a separable Hilbert space $\mathcal H$ . It determines a Green system $\{{\mathcal H}, {\mathcal B}; L_0, \Gamma_1, \Gamma_2\}$ , where ${\mathcal B}$ is a Hilbert space, and the $\Gamma_i: {\mathcal H} \to \mathcal B$ are operators connected by the Green formula $$ (L_0^*u, v)_{\mathcal H}-(u,L_0^*v)_{\mathcal H} =(\Gamma_1 u, \Gamma_2 v)_{\mathcal B} - (\Gamma_2 u, \Gamma_1 v)_{\mathcal B}. $$ The boundary space $\mathcal B$ and the boundary operators Γ i are chosen canonically in the framework of the Vishik theory. With the Green system one associates a dynamical system with boundary control (DSBC): $$ \begin{array}{lll} && u_{tt}+L_0^*u = 0, \quad u(t) \in {\mathcal H}, \quad t>0,\\ && u\big|_{t=0}=u_t\big|_{t=0}=0, \\ && \Gamma_1 u = f, \quad f(t) \in {\mathcal B},\quad t \geq 0. \end{array} $$ We show that this system is controllable if and only if the operator L 0 is completely non-self-adjoint. A version of the notion of wave spectrum of L 0 is introduced. It is a topological space determined by L 0 and constructed from reachable sets of the DSBC. Bibliography: 15 titles.      20. Multiple positive solutions for fractional differential systems      Nemat Nyamoradi 《Annali dell'Universita di Ferrara》2012,58(2):359-369 In this paper, we study the existence of positive solution to boundary value problem for fractional differential system $$\left\{\begin{array}{ll}D_{0^+}^\alpha u (t) + a_1 (t) f_1 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1),\\D_{0^+}^\alpha v (t) + a_2 (t) f_2 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1), \;\; 2 < \alpha < 3,\\u (0)= u' (0) = 0, \;\;\;\; u' (1) - \mu_1 u' (\eta_1) = 0,\\v (0)= v' (0) = 0, \;\;\;\; v' (1) - \mu_2 v' (\eta_2) = 0,\end{array}\right.$$ where ${D_{0^+}^\alpha}$ is the Riemann-Liouville fractional derivative of order ??. By using the Leggett-Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.

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).     

Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate

In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition $$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$ where Ω is a bounded smooth domain in ${\mathbb{R}^N}$ for ${N \ge 1, p > 1}$ , and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P _λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.     

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 $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$      

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.     

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain ??. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.     

Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in {\mathbb{R}^{3}}

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{array}\right.$$ where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a(x) has at least one minimum and b(x) has at least one maximum. We first prove the existence of least energy solution (u _ε , φ _ε ) for λ ≠ 0 and ε > 0 sufficiently small. Then we show that u _ε converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.     

Positive solutions of a Lidstone boundary value problem with variable coefficient function

We establish the existence of positive solutions of the Lidstone boundary value problem $$\begin{array}{rcl}(-1)^{n}u^{(2n)}&=&\lambda a(t)f(u),\quad 0<t<1,\\[3pt]u^{(2i)}(0)&=&u^{(2i)}(1)=0,\quad 0\leq i\leq n-1\end{array}$$ for all sufficiently small positive real λ, where the function a may change sign in [0,1] and the function f:[0,∞)→R satisfies f(0)>0. We also show that our assumption is not vacuous.     

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.     

Parabolic power concavity and parabolic boundary value problems

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$ where $\varOmega $ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$ . We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$ .     

Discontinuous superlinear elliptic equations of divergence form

We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem $\left\{ {\begin{array}{lc} {{\rm div} \left( a^{ij} (x,u)D_{j} u \right) = b(x,u,Du) \quad {\rm in}\, \Omega \subset {\mathbb R}^{n}, \, n \ge 2,} \\ {u = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,{\rm on}\, \partial\Omega \in C^{1}. } \\ \end{array}} \right.$ The coefficients a ^ij (x, u) are supposed to be VMO functions with respect to x while the term b(x, u, Du) allows controlled growth with respect to the gradient Du and satisfies a sort of sign-condition with respect to u. Our results correct and generalize the announcements in Ragusa (Nonlinear Differ Equ Appl 13:605–617, 2007, Erratum in Nonlinear Differ Equ Appl 15:277–277, 2008).     

A remark on global existence of solutions of shadow systems

Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d ₁, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].     

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .     

On the Fučik spectrum of non-local elliptic operators

In this article, we study the Fu?ik spectrum of the fractional Laplace operator which is defined as the set of all \({(\alpha, \beta)\in \mathbb{R}^2}\) such that $$\quad \left.\begin{array}{ll}\quad (-\Delta)^s u = \alpha u^{+} - \beta u^{-} \quad {\rm in}\;\Omega \\ \quad \quad \quad u = 0 \quad \quad \quad \qquad {\rm in}\; \mathbb{R}^n{\setminus}\Omega.\end{array}\right\}$$ has a non-trivial solution u, where \({\Omega}\) is a bounded domain in \({\mathbb{R}^n}\) with Lipschitz boundary, n > 2s, \({s \in (0, 1)}\) . The existence of a first nontrivial curve \({\mathcal{C}}\) of this spectrum, some properties of this curve \({\mathcal{C}}\) , e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fu?ik spectrum.     

Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions $$\left\{ \begin{array}{lll} u_{tt} ={\rm div} (\mathcal{A} \nabla u)-\gamma (x,u_t), && \quad {\rm in} \; (0, \infty) \times \Omega,\\ u(0, \cdot)=f, \, u_t(0,\cdot)=g, && \quad {\rm in}\; \Omega, \\ u_{tt} + \beta \partial^ \mathcal{A}_\nu u+c(x)u+ \delta (x,u_t)-q \beta \Lambda_{\rm LB} u=0,&& \quad {\rm on} \;(0, \infty ) \times \partial \Omega . \end{array}\right. $$ for t ≥  0 and ${x \in \Omega \subset \mathbb{R}^N}$ ; the last equation holds on the boundary ?Ω. Here ${\mathcal{A}= \{a_{ij}(x)\}_{ij}}$ is a real, hermitian, uniformly positive definite N × N matrix; ${\beta \in C(\partial \Omega)}$ , with β > 0; ${\gamma:\Omega \times \mathbb{R} \to \mathbb{R}; \delta:\partial \Omega \times \mathbb{R} \to \mathbb{R}; \,c:\partial \Omega \to \mathbb{R}; \, q \ge 0, \Lambda_{\rm LB}}$ is the Laplace–Beltrami operator on ?Ω, and ${\partial^\mathcal{A}_\nu u}$ is the conormal derivative of u with respect to ${\mathcal{A}}$ ; everything is sufficiently regular. We prove explicit stability estimates of the solution u with respect to the coefficients ${\mathcal{A},\,\beta,\,\gamma,\,\delta,\,c,\,q}$ , and the initial conditions f, g. Our arguments cover the singular case of a problem with q = 0 which is approximated by problems with positive q.     

Dynamical system with boundary control associated with a symmetric semibounded operator

Let L ₀ be a closed densely defined symmetric semibounded operator with nonzero defect indices in a separable Hilbert space $\mathcal H$ . It determines a Green system $\{{\mathcal H}, {\mathcal B}; L_0, \Gamma_1, \Gamma_2\}$ , where ${\mathcal B}$ is a Hilbert space, and the $\Gamma_i: {\mathcal H} \to \mathcal B$ are operators connected by the Green formula $$ (L_0^*u, v)_{\mathcal H}-(u,L_0^*v)_{\mathcal H} =(\Gamma_1 u, \Gamma_2 v)_{\mathcal B} - (\Gamma_2 u, \Gamma_1 v)_{\mathcal B}. $$ The boundary space $\mathcal B$ and the boundary operators Γ _i are chosen canonically in the framework of the Vishik theory. With the Green system one associates a dynamical system with boundary control (DSBC): $$ \begin{array}{lll} && u_{tt}+L_0^*u = 0, \quad u(t) \in {\mathcal H}, \quad t>0,\\ && u\big|_{t=0}=u_t\big|_{t=0}=0, \\ && \Gamma_1 u = f, \quad f(t) \in {\mathcal B},\quad t \geq 0. \end{array} $$ We show that this system is controllable if and only if the operator L ₀ is completely non-self-adjoint. A version of the notion of wave spectrum of L ₀ is introduced. It is a topological space determined by L ₀ and constructed from reachable sets of the DSBC. Bibliography: 15 titles.     

Multiple positive solutions for fractional differential systems

In this paper, we study the existence of positive solution to boundary value problem for fractional differential system $$\left\{\begin{array}{ll}D_{0^+}^\alpha u (t) + a_1 (t) f_1 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1),\\D_{0^+}^\alpha v (t) + a_2 (t) f_2 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1), \;\; 2 < \alpha < 3,\\u (0)= u' (0) = 0, \;\;\;\; u' (1) - \mu_1 u' (\eta_1) = 0,\\v (0)= v' (0) = 0, \;\;\;\; v' (1) - \mu_2 v' (\eta_2) = 0,\end{array}\right.$$ where ${D_{0^+}^\alpha}$ is the Riemann-Liouville fractional derivative of order ??. By using the Leggett-Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.