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In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ \Delta^m u = \pm u^{\alpha} \quad \text{ in } \mathbb R^n \] with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants $n$, $m$ and $\alpha$. Remarkably, we are able to provide necessary and sufficient conditions on the exponent $\alpha$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive., Comment: 18 pages, 0 figure

A Sobolev type embedding for radially symmetric functions on the unit ball $B$ in $\mathbb R^n$, $n\geq 3$, into the variable exponent Lebesgue space $L_{2^\star + |x|^\alpha} (B)$, $2^\star = 2n/(n-2)$, $\alpha>0$, is known due to J.M. do \'O, B. Ruf, and P. Ubilla, namely, the inequality \[ \sup\Big\{\int_B |u(x)|^{2^\star+|x|^\alpha} dx : u\in H^1_{0,{\rm rad}}(B), \|\nabla u\|_{L^2(B)} =1\Big\} < +\infty \] holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on $B$, namely, the embedding \[ H^m_{0,{\rm rad}}(B) \hookrightarrow L_{2_m^\star + |x|^\alpha} (B) \] with $2\leq m < n/2$, $2_m^* = 2n/(n-2m)$, and $\alpha>0$ holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime., Comment: 26 pages, 0 figure

Of interest in this note is the following geometric equation, $\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$. It was found by Choi--Xu (J. Differential Equations, 246, 216–234) and McKenna–Reichel (Electron. J. Differential Equations, 37 (2003)) that the condition $q > 1$ is necessary and any positive radially symmetric solution grows at least linearly and at most quadratically at infinity for any $q > 1$. In addition, when $q > 3$ any positive radially symmetric solution is either exactly linear growth or exactly quadratic growth at infinity. Recently, Guerra (J. Differential Equations, {253}, 3147–3157) has shown that the equation always admits a unique positive radially symmetric solution of exactly given linear growth at infinity for any $q > 3$ which is also necessary. In this note, by using the phase-space analysis, we show the existence of infinitely many positive radially symmetric solutions of exactly given quadratic growth at infinity for any $q > 1$, hence completing the picture of positive radially symmetric solutions of the equation.

In this paper, we study global positive $C^{2N}$-solutions of the geometrically interesting equation $(-\Delta)^N u + u^{-(4N-1)}= 0$ in $\mathbf R^{2N-1}$. We prove that any $C^{2N}$-solution $u$ of the equation having linear growth at infinity must satisfy the integral equation \[ u(x) = c_0 \int_{\mathbf R^{2N-1}} {|x - y|{u^{-(4N-1)}}(y)dy} \] for some positive constant $c_0$ and hence takes the following form \[ u(x) = (1+|x|^2)^{1/2} \] in $\mathbf R^{2N-1}$ up to dilations and translations. We also provide several non-existence results for positive $C^{2N}$-solutions of $(-\Delta)^N u = u^{-(4N-1)}$ in $\mathbf R^{2N-1}$., Comment: 15 pages, 0 figure

Using variational methods, we study the non-existence and multi- plicity of non-negative solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions of the form div(jruj p(x) 2 ru) +juj p(x) 2 u = 0 in , jruj p(x) 2 ¶u ¶n = g(x, u) on ¶ , where is a bounded domain with smooth boundary, n the outer unit normal to ¶ and a parameter. Besides, we want to emphasize that g : ¶ (0,1)! R is just a continuous function which may satisfy the Ambrosetti-Rabinowitz type condition or not. The operator div(jruj p(x) 2 ru) is called the p(x)-Laplace operator and it is a nat- ural generalization of the p-Laplace operator in which p(x) = p > 1 is a constant. For this reason the equations studied in the case in which the p(x)-Laplace opera- tor is involved are, in general, extensions of p-Laplacian problems. However, we point out that such generalizations are not trivial since the p(x)-Laplace operator possesses more complicated nonlinearity; for example it is inhomogeneous.

This paper deals with the multiplicity of weak solutions in W 0 1 ( Ω ) to a class of nonuniformly elliptic equations of the form − div ( a ( x , ∇ u ) ) = h ( x ) | u | r − 1 u + g ( x ) | u | s − 1 u in a bounded domain Ω of R N . Here a satisfies | a ( x , ξ ) | ≦ c 0 ( h 0 ( x ) + h 1 ( x ) | ξ | p − 1 ) for all ξ ∈ R N , a.e. x ∈ Ω , h 0 ∈ L p p − 1 ( Ω ) , h 1 ∈ L loc 1 ( Ω ) , h 1 ( x ) ≧ 1 for a.e. x in Ω , 1 r p − 1 s ( N p − N + p ) / ( N − p ) .

In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15, 16, 24], namely existence when the so called TT-tensor $\sigma$ is small and the Yamabe invariant of the manifold is positive. The method we describe is however much simpler than the original method and allows easy extensions to several other problems. Some non-existence results are also considered., Comment: 17 pages, no figure

In this short paper, an integral inequality posed in the 11th International Mathematical Competition for University Students, Skopje, Macedonia, 25–26 July 2004 is generalized.