Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight

We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODE$u^{″}+\lambda a(t)g(u)=0,\lambda >0,$ where $g(x)$ is a positive function on ${\mathbb{R}}^{+}$, superlinear at zero and sublinear at infinity, and $a(t)$ is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities $g(x)$ when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.     

Positive mountain pass solutions for a semilinear elliptic equation with a sign-changing weight function

We use the mountain pass theorem to study the existence and multiplicity of positive solutions of the generalisation of the well-known logistic equation -Δu=λg(x)u(x)(1-u(x))$-\mathrm{\Delta }u=\lambda g(x)u(x)(1-u(x))$ with Dirichlet boundary conditions to the case where g changes sign.     

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation

$\lambda u?\text{div}(A(x)Du)=H(x,Du)\text{ in }\mathrm{\Omega }$

where $A(x)$ is a bounded coercive matrix with measurable ingredients, $\lambda \ge 0$ and $\xi ?H(x,\xi )$ has a super linear growth and is convex at infinity. We improve earlier results where the convexity of $H(x,?)$ was required to hold globally.     

Global existence to the initial–boundary value problem for a system of nonlinear diffusion and wave equations

We prove the global existence of weak solution pair to the initial boundary value problem for a system of m-Laplacian type diffusion equation and nonlinear wave equation. The interaction of two equations is given through nonlinear source terms $f(u,v)$ and $g(u,v)$.