Dao, Nguyen Anh Díaz, Jesus Ildefonso Nguyen, Quan Ba Hong
Published in
Advanced Nonlinear Studies

We consider the high-dimensional equation ∂ t u - Δ u m + u - β χ { u > 0 } = 0 {\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0} , extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution u ∈ 𝒞 ( [ 0 , T ] ; L δ 1 ( Ω ) ) {u\in\mathca...

Ma, Li
Published in
Advanced Nonlinear Studies

In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete...

Feng, Zhenping Du, Zhuoran
Published in
Advanced Nonlinear Studies

We consider periodic solutions of the following problem associated with the fractional Laplacian: ( - ∂ x x ) s u ( x ) + ∂ u F ( x , u ( x ) ) = 0 {(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in ℝ {\mathbb{R}} . The smooth function F ( x , u ) {F(x,u)} is periodic about x and is a double-well potential with respect to u with we...

Gkikas, Konstantinos T. Nguyen, Phuoc-Tai
Published in
Advanced Nonlinear Studies

Let Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) be a C 2 {C^{2}} bounded domain, and let δ be the distance to ∂ Ω {\partial\Omega} . We study equations ( E ± ) {(E_{\pm})} , - L μ u ± g ( u , | ∇ u | ) = 0 {-L_{\mu}u\pm g(u,\lvert\nabla u\rvert)=0} in Ω, where L μ = Δ + μ δ 2 {L_{\mu}=\Delta+\frac{\mu}{\delta^{2}}} , μ ∈ ( 0 ,...

Pratelli, Aldo Saracco, Giorgio
Published in
Advanced Nonlinear Studies

We prove the validity of the ε - ε β {\varepsilon-\varepsilon^{\beta}} property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

Mouajria, Hattab Tayachi, Slim Weissler, Fred B.
Published in
Advanced Nonlinear Studies

In this paper, we study global well-posedness and long-time asymptotic behavior of solutions to the nonlinear heat equation with absorption, u t - Δ u + | u | α u = 0 {u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0} , where u = u ( t , x ) ∈ ℝ {u=u(t,x)\in\mathbb{R}} , ( t , x ) ∈ ( 0 , ∞ ) × ℝ N {(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and α > 0 ...

Chen, Huyuan Huang, Xia Zhou, Feng
Published in
Advanced Nonlinear Studies

Our purpose in this paper is to study positive solutions of the Lane–Emden equation - Δ u = V u p in ℝ N ∖ { 0 } , -\Delta u=Vu^{p}\quad\text{in }\mathbb{R}^{N}\setminus\{0\}, perturbed by a nonhomogeneous potential V, with p ∈ ( N N - 2 , p c ) p\in(\frac{N}{N-2},p_{c}) , where p c {p_{c}} is the Joseph–Ludgren exponent. We construct a seque...

Filippucci, Roberta Pucci, Patrizia Souplet, Philippe
Published in
Advanced Nonlinear Studies

We consider the elliptic equation - Δ u = u q | ∇ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} for any p > 2 {p>2} and q > 0 {q>0} . We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and su...

Lazzo, Monica Pisani, Lorenzo
Published in
Advanced Nonlinear Studies

We study a Klein–Gordon–Maxwell system in a bounded spatial domain under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many static solutions.

Aouaoui, Sami Jlel, Rahma
Published in
Advanced Nonlinear Studies

In this paper, we establish a new singular Trudinger–Moser type inequality for radial Sobolev spaces with logarithmic weights. The existence of nontrivial solutions is proved for an elliptic equation defined in ℝ n {\mathbb{R}^{n}} , relying on variational methods and involving a nonlinearity with doubly exponential growth at infinity.