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In this paper, we consider the following quasilinear Schrödinger equation involving Stein–Weiss type nonlinearity: - Δ u + V (x) u - Δ (u 2) u = 1 | x | α ∫ R N G (u (y)) | y - x | μ | y | α d y g (u (x)) , in R N , where N ≥ 3 , 0 < μ < N , α ≥ 0 and 2 α + μ < min { N + 2 2 , 4 } and G is the primitive of function g. The potential V : R N → R may decay to zero at infinity. By using variational methods, penalization technique and L ∞ -estimates, we obtain the existence of a positive solution for the above quasilinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

This paper focuses on the following class of fractional magnetic Schrödinger equations: \[ (-\Delta)_{A}^{s}u+V(x)u=g(\vert u\vert^{2})u+\lambda\vert u\vert^{q-2}u, \quad \hbox{in } \mathbb{R}^{N}, \] (− Δ) A s u + V (x) u = g (| u | 2) u + λ | u | q − 2 u , in R N , where $ (-\Delta)_{A}^{s} $ (− Δ) A s is the fractional magnetic Laplacian, $ A :\mathbb {R}^N \rightarrow \mathbb {R}^N $ A : R N → R N is the magnetic potential, $ s\in (0,1) $ s ∈ (0 , 1) , N>2s, $ \lambda \geq 0 $ λ ≥ 0 is a parameter, $ V:\mathbb {R}^N \rightarrow \mathbb {R} $ V : R N → R is a potential function that may decay to zero at infinity and $ g: \mathbb {R}_{+} \rightarrow \mathbb {R} $ g : R + → R is a continuous function with subcritical growth. We deal with supercritical case $ q\geq ~2^*_s:=2N/(N-2s) $ q ≥ 2 s ∗ := 2 N / (N − 2 s). Our approach is based on variational methods combined with penalization technique and $ L^{\infty } $ L ∞ -estimates. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of standing wave solutions for a class of nonlinear fractional Schrödinger-Poisson system involving nonlinearity with subcritical and critical growth. We suppose that the potential V satisfies either Palais-Smale type condition or there exists a bounded domain Ω such that V has no critical point in ∂ Ω . To overcome the "lack of compactness" of the problem, we combine Del Pino-Felmer's penalization technique with Moser's iteration method and some ideas from Alves [1]. [ABSTRACT FROM AUTHOR]

We focus on the following fractional (p, q)-Choquard problem: (−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=1|x|μ*F(u)f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=\left(\frac{1}{\vert x{\vert }^{\mu }}{\ast}F\left(u\right)\right)f\left(u\right) \,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \\ u\in {W}^{s,p}\left({\mathbb{R}}^{N}\right)\cap {W}^{s,q}\left({\mathbb{R}}^{N}\right), u{ >}0\,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \end{cases}$ where ɛ > 0 is a small parameter, 0 < s < 1, 1

This paper concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional p-Laplace operator in ℝ N {\mathbb{R}^{N}} . Precisely, we study the Kirchhoff-type problem ( a + b ⁢ ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ d x ⁢ d y ) ⁢ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = f ⁢ ( x , u )   in  ⁢ ℝ N , \Biggl{(}a+b\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,% \mathrm{d}x\,\mathrm{d}y\Biggr{)}(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u)\quad% \text{in }\mathbb{R}^{N}, where a , b > 0 {a,b>0} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian with 0 < s < 1 < p < N s {0V can have negative values and f fulfills suitable growth assumptions. According to the interaction between the attenuation of the potential at infinity and the behavior of the nonlinear term at the origin, using a penalization argument along with L ∞ {L^{\infty}} -estimates and variational methods, we prove the existence of a positive solution. In addition, we also establish the existence of infinitely many solutions provided the nonlinear term is odd. [ABSTRACT FROM AUTHOR]

This paper is devoted to a class of fractional p-Laplacian equations with the form ε ps (- Δ) p s u + V (x) | u | p - 2 u = f (u) in R N , where s ∈ (0 , 1) , N ≥ 2 , 1 < p < + ∞ , N > p s , (- Δ) p s stands for the fractional p-Laplacian. When f is superlinear and subcritical, we apply the variational gluing method and penalization technique to show the existence of families of positive multi-peak solutions concentrating around isolated components of local minimum of the potential V(x) as ε → 0 . [ABSTRACT FROM AUTHOR]

In this work, we study the existence of positive solutions to the following class of Schrödinger equations involving Stein–Weiss type nonlinearity - Δ u + V (x) u = 1 | x | α ∫ R N F (u (y)) | y - x | μ | y | α d y f (u (x)) , in R N , where N ≥ 3 , 0 < μ < N , α ≥ 0 and 0 < 2 α + μ < min { N + 2 2 , 4 } and F is the primitive of function f. We consider a potential V : R N → R that may decay to zero at infinity. Our approach is based on variational methods combined with penalization technique and L ∞ -estimates. [ABSTRACT FROM AUTHOR]

In this paper, we complete the study started in Ambrosio and Rădulescu (J Math Pures Appl (9) 142:101–145, 2020) on the concentration phenomena for a class of fractional (p, q)-Schrödinger equations involving the fractional critical Sobolev exponent. More precisely, we focus our attention on the following class of fractional (p, q)-Laplacian problems: (- Δ) p s u + (- Δ) q s u + V (ε x) (u p - 1 + u q - 1) = f (u) + u q s ∗ - 1 in R N , u ∈ W s , p (R N) ∩ W s , q (R N) , u > 0 in R N , where ε > 0 is a small parameter, s ∈ (0 , 1) , 1 < p < q < N s , q s ∗ = Nq N - s q is the fractional critical Sobolev exponent, (- Δ) r s , with r ∈ { p , q } , is the fractional r-Laplacian operator, V : R N → R is a positive continuous potential such that inf ∂ Λ V > inf Λ V for some bounded open set Λ ⊂ R N , and f : R → R is a continuous nonlinearity with subcritical growth. With the aid of minimax theorems and the Ljusternik–Schnirelmann category theory, we obtain multiple solutions by employing the topological construction of the set where the potential V attains its minimum. We also establish a multiplicity result when f (t) = t γ - 1 + μ t τ - 1 , with 1 < p < q < γ < q s ∗ < τ and μ > 0 sufficiently small, by combining a truncation argument with a Moser-type iteration. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following class of fractional (p, q)-Laplacian Kirchhoff type problem: 1 + [ u ] s , p p (- Δ) p s u + 1 + [ u ] s , q q (- Δ) q s u + V (ε x) (| u | p - 2 u + | u | q - 2 u) = f (u) in R N , u ∈ W s , p (R N) ∩ W s , q (R N) , u > 0 in R N , where ε > 0 , s ∈ (0 , 1) , 1 < p < q < N s < 2 q , (- Δ) t s , with t ∈ { p , q } , is the fractional t-Laplacian operator, V : R N → R is a positive continuous potential such that inf ∂ Λ V > inf Λ V for some bounded open set Λ ⊂ R N , and f : R → R is a superlinear continuous nonlinearity with subcritical growth at infinity. By combining the method of Nehari manifold, a penalization technique, and the Lusternik–Schnirelman category theory, we study the multiplicity and concentration properties of solutions for the above problem when ε → 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values. [ABSTRACT FROM AUTHOR]

In this paper, we study the following (p , q) -Kirchhoff problem with Choquard nonlinearity: − (1 + a ∫ R N | ∇ u | p d x) Δ p u − (1 + b ∫ R N | ∇ u | q d x) Δ q u + V ε (x) (| u | p − 2 u + | u | q − 2 u) = (| x | − μ ⁎ F (u)) f (u) in R N , where ε is a small positive parameter, a , b are positive constants, 1 < p < q < N , q < 2 p , Δ s u = div (| ∇ u | s − 2 ∇ u) with s ∈ { p , q } is the s -Laplacian, the potential V : R N → R is continuous, V ε (x) = V (ε x) , 0 < μ < q , f is a continuous nonlinearity, and F is the primitive of f. The main result in this paper establishes multiplicity and concentration properties of positive solutions under weaker hypotheses. The proofs combine nonstandard Nehari manifold methods, penalization techniques and the Ljusternik-Schnirelmann category theory. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of nontrivial solutions for the fractional Laplacian resonance problem with a bounded nonlinearity via Morse theory and a penalization technique. [ABSTRACT FROM AUTHOR]

We consider the following class of fractional problems with unbalanced growth: { (− Δ) p s u + (− Δ) q s u + V (ε x) (| u | p − 2 u + | u | q − 2 u) = f (u) in R N , u ∈ W s , p (R N) ∩ W s , q (R N) , u > 0 in R N , where ε > 0 is a small parameter, s ∈ (0 , 1) , 2 ≤ p < q < N s , (− Δ) t s (with t ∈ { p , q }) is the fractional t -Laplacian operator, V : R N → R is a continuous potential satisfying local conditions, and f : R → R is a continuous nonlinearity with subcritical growth. Applying suitable variational and topological arguments, we obtain multiple positive solutions for ε > 0 sufficiently small as well as related concentration properties, in relationship with the set where the potential V attains its minimum. [ABSTRACT FROM AUTHOR]

We deal with the existence of positive solutions for the following fractional Schrodinger equation: e2s(-Δ)su + V (x)u = f(u) in RN, where ε > 0 is a parameter, s ∈ (0, 1), N = 2, (-Δ)s is the fractional Laplacian operator, and V : RN → R is a positive continuous function. Under the assumptions that the nonlinearity f is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of V as e tends tozero. [ABSTRACT FROM AUTHOR]

In this work, a penalization method is discussed in the context of vortex methods for incompressible flows around complex geometries. In particular, we illustrate the method in two cases: the flow around a rotating blade for Reynolds numbers 1000 and 10,000 and the flow past a semi-circular body consisting of a porous layer surrounding a rigid body at Reynolds numbers 550 and 3000. In the latter example, the results are interpreted in terms of control strategy. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper deals with the implementation of a volume penalization technique in a lattice Boltzmann model, in order to compute flows around obstacles. The penalization term was introduced into the lattice Boltzmann equation via a forcing term. This approach was applied to the one dimensional Burgers equation for motionless and moving obstacles (forced motion, and coupling between the fluid force calculated with the penalized Burgers equation and the motion of the obstacle), and to the two dimensional Navier-Stokes equation, for motionless obstacles (flows over a square obstacle, and past a circular cylinder). A good agreement with numerical results obtained with other techniques, and with results found in literature was obtained. [ABSTRACT FROM AUTHOR]

Abstract: The interest on embedded boundary methods increases in Computational Fluid Dynamics (CFD) because they simplify the mesh generation problem in the case of the Navier–Stokes equations. The same simplifications occur for the simulation of multi-physics flows, the coupling of fluid–solid interactions in situation of large motions or deformations, to give a few examples. Nevertheless an accurate treatment of the wall boundary conditions remains an issue of the method. In this work, the wall boundary conditions are easily taken into account through a penalization technique, and the accuracy of the method is recovered using mesh adaptation, thanks to the potential of unstructured meshes. Several classical examples are used to demonstrate that claim. [Copyright &y& Elsevier]

In this paper we consider an asymptotically linear cooperative elliptic system at resonance. By a penalization technique and by the use of Morse theory, we prove, firstly, the existence of a non-zero solution if the difference between the Morse index at zero and the Morse index at infinity is big enough. Later on we will prove the existence of N−1 pairs of non-trivial solutions, where N is, roughly speaking, the difference between the Morse index at zero and the Morse index at infinity. [Copyright &y& Elsevier]