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This article is concerned with the following Kirchhoff equation: −a+b∫R3∣∇u∣2dxΔu=g(u)+h(x)inR3,-\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=g\left(u)+h\left(x)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where aa and bb are positive constants and h≠0h\ne 0. Under the Berestycki-Lions type conditions on gg, we prove that the equation has at least two positive solutions by using variational methods. Furthermore, we obtain the existence of ground state solutions.

For the following quasilinear Choquard-type equation: −Δu−Δ(u2)u+V(x)u=(Iμ*∣u∣p)∣u∣p−2u,x∈RN,-\Delta u-\Delta \left({u}^{2})u+V\left(x)u=\left({I}_{\mu }* {| u| }^{p}){| u| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N≥3,0

In this paper, we study the following Kirchhoff-type problem with critical nonlinearity - a + b ∫ Ω | ∇ u | 2 d x Δ u = λ f (x) | u | p - 2 u + | u | 4 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R 3 , a > 0 is a constant, b , λ are positive parameters and 2 < p < 4 . Under different assumptions on the nonlinearity, the equation has been extensively considered in the case 4 < p < 6 . By contrast, there is no existence result of solutions for the case 2 < p < 4 since the appearance of the nonlocal term. By using some innovative analytical skills, we obtain the existence results about the sign-changing solutions of this problem. Furthermore, we also present asymptotic behaviors of the sign-changing solutions as b ↘ 0 or λ ↘ 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities \[ -\left(a+b\displaystyle\int_{\mathbb{R}^N} |\nabla u|^2\right)\Delta u=\lambda u+|u|^{p-2}u+\mu |u|^{q-2}u \ \hbox{in}\ \mathbb{R}^N, \quad 1\leq N\leq 3 \] − (a + b ∫ R N | ∇u | 2) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass $ \int _{\mathbb {R}^N} u^2=c^2 $ ∫ R N u 2 = c 2 , where $ a{ \gt }0,b{ \gt }0,c{ \gt }0 $ a > 0 , b > 0 , c > 0 , $ \mu { \lt }0 $ μ < 0 , $ 2{ \lt }q\leq 2+\frac {8}{N}{ \lt }p{ \lt }2^{*} $ 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution $ (\tilde {u},\lambda _c)\in S_c \times \mathbb {R} $ (u ~ , λ c) ∈ S c × R , where $ S_c=\{ u\in H^1(\mathbb {R}^N): \int _{\mathbb {R}^N} u^2=c^2 \} $ S c = { u ∈ H 1 (R N) : ∫ R N u 2 = c 2 }. [ABSTRACT FROM AUTHOR]

We study the Schrödinger–Poisson–Slater equation where p ∈ (2 , 3) , λ > 0 , V (x) ∈ C (R 3 , R +) and I 2 (x) : = (4 π | x |) - 1 is the Newtonian potential. We prove the nonexistence of nontrivial solutions, existence of positive nonradial (and radial) ground-state solutions, and mountain-pass-type solutions to (SPS), depending on the values of parameters p and λ . To our knowledge, this is the first study of the existence of ground-state solutions at positive energy levels when p ∈ (2 , 3) . Furthermore, we show that a symmetry breaking occurs for the ground-state solutions, which is a purely nonlocal phenomenon that cannot be observed in the local prototype case. [ABSTRACT FROM AUTHOR]

The existence of L 2–normalized solutions is studied for the equation −Δu+μu=f(x,u) inRN,∫RNu2dx=m. $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: (−Δ+m2)su+V(εx)u=f(u)+u2s*−1inRN,u∈Hs(RN),u>0inRN,\left\{\begin{array}{ll}{\left(-\Delta +{m}^{2})}^{s}u+V\left(\varepsilon x)u=f\left(u)+{u}^{{2}_{s}^{* }-1}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{s}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε>0\varepsilon \gt 0 is a small parameter, s∈(0,1)s\in \left(0,1), m>0m\gt 0, N>2sN\gt 2s, 2s*=2NN−2s{2}_{s}^{* }=\frac{2N}{N-2s} is the fractional critical exponent, (−Δ+m2)s{\left(-\Delta +{m}^{2})}^{s} is the fractional relativistic Schrödinger operator, V:RN→RV:{{\mathbb{R}}}^{N}\to {\mathbb{R}} is a continuous potential, and f:R→Rf:{\mathbb{R}}\to {\mathbb{R}} is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential VV, we construct a family of positive solutions uε∈Hs(RN){u}_{\varepsilon }\in {H}^{s}\left({{\mathbb{R}}}^{N}), with exponential decay, which concentrates around a local minimum of VV as ε→0\varepsilon \to 0.

In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: −Δu−12(x⋅∇u)=μh(x)uq−1+λu−up,x∈RN,u(x)→0,as∣x∣→+∞,\left\{\begin{array}{l}-\Delta u-\frac{1}{2}\left(x\cdot \nabla u)=\mu h\left(x){u}^{q-1}+\lambda u-{u}^{p},\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to +\infty ,\end{array}\right. where N⩾3N\geqslant 3, 00\lambda \gt 0, p>1p\gt 1, μ>0\mu \gt 0 is a parameter and the function h(x)h\left(x) satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and uniqueness of the positive solution for the aforementioned singular elliptic differential equation as λ>N2\lambda \gt \frac{N}{2}. In addition, there is no solution as λ⩽N2\lambda \leqslant \frac{N}{2}. Later, from an experimental point of view, we give the numerical solution of the aforementioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the nonsingular case.

The existence of L2–normalized solutions is studied for the equation − Δ u + μ u = f (x , u) in R N , ∫ R N u 2 d x = m. Here m > 0 and f(x, s) are given, f(x, s) has the L2-subcritical growth and (μ, u) ∈ R × H1(RN) are unknown. In this paper, we employ the argument in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry. [ABSTRACT FROM AUTHOR]

Here, considering - ∞ < a < N - p p , a ≤ e ≤ a + 1 , d = 1 + a - e and p ∗ : = p ∗ (a , e) = Np N - d p , the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials - Δ ap u + V (x) | x | - e p ∗ | u | p - 2 u = | x | - e p ∗ f (u) in R N is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class C 1 . We use Del Pino & Felmer’s arguments to overcome the lack of compactness and the Moser iteration method with Caffarelli–Kohn–Nirenberg inequality to obtain estimates of the solution in L ∞ (R N). [ABSTRACT FROM AUTHOR]

We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity (- Δ) s u (x) = K (x) u - α (x) + μ u p - 1 (x) in Ω , u > 0 in Ω , u = 0 in Ω c : = R N \ Ω , where s ∈ (0 , 1) , α > 0 and Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω and N > 2 s. Under some appropriate assumptions of α , p , μ and K, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of C loc 1 , 1 ∩ L ∞ solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of C 1 , 1 ∩ L ∞ solutions are obtained for star-shaped domain under a condition of K. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following k-coupled nonlinear Schrödinger systems in the critical case: { − Δ u i + λ i u i = μ i u i 3 + ∑ j = 1 , j ≠ i k β i j u i u j 2 in Ω , u i ⩾ 0 in Ω , u i (x) = 0 on ∂ Ω , i = 1 , 2 , ⋯ , k. Here, Ω ⊂ R 4 is a smooth bounded domain, − λ 1 (Ω) < λ i < 0 , μ i > 0 and β i j = β j i ≠ 0 for every i ≠ j , where λ 1 (Ω) is the first eigenvalue of − Δ with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (i.e. 2 N / (N − 2) = 4 when N = 4). We call the couplings β i j are attractive if β i j > 0 , while repulsive stands for β i j < 0. Under the assumption that all the couplings β i j are purely attractive and large enough, we first show that this critical system has a fully nontrivial ground state solution, that is, a solution u → = (u 1 , ... , u k) has all components nontrivial, under conditions providing additional | λ i − λ i ′ | << 1 , | β i j − β i ′ j ′ | << 1 , while ground state solution may be semitrivial ( u → has null components) without the above additional conditions. When the systems admit mixed couplings, i.e. there exist (i , j) and (i ′ , j ′) such that β i j > 0 and β i ′ j ′ < 0 , we establish the existence of least energy positive solutions. The purpose of this paper is to solve some remaining open questions on Tavares and You (Calc. Var. Partial Differential Equations 59:26, 2020). [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of solutions for two classes of singular quasilinear elliptic problems in R N. For the first class of problems, we use a change of variable with global minimization argument and prove the existence and uniqueness of solution. For the second class of problems, we use a change of variable with local minimization argument and an application of a version of the mountain pass theorem combined with a truncation argument to obtain two solutions. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence of normalized solutions (u c , λ c) ∈ H 1 (R 3) × R to the following Kirchhoff problem - a + b ∫ R 3 | ∇ u | 2 d x Δ u + V (x) u + λ u = g (u) + | u | 4 u in R 3 , satisfying the normalization constraint ∫ R 3 u 2 d x = c , where a , b , c > 0 are prescribed constants, and the nonlinearities g(s) are very general and of mass super-critical. Under some suitable assumptions on V(x) and g(u), we will prove that the above problem has a ground state normalized solutions for any given c > 0 , by studying a constraint problem on a Nehari–Pohozaev manifold. [ABSTRACT FROM AUTHOR]

We consider three kinds of anisotropic double phase problems with Dirichlet boundary condition. In two of them we have strong singularity and an unbounded coefficient. Using variational method together with truncation, comparison and approximation techniques, we prove existence and multiplicity theorems for our problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of normalized positive solutions to the following Schrödinger equation with a perturbation - Δ u + λ u = | u | p - 2 u + h (x) in R N , ∫ R N u 2 d x = a , u ∈ H 1 (R N). We treat two cases. First, for the mass-subcritical case 2 < p < 2 + 4 N , we show that there exists a global minimizer with negative energy for arbitrarily positive perturbation. Secondly, for the mass-supercritical case 2 + 4 N < p < 2 ∗ where 2 ∗ = 2 N N - 2 if N ⩾ 3 and 2 ∗ = + ∞ if N = 1 , 2 , when h is small radial positive function, we prove the existence of a mountain pass solution with positive energy. It seems this is the first contribution to the normalized solution for such a perturbed equation. [ABSTRACT FROM AUTHOR]

We study the following nonlinear mass supercritical Schrödinger–Poisson–Slater equation - ▵ u + μ u - γ (| x | - 1 ∗ | u | 2) u - a f (u) = 0 in R 3 satisfying the normalized constrain ∫ R 3 | u | 2 d x = m , and m > 0 prescribed. The nonlinearity f is super-critical. Under some mild assumptions, we establish the existence of ground state and infinitely many radial solutions by constructing a particular bounded Palais-Smale sequence when γ < 0 , a > 0 . Meanwhile, we obtain the non-existence result in the case γ < 0 , a < 0 and the existence result when γ > 0 , a < 0 via variational methods. [ABSTRACT FROM AUTHOR]

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation - Δ p u = λ | u | p - 2 u + a (x) | u | q - 2 u in a bounded domain Ω ⊂ R N , where 1 < q < p , λ ∈ R , and a is a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in { x ∈ Ω : a (x) > 0 } , when the parameter λ lies in a neighborhood of the critical value λ ∗ : = inf ∫ Ω | ∇ u | p d x / ∫ Ω | u | p d x : u ∈ W 0 1 , p (Ω) \ { 0 } , ∫ Ω a | u | q d x ≥ 0 . Among main results, we show that if p > 2 q and either ∫ Ω a φ p q d x = 0 or ∫ Ω a φ p q d x > 0 is sufficiently small, then such solutions do exist in a right neighborhood of λ ∗ . Here φ p is the first eigenfunction of the Dirichlet p-Laplacian in Ω . This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case q > p or in the sublinear case q < p = 2 the nonexistence takes place for any λ ≥ λ ∗ . Moreover, we prove that if p > 2 q and ∫ Ω a φ p q d x > 0 is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of λ ∗ , two of which are strictly positive in { x ∈ Ω : a (x) > 0 } . [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff-type equation: - a + b ∫ R N | ∇ u | 2 d x Δ u + λ V (x) u + μ ϕ | u | p - 2 u = | u | q - 2 u , in R N , (- Δ) α 2 ϕ = | u | p , in R N , where a , b > 0 , 0 < α < N , N ≥ 3 , 2 ≤ p < N + α N - 2 , 2 < q < min { 4 , 2 ∗ } , 2 ∗ = 2 N / (N - 2) and λ , μ > 0 are parameters. Using the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for the above problem when b , μ are sufficiently small, and λ is large enough. Furthermore, the decay rate and the asymptotic behavior of the positive solutions are also explored. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Caffarelli–Kohn–Nirenberg (CKN) inequality: (∫ R N | x | - b (p + 1) | u | p + 1 d x) 2 p + 1 ≤ C a , b , N ∫ R N | x | - 2 a | ∇ u | 2 d x where N ≥ 3 , a < N - 2 2 , a ≤ b ≤ a + 1 and p = N + 2 (1 + a - b) N - 2 (1 + a - b) . It is well-known that up to dilations τ N - 2 2 - a u (τ x) and scalar multiplications Cu(x), the CKN inequality has a unique extremal function W(x) that is positive and radially symmetric in the parameter region b FS (a) ≤ b < a + 1 with a < 0 and a ≤ b < a + 1 with a ≥ 0 and a + b > 0 , where b FS (a) is the Felli–Schneider curve. We prove that in the above parameter region the following stabilities hold: stability of CKN inequality in the functional inequality setting d i s t D a 1 , 2 2 (u , Z) ≲ ‖ u ‖ D a 1 , 2 (R N) 2 - C a , b , N - 1 ‖ u ‖ L p + 1 (| x | - b (p + 1) , R N) 2 where Z = { c W τ ∣ c ∈ R \ { 0 } , τ > 0 } ; stability of CKN inequality in the critical point setting (in the class of nonnegative functions) d i s t D a 1 , 2 (u , Z 0 ν) ≲ Γ (u) , p > 2 or ν = 1 , Γ (u) | log Γ (u) | 1 2 , p = 2 and ν ≥ 2 , Γ (u) p 2 , 1 < p < 2 and ν ≥ 2 , where Γ (u) = ‖ d i v (| x | - a ∇ u) + | x | - b (p + 1) | u | p - 1 u ‖ H - 1 and Z 0 ν = { (W τ 1 , W τ 2 , ... , W τ ν) ∣ τ i > 0 }. Our results generalize the recent work in [7, 11, 15] on the sharp stability of profile decompositions for the special case a = b = 0 (the Sobolev inequality) to the above parameter region of the Caffarelli–Kohn–Nirenberg inequality. This parameter region is optimal for such stabilities in the sense that in the region a < b < b FS (a) with a < 0 , the nonnegative solution of the Euler–Lagrange equation of CKN inequality is no longer unique. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following autonomous Choquard equation − Δ u + u = (I α ∗ F (u)) F ′ (u) in R N , where N ≥ 3 , I α denotes the Riesz potential of order α ∈ (0 , N) and F satisfies general critical growth conditions. By using the variational methods and the Pohožaev manifold techniques, we prove the existence of radially symmetric positive ground state solution. [ABSTRACT FROM AUTHOR]

We study the following Kirchhoff type equation:

In this paper, we study the nonlinear Choquard equation - Δ u + u = | x - y | - μ ∗ G (u) g (u) , in R N , where 0 < μ < min { N , 4 } , N ≥ 3 , g (u) ∈ C (R , R) satisfies very general critical growth conditions in the sense of the Hardy–Littlewood–Sobolev inequality and G (u) = ∫ 0 u g (s) d s . Using the Pohozaev constraint, we find that the above problem admits a ground state solution of Pohozaev type. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the 'charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when 'charge' function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions. [ABSTRACT FROM AUTHOR]

In this study, we consider the following quasilinear Choquard equation with singularity −Δu+V(x)u−uΔu2+λ(Iα∗∣u∣p)∣u∣p−2u=K(x)u−γ,x∈RN,u>0,x∈RN,\left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where Iα{I}_{\alpha } is a Riesz potential, 00\lambda \gt 0. Under suitable assumption on VV and KK, we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ→0\lambda \to 0.

This paper deals with a class of nonlinear scalar field equations with an inhomogeneous perturbation. Two positive solutions were obtained using the variational methods. [ABSTRACT FROM AUTHOR]

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term

In this paper, we investigate the following Chern–Simons–Schrödinger system − Δ u + u + λ ∫ | x | ∞ h (s) s u 2 (s) d s + h 2 (| x |) | x | 2 u = f (x , u) i n R 2 , where λ > 0 , h (s) = ∫ 0 s t 2 u 2 (t) d t and the nonlinearity f (x , s) ∈ C (R 2 × R , R) behaves like e 4 π s 2 as | s | → + ∞. By using the variational methods and the Trudinger–Moser inequality, we obtain the existence of positive solutions for this system. Moreover, we observe the concentrate behavior of mountain-pass type solutions of this system as λ → 0. [ABSTRACT FROM AUTHOR]

This paper investigates the following critical elliptic problem with singularity - Δ u = λ f (x) | u | - γ + Q (x) | u | 4 u , x ∈ R 3 , u > 0 , x ∈ R 3 , where 0 < γ < 1 , λ > 0 , f ∈ L 6 5 + γ (R 3) is a positive function and 0 < Q (x) ∈ C (R 3) . By variational method and Nehari type constraint, we establish a relation between the number of maxima of Q(x) and the number of positive solutions. [ABSTRACT FROM AUTHOR]

We establish the existence of a positive solution to the problem

In this paper, we consider the following Kirchhoff equation:

In this paper, by using variational approach, Mountain Pass Theorem and Krasnoselskii's genus theory, we show the existence and multiplicity of solutions for a Schrödinger–Kirchhoff type equation involving the fractional p. ,. -Laplacian in fractional Sobolev space with variable exponent. We also establish a Bartsch–Wang type compact embedding theorem for fractional Sobolev space with variable exponent. [ABSTRACT FROM AUTHOR]

We consider the Null Mass nonlinear field equation () $$\left\{ {\matrix{ {-\Delta u = f(u){\rm in}\;\;\Omega } \hfill \hfill \hfill \hfill \cr {u > 0} \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \cr {u \vert_{\partial \Omega } = 0} \cr } } \right.$$ where ${\open R}^N \setminus \Omega $ is a bounded regular domain. The existence of a bound state solution is established in situations where this problem does not have a ground state. [ABSTRACT FROM AUTHOR]

We are concerned with the following Schrödinger–Newton problem - ε 2 Δ u + V (x) u = 1 8 π ε 2 ∫ R 3 u 2 (ξ) | x - ξ | d ξ u , x ∈ R 3. For ε small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of V(x). The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schrödinger–Newton problem is quite different from those of Schrödinger equations, which is mainly caused by the nonlocal term. [ABSTRACT FROM AUTHOR]

In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in ℝN{\mathbb{R}^{N}}. After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in H1⁢(ℝN){H^{1}(\mathbb{R}^{N})}. The proofs are based on the Pohozaev manifold and a linking theorem.

We consider the fractional nonlinear Schrödinger equation in this paper. Applying the finite reduction method, we prove that the equation has positive solutions with peaks on a Clifford torus under some suitable conditions. [ABSTRACT FROM AUTHOR]

In this work, we study the following Kirchhoff type problem -(a+b∫Ω|∇u|pdx)Δpu=g(x)u-γ+λf(x,u),inΩ,u=0,on∂Ω,where p≥2, Ω is a regular bounded domain in RN, (N≥3). Firstly, for p>2, we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For p=2, we show, under supplementary condition, the positivity of this solution. Moreover, in the case λ=0 we prove an uniqueness result. We use the variational method to prove our main results. [ABSTRACT FROM AUTHOR]

We investigate the problem

Let A={x∈ℝ2N+2:0 0 in A, ∂u∂ν=0$\frac{\partial u}{\partial \nu } = 0$ on ∂A$\partial A$, where 1

This paper is concerned with the concentration of positive ground states solutions for a modified Schrödinger equation where $$40$$ is a parameter and $$2^{*}:=\frac{2N}{N-2}(N\ge 3)$$ is the critical Sobolev exponent. We prove the existence of a positive ground state solution $$v_{\varepsilon }$$ and $$\varepsilon $$ sufficiently small under some suitable conditions on the nonnegative functions V( x) and K( x). Moreover, $$v_{\varepsilon }$$ concentrates around a global minimum point of V as $$\varepsilon \rightarrow 0$$ . The proof of the main result is based on minimax theorems and concentration compact theory. [ABSTRACT FROM AUTHOR]

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on R N , when N ≥ 2 , (0.1) − Δ N u + V ( x ) | u | N − 2 u = λ | u | r − 2 u + f ( x , u ) . Here, V ( x ) > 0 : R N → R is a suitable potential function, r ∈ ( 1 , N ) , f ( x , u ) is a continuous function of N -superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition , while λ > 0 is a constant. A suitable Moser–Trudinger inequality and the compact embedding W V 1 , N ( R N ) ↪ L r ( R N ) are proved to study problem (0.1) . Moreover, the compact embedding H V 1 ( R N ) ↪ L K t ( R N ) is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation (0.2) − Δ u + V ( x ) u = K ( x ) g ( u ) with potentials vanishing at infinity in a measure-theoretic sense when N ≥ 3 . [ABSTRACT FROM AUTHOR]

We investigate an indefinite superlinear elliptic equation coupled with a sublinear Neumann boundary condition (depending on a positive parameter $$\lambda $$ ), which provides a concave-convex nature to the problem. We establish a global multiplicity result for positive solutions in the spirit of Ambrosetti-Brezis-Cerami and obtain their asymptotic profiles as $$\lambda \rightarrow 0$$ . Furthermore, we also analyse the case where the nonlinearity is concave. Our arguments are based on a bifurcation analysis, a comparison principle, and variational techniques. [ABSTRACT FROM AUTHOR]