Search

In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem (0.1) { (− Δ) s u = β (x) u p − ε in Ω , u = 0 in R n ﹨ Ω , where s ∈ (0 , 1) , n > 2 s , p : = n + 2 s n − 2 s is the Sobolev critical exponent, Ω ⊂ R n is a smooth bounded domain with Lipschitz boundary, (− Δ) s is the fractional Laplace operator and β ∈ C 2 (Ω ‾) is a bounded positive continuous function. We assume that there exists a nondegenerate critical point ξ ⁎ ∈ ∂ Ω of the restriction of β to the boundary ∂Ω such that ∇ (β (ξ ⁎) − n − 2 s 2 s ) ⋅ η (ξ ⁎) > 0. Given any integer k ≥ 1 , we show that for ε > 0 small enough, problem (0.1) has a positive solution, which is a sum of k bubbles which concentrate at ξ ⁎ as ε tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point ξ ⁎. This work can be seen as a nonlocal analogue of the result by Dávila, Faya and Mahmoudi, see [28]. [ABSTRACT FROM AUTHOR]

We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem \begin{document}$ \begin{align} \left\{\begin{array}{ll} (-1)^m (u^{(2m)}(t)+c u^{(2m-2)}(t)) = \lambda a(t)f(u), \; \; \ \ \ t\in (0, r), \\ u^{(2i)}(0) = u^{(2i)}(r) = 0, \ \ i = 0, 1, \cdots, m-1, \end{array} \right.~~(P) \end{align} $\end{document} where $ \lambda > 0 $ is a parameter, $ c $ is a constant, $ m\geq1 $ is an integer, $ a :[0, r]\rightarrow [0, \infty) $ is continuous with $ a\not\equiv0 $ on the subinterval within $ [0, r] $, and $ f: \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function. We analyze the spectrum structure of the corresponding linear eigenvalue problem via the disconjugacy theory and Elias's spectrum theory. As applications of our spectrum results, we show that problem $ (P) $ has nodal solutions under some suitable conditions. The bifurcation technique is used to obtain the main results of this paper.

In this paper, we are interested in the following Kirchhoff type equation 0.1 [ a + λ (∫ R 3 (| ∇ u | 2 + V (| x |) u 2) d x ) α ] (- Δ u + V (| x |) u) = | u | p - 2 u in R 3 , u ∈ H 1 (R 3) , where a , λ > 0 , α ∈ (0 , 2) and p ∈ (2 α + 2 , 6). The potential V(|x|) is radial and bounded below by a positive number. By introducing the Gersgorin Disc’s theorem, we prove that for each positive integer k, Eq. (0.1) has a radial nodal solution U k λ with exactly k nodes. Moreover, the energy of U k λ is strictly increasing in k and for any sequence { λ n } with λ n → 0 + , up to a subsequence, U k λ n converges to U k 0 in H 1 (R 3) , which is also a radial nodal solution with exactly k nodes to the classical Schrödinger equation - a Δ u + a V (| x |) u = | u | p - 2 u in R 3 , u ∈ H 1 (R 3). Our results can be viewed as an extension of Kirchhoff equation concerning the existence of nodal solutions with any prescribed numbers of nodes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified Kirchhoff type equation, that is, the Kirchhoff type equation with a quasilinear term (1) $$\begin{equation} \begin{cases} -\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\right)\Delta u+V(|x|)u-\dfrac{1}{2}\Delta(|u|^2)u\\\noalign{\vskip2.6pt} \quad=|u|^{p-2}u\quad\mbox{in }\mathbb{R}^3,\\ u\to 0 \qquad as\ |x|\to\infty, \end{cases} \end{equation}$$ { − (a + b ∫ R 3 | ∇u | 2 d x) Δu + V (| x |) u − 1 2 Δ (| u | 2) u = | u | p − 2 u in R 3 , u → 0 as | x | → ∞ , where a, b>0, $ p\in (4,22^*) $ p ∈ (4 , 22 ∗) and $ V(|x|) $ V (| x |) is a radial potential function and bounded below by a positive number. The appearance of nonlocal term $ b\int _{\mathbb {R}^3}|\nabla u|^2{\rm d}x\Delta u $ b ∫ R 3 | ∇u | 2 d x Δu and quasilinear term $ \frac {1}{2}\Delta (|u|^2)u $ 1 2 Δ (| u | 2) u makes the variational functional of (1) totally different from the classical Schrödinger equation. By introducing the Miranda theorem, via a construction and gluing method, for any given integer $ k\geq ~1 $ k ≥ 1 , we prove that Equation (1) admits a radial nodal solution $ U_{k}^b $ U k b having exactly k nodes. Moreover, the energy of $ U_{k}^b $ U k b is monotonically increasing in k and for any sequence $ \{b_n\} $ { b n } , and up to a subsequence, $ U_{k}^{b_n} $ U k b n converges strongly to some $ U_{k}^0\neq ~0 $ U k 0 ≠ 0 as $ b_n\to 0_+ $ b n → 0 + , which is a nodal solution with exactly k nodes to the local quasilinear Schrödinger equation (2) $$\begin{equation} \begin{cases} -a\Delta u+V(|x|)u-\dfrac{1}{2}\Delta(|u|^2)u=|u|^{p-2}u\quad\mbox{in }\mathbb{R}^3,\\\noalign{\vskip2.6pt} u\to 0 \qquad as\ |x|\to\infty. \end{cases} \end{equation}$$ { − a Δu + V (| x |) u − 1 2 Δ (| u | 2) u = | u | p − 2 u in R 3 , u → 0 as | x | → ∞. These results improve and generalize the previous results in the literature from the local quasilinear Schrödinger equation to the nonlocal quasilinear Kirchhoff equation. [ABSTRACT FROM AUTHOR]

In this work, we will study the existence of solutions for a class of problems involving the p-Laplacian operator with weight and a nonlinearity that changes sign in a domain $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N with $ \partial \Omega $ ∂Ω smooth and $ N\geq 3 $ N ≥ 3. For this class of problems with subcritical growth, we can show the existence of two solutions, one nodal and one that does not change sign in Ω. To class with critical case, we prove the existence of nontrivial solution which does not change sign in Ω. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following planar Schrödinger–Poisson system 0.1 - Δ u + a (x) u + 2 π ϕ u = | u | p - 2 u e α 0 | u | γ , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where p > 2 , α 0 > 0 and 0 < γ ≤ 2 , the potential a : R 2 → R is invariant under the action of a closed subgroup of the orthogonal transformation group O(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if 0 < γ < 2 and p > 2 . Furthermore, in the critical case γ = 2 and p > 4 , we prove that equation (0.1) possesses a positive solution which is invariant under the same group action. [ABSTRACT FROM AUTHOR]

We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem { (− 1) m (u (2 m) (t) + c u (2 m − 2) (t)) = λ a (t) f (u) , t ∈ (0 , r) , u (2 i) (0) = u (2 i) (r) = 0 , i = 0 , 1 , ⋯ , m − 1 , (P) where λ > 0 is a parameter, c is a constant, m ≥ 1 is an integer, a : [ 0 , r ] → [ 0 , ∞) is continuous with a ≢ 0 on the subinterval within [ 0 , r ] , and f : R → R is a continuous function. We analyze the spectrum structure of the corresponding linear eigenvalue problem via the disconjugacy theory and Elias's spectrum theory. As applications of our spectrum results, we show that problem (P) has nodal solutions under some suitable conditions. The bifurcation technique is used to obtain the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper is concerned to studying the quasilinear Schrödinger equation: - Δ u + V (x) u - γ 2 Δ (u 2) u = | u | p - 2 u , x ∈ R N , where V(x) is a given potential, γ > 0 and either p ∈ (2 , 2 ∗) , 2 ∗ = 2 N N - 2 for N ⩾ 4 or p ∈ (2 , 4) for N = 3 . We establish the existence of arbitrary multiple nodal solutions for the above equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger-Poisson system with a cubic term ... where λ > 0 and the radial function V (x) is an external potential. By taking advantage of the Gersgorin disc theorem and Miranda theorem, via the variational method and blow up analysis, we prove that for each positive integer k, problem (0.1) admits a radial nodal solution U λ k, 4 that changes sign exactly k times. Furthermore, the energy of U λ k, 4 is strictly increasing in k and the asymptotic behavior of U λ k, 4 as λ → 0 + is established. These results extend the existing ones from the super-cubic case in [17] to the cubic case. [ABSTRACT FROM AUTHOR]

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L ∞ and in the Musielak–Orlicz Sobolev space. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of nodal solutions for a third order boundary value problem u ′ ′ ′ (x) = g (u (x)) + p (x , u (x) , u ′ (x) , u ′ ′ (x)) , x ∈ (0 , 1) , u (0) = u (1) = u ′ (1) = 0 , where g : R → R is continuous and satisfies lim | ξ | → ∞ g (ξ) / ξ = ∞ (g is superlinear as | ξ | → ∞ ), p : [ 0 , 1 ] × R 3 → R is continuous and satisfies | p (x , ξ 0 , ξ 1 , ξ 2) | ≤ C + | ξ 0 | / 3 , x ∈ [ 0 , 1 ] , (ξ 0 , ξ 1 , ξ 2) ∈ R 3 , for some C > 0 . We obtain infinitely many solutions having specified nodal properties. The proof of our main result is based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with elliptic problems { − Δ u = f ( u ) + g ( | x | , u , x | x | ⋅ ∇ u ) , x ∈ Ω , u | ∂ Ω = 0 , $$ \textstyle\begin{cases} -\Delta u= f(u)+ g( \vert x \vert ,u,\frac{x}{ \vert x \vert }\cdot \nabla u),&x\in \Omega , \\ u|_{\partial \Omega}=0, \end{cases} $$ where Ω = { x ∈ R N : R 1 < | x | < R 2 } $\Omega =\{x\in \mathbb{R}^{N}:R_{1} 2 $N>2$ , 0 < R 1 < R 2 < ∞ $0< R_{1} 0 $C>0$ , 0 < β < 1 4 ( R 2 − R 1 ) 2 $0

By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k-th mean curvature problem in Minkowski spacetime r N − k v ′ 1 − v ′ 2 k ′ = λ N C N k r N − 1 H k (r , v) in (0 , R) , | v ′ | < 1 in (0 , R) , v ′ (0) = v (R) = 0. As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ. [ABSTRACT FROM AUTHOR]

In this article, we study the following problem - d i v (ω (x) | ∇ u | N - 2 ∇ u) = λ f (x , u) in B , u = 0 on ∂ B , where B is the unit ball in R N , N ≥ 2 and w(x) a singular weight of logarithm type. The reaction source f(x, u) is a radial function with respect to x and is subcritical or critical with respect to a maximal growth of exponential type. By using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory, we prove the existence of nodal solutions. [ABSTRACT FROM AUTHOR]

In this paper, a class of (p , q) -Laplacian equations with critical growth is taken into consideration: − Δ p u − Δ q u + (| u | p − 2 + | u | q − 2) u + λ ϕ | u | q − 2 u = μ g (u) + | u | q ∗ − 2 u , x ∈ R 3 , − Δ ϕ = | u | q , x ∈ R 3 , where Δ ξ u = div (| ∇ u | ξ − 2 ∇ u) is the ξ-Laplacian operator (ξ = p , q) , 3 2 < p < q < 3 , λ and μ are positive parameters, q ∗ = 3 q / (3 − q) is the Sobolev critical exponent. We use a primary technique of constrained minimization to determine the existence, energy estimate and convergence property of nodal (that is, sign-changing) solutions under appropriate conditions on g, and thus generalize the existing results. [ABSTRACT FROM AUTHOR]

We show the existence of nodal solutions of the second-order nonlinear boundary value problem - u ″ x = λ g u x + p x , u x , u ′ x , x ∈ 0 , 1 , u 0 = u 1 = 0 , (P) where λ > 0 is a parameter, p : [0, 1]×ℝ2 → ℝ and g : ℝ →ℝ are continuous functions, and g(0) = 0. For a nonnegative integer k, we say that a solution is nodal if it has only simple zeros in (0, 1) and has exactly k-1 such zeros. Under some suitable conditions, we obtain that there exists λ∗ > 0 (or λ∗ > 0) such that for fixed k ∈ {1, 2,...}, problem (P) has at least one nodal solution for λ ∈ (k2π2/g∞, λ∗) (or λ ∈ (λ∗, k2π2/g∞)), where g∞ = lim|s|→∞g(s)/s. The proof of our main results relies on the bifurcation technique. [ABSTRACT FROM AUTHOR]

In the present paper, we consider nonlinear boundary value problems for fourth-order ordinary differential equations with a boundary condition containing a parameter. Using previously obtained results on the global bifurcation of solutions from zero and infinity to nonlinear fourth-order eigenvalue problems with a spectral parameter in the boundary condition, we establish the existence of nodal solutions to these nonlinear boundary value problems. [ABSTRACT FROM AUTHOR]

In this article, we study the following Kirchhoff equation: (0.1)−(a+b‖∇u‖L2(R3)2)Δu+V(∣x∣)u=f(u)inR3,-(a+b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2})\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where a,b>0a,b\gt 0, VV is a positive radial potential function, and f(u)f\left(u) is an asymptotically cubic term. The nonlocal term b‖∇u‖L2(R3)2Δub\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u is 3-homogeneous in the sense that b‖∇tu‖L2(R3)2Δ(tu)=t3b‖∇u‖L2(R3)2Δub\Vert \nabla tu{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta \left(tu)={t}^{3}b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u, so it competes complicatedly with the asymptotically cubic term f(u)f\left(u), which is totally different from the super-cubic case. By using the Miranda theorem and classifying the domain partitions, via the gluing method and variational method, we prove that for each positive integer kk, equation (0.1) has a radial nodal solution Uk,4b{U}_{k,4}^{b}, which has exactly k+1k+1 nodal domains. Moreover, we show that the energy of Uk,4b{U}_{k,4}^{b} is strictly increasing in kk, and for any sequence {bn}→0+,\left\{{b}_{n}\right\}\to {0}_{+}, up to a subsequence, Uk,4bn{U}_{k,4}^{{b}_{n}} converges strongly to Uk,40{U}_{k,4}^{0} in H1(R3){H}^{1}\left({{\mathbb{R}}}^{3}), where Uk,40{U}_{k,4}^{0} also has k+1k+1 nodal domains exactly and solves the classical Schrödinger equation: −aΔu+V(∣x∣)u=f(u)inR3.-a\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}. Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.

In this paper, we are concerned with elliptic problems { − Δ u = f (u) + g (| x | , u , x | x | ⋅ ∇ u) , x ∈ Ω , u | ∂ Ω = 0 , where Ω = { x ∈ R N : R 1 < | x | < R 2 } is an annular domain, N > 2 , 0 < R 1 < R 2 < ∞ , 4 (R 2 − R 1) (N − 1) ≤ R 1 . f : R → R is continuous and satisfies lim | ξ | → ∞ f (ξ) ξ = ∞ . g : [ R 1 , R 2 ] × R 2 → R is continuous, | g (r , ξ 0 , ξ 1) | ≤ C + β | ξ 0 | for some C > 0 , 0 < β < 1 4 (R 2 − R 1) 2 . We obtain infinitely many radial solutions with prescribed nodal properties using bifurcation techniques. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems − (a + b ∫ R N | ∇ u | 2 d x) △ u + u = f (u) , in R N , where N ≥ 3 , a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and N ≥ 4 , b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for N ≥ 4 and b large enough. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems − (a + b ∫ R N | ∇ u | 2 d x) △ u + u = f (u) , in R N , where N ≥ 3 , a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and N ≥ 4 , b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for N ≥ 4 and b large enough. [ABSTRACT FROM AUTHOR]

We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of Hénon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Robin problem driven by a general nonhomogeneous differential operator plus an indefinite potential term. The reaction is of generalized logistic type. Using variational tools we prove a multiplicity theorem producing three nontrivial solutions with sign information (positive, negative and nodal). In the particular case of (p, 2)-equations, employing also critical groups, we produce a second nodal solution. Our results extend earlier multiplicity results for coercive problems. [ABSTRACT FROM AUTHOR]

This paper studies the global structure of the set of nodal solutions of a generalized Sturm–Liouville boundary value problem associated to the quasilinear equation$ -(\phi(u'))' = \lambda u + a(t)g(u), \quad \lambda\in {\mathbb R}, $where \begin{document}$ a(t) $\end{document} is non-negative with some positive humps separated away by intervals of degeneracy where \begin{document}$ a\equiv 0 $\end{document}. When \begin{document}$ \phi(s) = s $\end{document} this equation includes a generalized prototype of a classical model going back to Moore and Nehari [35], 1959. This is the first paper where the general case when \begin{document}$ {\lambda}\in\mathbb{R} $\end{document} has been addressed when \begin{document}$ a\gneq 0 $\end{document}. The semilinear case with \begin{document}$ a\lneq 0 $\end{document} has been recently treated by López-Gómez and Rabinowitz [28,29,30]. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system: $ \begin{equation*} \left\{\begin{array}{ll} - \left(a+b \int_{ \mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u+V(|x|) u+\lambda\phi u = K(|x|)f(u), & x \in \mathbb{R}^{3}, \\ -\Delta \phi = u^2, & x \in \mathbb{R}^{3}, \end{array}\right. \end{equation*} $ where $ V, K $ are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity $ f $, we develop a direct approach to establish the existence of infinitely many nodal solutions $ \{u_k^{b, \lambda}\} $ with a prescribed number of nodes $ k $, by using the Gersgorin disc's theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of $ \{u_k^{b, \lambda}\} $ is strictly increasing in $ k $, and give a convergence property of $ \{u_k^{b, \lambda}\} $ as $ b\rightarrow 0 $ and $ \lambda \rightarrow 0 $.

In this paper we consider a non-linear Robin problem driven by the Orlicz g -Laplacian operator. Using variational technique combined with a suitable truncation and Morse theory (critical groups), we prove two multiplicity theorems with sign information for all the solutions. In the first theorem, we establish the existence of at least two non-trivial solutions with fixed sign. In the second, we prove the existence of at least three non-trivial solutions with sign information (one positive, one negative, and the other change sign) and order. The result of the nodal solution is new for the non-linear g -Laplacian problems with the Robin boundary condition. [ABSTRACT FROM AUTHOR]

We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation u ′ ′ ′ ′ − M ( ∫ 0 1 | u ′ | 2 d x) u ′ ′ = λ f (x , u) , x ∈ (0 , 1) , u (0) = u (1) = u ′ ′ (0) = u ′ ′ (1) = 0 , where λ ∈ R is a parameter, M : [ 0 , ∞) → [ 0 , ∞) and f : [ 0 , 1 ] × R → R are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

In this article, we are interested in the following Kirchhoff-type problem (0.1)−a+b∫RN∣∇u∣2dxΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN),\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where a,b>0,N=2a,b\gt 0,N=2 or 3, the potential function VV is radial and bounded from below by a positive number. Because the nonlocal b∣∇u∣L2(RN)2Δub| \nabla u\hspace{-0.25em}{| }_{{L}^{2}\left({{\mathbb{R}}}^{N})}^{2}\Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣u∣2u| u\hspace{-0.25em}{| }^{2}u. This causes that not all function in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer kk, equation (0.1) admits a radial nodal solution Uk,4b{U}_{k,4}^{b} having exactly kk nodes. Moreover, we show that the energy of Uk,4b{U}_{k,4}^{b} is strictly increasing in kk and for any sequence {bn}\left\{{b}_{n}\right\} with bn→0+,{b}_{n}\to {0}_{+}, up to a subsequence, Uk,4bn{U}_{k,4}^{{b}_{n}} converges to Uk,40{U}_{k,4}^{0} in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}), which is a radial nodal solution with exactly kk nodes of the classical Schrödinger equation −aΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN).\left\{\begin{array}{l}-a\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}).\end{array}\right. Our results extend the existence result from the super-cubic case to the cubic case.

In this paper, we study the existence of least-energy nodal (sign-changing) weak solutions for a class of fractional Orlicz equations given by (− △ g) α u + g (u) = K (x) f (u) , in R N , where N ⩾ 3 , (− △ g) α is the fractional Orlicz g-Laplace operator, while f ∈ C 1 (R) and K is a positive and continuous function. Under a suitable conditions on f and K, we prove a compact embeddings result for weighted fractional Orlicz–Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show the existence of at least one nodal (sign-changing) weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global bifurcation results for quasilinear elliptic problem − d i v (φ p (∇ y)) = λ a (x) φ p (y) + a (x) f (x , y , λ) + g (x , y , λ) , i n B , y = 0 , o n ∂ B , where λ is a parameter, f , g ∈ C (B × R × R , R). Let B be a unit open ball of R N with a smooth boundary ∂ B. We shall show that there are two distinct unbounded continua C k + and C k − , consisting of the bifurcation branch C k if f is not necessarily differentiable at the origin with respect to φ p (y) , and there are two distinct unbounded continua D k + and D k − , consisting of the bifurcation branch D k if f is not necessarily differentiable at infinity with respect to φ p (y). As the applications of the above result, we shall prove more details about the existence and multiplicity results of sign-changing solutions for the elliptic problem − d i v (φ p (∇ y)) = λ a (x) f (y) + a (x) g (y) , i n B , y = 0 , o n ∂ B , where f , g ∈ C (R , R) and g is not necessarily differentiable at the origin and infinity with respect to φ p (y). Furthermore, by using a comparison theorem, we also obtain a non-existence result of nodal solutions to the above problem. [ABSTRACT FROM AUTHOR]

We study the behavior of solutions for the parametric equation $ -\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0, $ under Dirichlet condition, where $ \Omega \subseteq \mathbb{R}^N $ is a bounded domain with a $ C^2 $-boundary $ \partial \Omega $, $ a_1, a_2 \in L^\infty(\Omega) $ with $ a_1(z), a_2(z) > 0 $ for a.a. $ z \in \Omega $, $ p, q \in (1, \infty) $ and $ \Delta_{p}^{a_1}, \Delta_{q}^{a_2} $ are weighted versions of $ p $-Laplacian and $ q $-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $ f(z, x) $ asymptotically as $ x \to \pm \infty $ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When $ \lambda $ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators.

This paper deals with the following Schrödinger–Poisson system 0.1−Δu+u+λϕu=f(u)inℝ3,−Δϕ=u2inℝ3,$$ \left\{\begin{array}{cc}\hfill &amp; -\Delta u&#x0002B;u&#x0002B;\lambda \phi u&#x0003D;f(u)\kern0.30em \mathrm{in}\kern0.4em {\mathbb{R}}&#x0005E;3,\hfill \\ {}\hfill &amp; -\Delta \phi &#x0003D;{u}&#x0005E;2\kern0.30em \mathrm{in}\kern0.4em {\mathbb{R}}&#x0005E;3,\hfill \end{array}\right. $$where λ>0$$ \lambda &gt;0 $$ and f(u)$$ f(u) $$ is a nonlinear term asymptotically cubic at the infinity. Taking advantage of the Miranda's theorem and deformation lemma, we combine some new analytic techniques to prove that for each positive integer k$$ k $$, system (0.1) admits a radial nodal solution Ukλ$$ {U}_k&#x0005E;{\lambda } $$, which has exactly k+1$$ k&#x0002B;1 $$ nodal domains and the corresponding energy is strictly increasing in k$$ k $$. Moreover, for any sequence {λn}→0+$$ \left\{{\lambda}_n\right\}\to {0}_{&#x0002B;} $$ as n→∞$$ n\to \infty $$, up to a subsequence, Ukλn$$ {U}_k&#x0005E;{\lambda_n} $$ converges to some Uk0∈Hr1(ℝ3)$$ {U}_k&#x0005E;0\in {H}_r&#x0005E;1\left({\mathbb{R}}&#x0005E;3\right) $$, which is a radial nodal solution with exactly k+1$$ k&#x0002B;1 $$ nodal domains of (0.1) for λ=0$$ \lambda &#x0003D;0 $$. These results give an affirmative answer to the open problem proposed in Kim and Seok (2012) about the existence of nodal solutions with prescribed number of nodal domains for the Schrödinger–Poisson system with an asymptotically cubic term. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system: { - ( a + b ∫R³ |∇u|²dx) ∆u + V(|x|)u + λφu = K(|x|)f(u), x ∈ R³, −∆φ = u², x ∈ R³, where V, K are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity f, we develop a direct approach to establish the existence of infinitely many nodal solutions {ub,λ k } with a prescribed number of nodes k, by using the Gersgorin disc’s theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of {u b,λ k } is strictly increasing in k, and give a convergence property of {u b,λ k } as b → 0 and λ → 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multiplicity of nodal solutions for the following Kirchhoff type equations: ⎧ ⎩ ⎨ − ε 2 M (ε 2 − N | | ∇ u | | 2 L 2) Δ u + u = f (x) | u | p − 2 u , in R N , u ∈ H 1 (R N) , { − ε 2 M (ε 2 − N | | ∇ u | | L 2 2) Δ u + u = f (x) | u | p − 2 u , in R N , u ∈ H 1 (R N) , where N ≥ 4 N ≥ 4 , ε > 0 ε > 0 is a small parameter, M (t) = a t + b (a , b > 0) M (t) = a t + b (a , b > 0) and 2 < p < 2 ∗ = 2 N N − 2 2 < p < 2 ∗ = 2 N N − 2. We assume that the weight function f ∈ C (R N , R +) f ∈ C (R N , R +) has k k maximum points in R N R N. By using a novel constraint approach as well as the barycenter map, k 2 k 2 nodal solutions are obtained when N ≥ 4 N ≥ 4 for ε , a ε , a sufficiently small. [ABSTRACT FROM AUTHOR]

We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem − (ε 2 a + ε b ∫ R 3 | ∇ v | 2 d x) Δ v + V (x) v = P (x) f (v) i n R 3 , − (ε 2 a + ε b ∫ R 3 | ∇ v | 2 d x) Δ v + V (x) v = P (x) f (v) i n R 3 , where ε ε is a small positive parameter, a , b > 0 a , b > 0 and V , P ∈ C 1 (R 3 , R) V , P ∈ C 1 (R 3 , R). Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential P P is a positive constant, our result generalizes the result obtained in [ 5 ] to Kirchhoff problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhoff-type problems: (1 + b ∥ u ∥ 2) [ (− Δ) 1 / 2 u + V (x) u ] = f (u) in  Ω , u = 0 in  ℝ ∖ Ω , where b ≥ 0 is a constant, Ω ⊂ ℝ is a bounded open interval, V : Ω ¯ → [ 0 , + ∞) is a continuous potential, the nonlinear term f (u) has exponential growth of Trudinger–Moser type, ∥ u ∥ 2 = 1 2 π [ u ] 1 , 2 2 + ∥ V 1 / 2 u ∥ 2 2 and [ u ] 1 , 2 denotes the standard Gagliardo seminorm of the fractional Sobolev space H 1 / 2 (ℝ). More precisely, by exploring a minimization argument and the quantitative deformation lemma, we establish the existence of a nodal (or sign-changing) solution and by means of the Mountain Pass Theorem, we get one nonpositive and one nonnegative ground state solution. Moreover, we show that the energy of the nodal solution is strictly larger than twice the ground state level. When we regard b as a positive parameter, we study the behavior of the nodal solutions as b → 0. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of least-energy nodal (sign-changing) solutions for a class of critical Schrödinger-Poisson system on the Heisenberg group given by - Δ H u + μ ϕ | u | q - 2 u = λ f (ξ , u) + | u | 2 u , in Ω , - Δ H ϕ = | u | q , in Ω , u = ϕ = 0 , on ∂ Ω , where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 , and Ω ⊂ H 1 is a smooth bounded domain, λ > 0 and μ ∈ R are some real parameters. Under the suitable conditions on f, together with the constraint variational method and the quantitative deformation lemma, we obtain the existence, energy estimates and the convergence property of the least energy sign-changing solution. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient as well as critical nonlinearities. [ABSTRACT FROM AUTHOR]

We investigate the global structure of nodal solutions for the Kirchhoff-type problem $ \left\{\begin{array}{ll} -(a+b\int_{0}^{1}|u'|^2dx)u'' = \lambda f(u),\ x\in (0,1),\\[2ex] u(0) = u(1) = 0, \end{array} \right. $ where $ a > 0, b > 0 $ are real constants, $ \lambda $ is a real parameter. $ f\in C(\mathbb{R}, \mathbb{R}) $ and there exist four constants $ s_1\leq s_2 < 0 < s_3\leq s_4 $ such that $ f(0) = f(s_i) = 0, i = 1, 2, 3, 4 $, $ f(s) > 0 $ for $ s\in(s_1, s_2)\cup(0, s_3)\cup(s_4, +\infty), f(s) < 0 $ for $ s\in(-\infty, s_1)\cup(s_2, 0)\cup(s_3, s_4) $. Under some suitable assumptions on nonlinear terms, we prove the existence of unbounded continua of nodal solutions of this problem which bifurcate from the line of trivial solutions or from infinity, respectively.

In this paper, we study the existence of infinitely many nodal solutions for the following quasilinear elliptic equation \begin{equation*} \begin{cases} -\nabla\cdot\left[\phi'(|\nabla u|^2)\nabla u\right]+|u|^{\alpha-2}u=f(u) , \quad x\in \mathbb{R}^N,\\ u(x)\rightarrow 0, \quad \mbox{as} ~|x|\rightarrow \infty, \end{cases} \end{equation*} where $N\geq2$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, $1

In this paper, we study the global structure of nodal solutions of $$ \begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0 0$ is a parameter, $h\in C([0,1],(0, \infty))$, $f\in C(\mathbb{R})$ and $sf(s)>0$ for $|s|>0$. We show the existence of $S$-shaped component of nodal solutions for the above problem. The proof is based on the bifurcation technique.

This paper deals with following Kirchhoff-type system with critical growth \[\begin{cases} -(a+ b\int _{\mathbb{R}^3}|\nabla u|^{2}dx)\Delta u+ V(x)u+\phi|u|^{p-2}u =|u|^{4}u+\mu f(u), ~\ x\in\mathbb{R}^3,\\ (-\Delta)^{\alpha/2}\phi=l|u|^p, ~\ x\in \mathbb{R}^3, \end{cases}\] where $a, \mu>0$, $b, l\geq0$, $\alpha\in(0,3)$, $p\in[2,3)$ and $\phi|u|^{p-2}u$ is a Hartree-type nonlinearity. By the minimization argument on the nodal Nehari manifold and the quantitative deformation lemma, we prove that the above system has a least energy nodal solution. Our result improve and generalize some interesting results which were obtained in subcritical case.

In this paper, we consider the following Kirchhoff equation with Choquard nonlinearity: 0.1 - a + b ∫ R 3 | ∇ u | 2 d x Δ u + V (x) u = ∫ R 3 | u (y) | p | x - y | 3 - α d y | u | p - 2 u + | u | q - 2 u , in R 3 , where V(x) is a smooth function, a and b are positive constants, α ∈ (1 , 3) , q ∈ (4 , 6) . By employing the results from the matrix theory, gluing approach and Brouwer degree theory, we prove that for any integer k, the above equation with p ∈ (4 , 3 + α) has a sign changing radial solution, which changes sign k times. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of nodal solutions of the following Kirchhoff problem (KP)-(a+b∫RN|∇u|2dx)Δu+u=g(u),inRN,where a,b>0 and N≥3. We first prove that (KP) is equivalent to the following system with respect to (u,λ): (Fλ)-Δu+u=g(u),λ-a-bλN-22∫RN|∇u|2dx=0,inRN×R+.For every integer k>0, radial solutions of (KP) with exactly k nodes are obtained by dealing with the system (Fλ) under some suitable assumptions. Moreover, nonexistence results are proved if N≥4 and b is sufficiently large. [ABSTRACT FROM AUTHOR]