Version Of Mountain Pass Theorem Research Articles

Multiplicity results for $p$-Kirchhoff modified Schrödinger equations with Stein-Weiss type critical nonlinearity in $\mathbb R^N$

In this article, we consider the following modified quasilinear critical Kirchhoff-Schrödinger problem involving Stein-Weiss type nonlinearity: $$ \mathcal K(u)= \lambda f(x) |u(x)|^{q-2} u(x)+ \Big ( \int_{\mathbb R^N}\frac{|u(y)|^{2p_{\beta,\mu}^{*}}} {|x-y|^{\mu}|y|^{\beta}}dy \Big ) \frac {|u(x)|^{2p_{\beta,\mu}^{*}-2} u(x)}{|x|^\beta} \; \text{ in }\; \mathbb R^N, $$ where $\lambda > 0$ is a parameter, $N\geq 3$, $$ \mathcal K(u) = \Big ( a+b \int_{\mathbb R^N}|\nabla u|^{p}dx \Big ) \Delta_{p} u - a u \Delta_{p}(u^2) $$ with $a > 0$, $b\geq 0$, $\beta\geq0,$ $0 < \mu < N$, $0 < 2\beta+ \mu < N$, $2\leq q < 2 p^*$. Here, $p^*=\frac{Np}{N-p}$ is the Sobolev critical exponent and $ p_{\beta,\mu}^{*}:= \frac p2\frac{(2N-2\beta-\mu)}{N-p} $ is the critical exponent with respect to the doubly weighted Hardy-Littlewood-Sobolev inequality (also called Stein-Weiss type inequality). Then by establishing a concentration-compactness argument for our problem, we show the existence of infinitely many nontrivial solutions to the equations with respect to the parameter $\lambda$ by using Krasnoselskii's genus theory, symmetric mountain pass theorem and $\mathbb Z_2$- symmetric version of mountain pass theorem for different ranges of $q$. We further show that these solutions belong to $L^\infty(\mathbb R^N)$.

Group invariant solutions for the planar Schrödinger-Poisson equations

&lt;abstract&gt;&lt;p&gt;This paper is concerned with the following planar Schrödinger-Poisson equations&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} -\Delta{u}+V(x)u+\left(\ln{|\cdot|}\ast |u|^p\right)|u|^{p-2}u = f(x,u),\; \; \; x\in\mathbb{R}^{2}, \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ p\geq2 $ is a constant, and $ V(x) $ and $ f(x, u) $ are continuous, mirror symmetric or rotationally periodic functions. The nonlinear term $ f(x, u) $ satisfies a certain monotonicity condition and has critical exponential growth in the Trudinger-Moser sense. We adopted a version of mountain pass theorem by constructing a Cerami sequence, which in turn leads to a ground state solution. Our method has two new insights. First, we observed that the integral $ \int_{\mathbb{R}^2}\int_{\mathbb{R}^2}\ln{(|x-y|)}|u(x)|^{p}|u(y)|^pdxdy $ is always negative if $ u $ belongs to a suitable space. Second, we built a new Moser type function to ensure the boundedness of the Cerami sequence, which further guarantees its compactness. In particular, by replacing the monotonicity condition with the Ambrosetti–Rabinowitz condition, our approach works also for the subcritical growth case.&lt;/p&gt;&lt;/abstract&gt;

Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t| p–2 t at infinity

Abstract We consider the following p-harmonic problem Δ ( | Δ u | p − 2 Δ u ) + m | u | p − 2 u = f ( x , u ) , x ∈ R N , u ∈ W 2 , p ( R N ) , $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m &gt; 0 is a constant, N &gt; 2p ≥ 4 and lim t → ∞ f ( x , t ) | t | p − 2 t = l $\begin{array}{} \displaystyle \lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l \end{array}$ uniformly in x, which implies that f(x, t) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f(x, u) ≡ f(u), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.

Existence of a radial solution to a 1-Laplacian problem in [formula omitted

We obtain the existence of a radial solution to the following 1-Laplacian problem −Δ1u+u|u|=Q(x)f(u),inRN,u∈BV(RN).(0.1) The work is carried out in the space of functions of bounded variation BV(RN). The proof of the main result relies on a version of mountain pass theorem without Palais–Smale condition to Lipschitz continuous functionals, and symmetric criticality principle of Palais for non-smooth functionals .

Existence of solution to singular Schrödinger systems involving the fractional p‐Laplacian with Trudinger–Moser nonlinearity in ℝN

In this paper, we study the existence of weak solution for singular fractional Schrödinger system in involving Trudinger–Moser nonlinearity as follows: where N ≥ 1, 0 &lt; s &lt; 1, N = ps, γ ∈ [0, N), and H has exponential growth and does not satisfy the Ambrosetti–Rabinowitz condition. Note that our problem is the lack of compactness. First, we give a version of vanishing lemma due to Lions; using that result and a version of Mountain pass theorem without (PS) condition, we obtain the existence of nontrivial solution to the above system. When H satisfies the Ambrosetti–Rabinowitz condition, motivated by the work of Chen et al., we study the existence of nontrivial solution to singular Schrödinger system involving the fractional (p1, p)‐Laplace and Trudinger–Moser nonlinearity. In our best knowledge, this is the first time the above problems are studied.

On the existence of a global diffeomorphism between Fréchet spaces

We provide sufficient conditions for the existence of a global diffeomorphism between tame Fr\'echet spaces. We prove a version of mountain pass theorem which plays a key ingredient in the proof of the main theorem.

Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion

<p style='text-indent:20px;'>This paper is concerned with a class of biharmonic elliptic differential inclusion in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^N $\end{document}</tex-math></inline-formula>. Under various growth conditions on the nonlinearity, new results on existence and multiplicity of solutions are derived. The main tools used in this paper are the nonsmooth version of mountain pass theorem and the generalized gradient for locally Lipschitz functionals.

Existencia de tres soluciones para el sistema hamiltoniano fraccionario

In this paper we consider the fractional Hamiltonian system given by (0.1) −tDα T(0Dα t u(t)) = ∇F(t,u(t)), a.e t ∈ [0,T] u(0) = u(T) = 0. where α ∈ (1/2,1), t ∈ [0,T], u ∈Rn, F : [0,T]×Rn →R is a given function and ∇F(t,u) is the gradient of F at u. The novelty of this paper is that, using a modified version of mountain pass theorem for functional bounded from below we prove the existence of at least three solutions for (0.2).

Bound state for fractional Schrödinger equation with saturable nonlinearity

In this paper, we study the existence of bound state for the following fractional Schrödinger equation(P)(−Δ)αu+V(x)u=f(u),x∈RN,N≥3,where (−Δ)α with α ∈ (0, 1) is the fractional Laplace operator defined as a pseudo-differential operator with the symbol |ξ|2α, V(x) is a positive potential function and the nonlinearity f is saturable, that is, f(u)/u→l∈(0,+∞) as |u|→+∞. By using a variant version of Mountain Pass Theorem, we prove that there exists a bound state and ground state of (P) when V and f satisfy suitable assumptions.

Existence and multiplicity results for p-laplacian Dirichlet problem with combined nonlinearities

A class of p-laplacian equations with combined nonlinearities is studied. First, when the nonlinear term is asymptotically linear at infinity, existence result of four non-trivial solutions is obtained by Ekeland variational principle and mountain pass theorem. Second, when the nonlinear term is superlinear at positive infinity but asymptotically linear at negative infinity, some existence results for non-trivial solution are established by mountain pass theorem and a variant version of mountain pass theorem.

Nontrivial Solutions for Asymmetric Kirchhoff Type Problems

We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at+∞and−∞inℝN(N=2,3). Namely, it is 4-linear at−∞and 4-superlinear at+∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by combining Mountain Pass Theorem and a variant version of Mountain Pass Theorem with Moser-Trudinger inequality.

Existence of homoclinic solutions for a class of second-order non-autonomous Hamiltonian systems

Abstract By using the variant version of Mountain Pass Theorem, the existence of homoclinic solutions for a class of second-order Hamiltonian systems is obtained. The result obtained generalizes and improves some known works.

Existence of homoclinic orbits for a class of asymptotically p-linear aperiodic p-Laplacian systems

By applying a variant version of Mountain Pass Theorem in critical point theory, the existence of homoclinic solutions is obtained for the following asymptotically p-linear aperiodic p-Laplacian systemddt(|u˙(t)|p-2u˙(t))+∇[-K(t,u(t))+W(t,u(t))]=0,where p∈(1,+∞),t∈R,u∈RN,K,W∈C1(R×RN,R) are not periodic in t and W is asymptotically p-linear at infinity.

Existence of homoclinic solutions for a class of asymptotically quadratic non‐autonomous Hamiltonian systems

AbstractBy using the variant version of Mountain Pass Theorem, the existence of homoclinic solutions for a class of asymptotically quadratic non‐autonomous Hamiltonian systems is obtained. The result obtained generalizes and improves some known works.

Existence of Homoclinic Orbits for a Class of Asymptotically p‐Linear Difference Systems with p‐Laplacian

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically p‐linear difference system with p‐Laplacian Δ(|Δu(n − 1)|p−2Δu(n − 1))+∇[−K(n, u(n)) + W(n, u(n))] = 0, where p ∈ (1, +∞), n ∈ ℤ, u ∈ ℝN, K, W : ℤ × ℝN → ℝ are not periodic in n, and W is asymptotically p‐linear at infinity.

Biharmonic equations with asymptotically linear nonlinearities

This article considers the equation Δ 2 u = f(x,u) with boundary conditions either u|∂Ω= ∂ u ∂ n | ∂ Ω = 0 or u|∂Ω= Δ u|∂Ω=0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R N , N >4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x,t).

Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations

By a variant version of mountain pass theorem, the existence and multiplicity of solutions are obtained for a class of superlinear -Laplacian equations: . In this paper, we suppose neither satisfies the superquadratic condition in Ambrosetti-Rabinowitz sense nor is nondecreasing with respect to .

A Robin problem for a class of quasilinear operators and a related minimizing problem

In this paper we establish the existence of multiple radial solutions for a class of quasilinear operators with nonlinear boundary Robin conditions. Besides other conditions, we consider the nonlinearities having a behavior at -∞ at least like a linearity of slope less than first eigenvalue λ1(R). The technical approach is by variational methods, which is mainly based on a version of Mountain Pass Theorem due to Ghoussoub and Preiss.

A Robin problem for a class of quasilinear operators and a related minimizing problem

In this paper we establish the existence of multiple radial solutions for a class of quasilinear operators with nonlinear boundary Robin conditions. Besides other conditions, we consider the nonlinearities having a behavior at - ∞ at least like a linearity of slope less than first eigenvalue λ 1 ( R ) . The technical approach is by variational methods, which is mainly based on a version of Mountain Pass Theorem due to Ghoussoub and Preiss.

Solutions to Semilinear Elliptic Problems with Combined Nonlinearities

We are concerned with the following nonlinear Dirichlet problem:−Δu=h(x)uq+f(x,u),0⩽u∈H01(Ω),0<q<1 , where Ω is a bounded smooth domain in RN (N⩾1) and h(x)∈L∞(Ω), f(x,s) is asymptotically linear with respect to s at infinity. By a variant version of Mountain Pass Theorem, we prove that there exist at least two nonnegative solutions under suitable assumptions on f(x,s). Our methods also work for the cases where f(x,s) is superlinear or linear in s.