Periodic solutions for nonlinear equations with mean curvature-like operators

AbstractWe give an existence result for a periodic boundary value problem involving mean curvature-like operators in the scalar case. Following [R. Manásevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998), 367–393], we use an approach based on the Leray–Schauder degree

Existencia e concentração de solução para o p-Laplaciano com condição de Neumann

Orientador: Yang JianfuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Neste trabalho, vamos estudar a existência de solucão de energia mínima e fenômeno de concentração para o seguinte problema de Neumann quasilinear perturbado, onde, é o operador p-Laplaciano, E é um parâmetro positivo, 1 < p < N, p < q < p* := um domínio limitado suave e 71 é o vetor normal unitário exterior à fronteira de O. No caso subcrítico, p < q < p* := vamos usar métodos variacionais para obter a existência de uma solução UE com energia mínima. Para mostrar que esta solução é não trivial, vamos comparar a energia de UE com a energia do ground state do problema limite. Primeiro vamos mostrar a existência de um ground state para este problema, e usando argumento de blow up estudamos o comportamento assintótico de UE e mostramos que o máximo de UE é assumido em um ponto PE que tende para P E 80, o ponto onde a curvatura generalizada é máxima. No caso crítico, ou seja, quando q = p* usamos uma desigualdade devido a Cherrier [14] para provar uma versão do Lema de concentração de compacidade. Usando este resultado juntamente com argumento de minimização, vamos mostrar a existência de uma solução com energia mínima e estudar o comportamento assintótico da solução por argumento de blow uAbstract: In this work, we study the existence of least energy solutions and phenomenon of concentration for the following Neumann perturbated Quasilinear problem where is the p-Laplacian operator, ¿ is a positive parameter, 1 < p < N, p < q ::; p* is a bounded smooth domain and TJ is the outer unit normal to ano. In the subcritical case p < q < p* := //!p we use variational methods to obtain the existence of solution UE with the least energy. To prove that UE is not trivial, we compare the energy of UE with the energy of ground state of the limit problem. First we show the existence of a ground state for this problem, and then using blow up argument, we study the asymptotic behavior of UE and show that the maximum of UE is assumed at point PE which tends to P E an, the point where generalized curvature maximizes. In the critical case, that is, when q = p* we use an inequality due to Cherrier [14], to prove a version of the compactness of concentration Lemma. Using this result together with the minimizing method we show the existence of a least energy solution and study the asymptotic behavior of the solution by the blow up argumentDoutoradoDoutor em Matemátic

On a class of nonhomogeneous elliptic problems involving exponential critical growth

In this paper we establish the existence of solutions for elliptic equations of the form $-\text{div}(|\nabla u|^{n-2}\nabla u) + V(x)|u|^{n-2}u=g(x,u)+\lambda h$ in $\mathbb{R}^n$ with $n\geq2$. Here the potential $V(x)$ can change sign and the nonlinearity $g(x,u)$ is possibly discontinuous and may exhibit exponential growth. The proof relies on the application of a fixed point result and a version of the Trudinger-Moser inequality

Multiplicity results for some quasilinear elliptic problems

In this paper, we study multiplicity of weak solutions for the following class of quasilinear elliptic problems of the form $$-\Delta_p u -\Delta u = g(u)-\lambda |u|^{q-2}u \quad \text{in } \Omega \text{ with } u=0 \text{ on } \partial\Omega,$$ where $\Omega$ is a bounded domain in ${\mathbb R}^n$ with smooth boundary $\partial\Omega$, $1< q< 2< p\leq n$, $\lambda$ is a real parameter, \Delta_p u = \dive(|\nabla u|^{p-2}\nabla u ) is the $p$-Laplacian and the nonlinearity $g(u)$ has subcritical growth. The proofs of our results rely on some linking theorems and critical groups estimates

Hamiltonian elliptic systems with nonlinearities of arbitrary growth

We study the existence of standing wave solutions for the following class of elliptic Hamiltonian-type systems: \begin{cases} -\hs^2\Delta u+ V(x)u = g(v) & \mbox{in } \mathbb{R}^N, \\ -\hs^2\Delta v+ V(x)v = f(u) & \mbox{in } \mathbb{R}^N, \end{cases} with $N\geq2$, where $\hbar$ is a positive parameter and the nonlinearities $f,g$ are superlinear and can have arbitrary growth at infinity. This system is in variational form and the associated energy functional is strongly indefinite. Moreover, in view of unboundedness of the domain $\mathbb{R}^N$ and the arbitrary growth of nonlinearities we have lack of compactness. We use a dual variational approach in combination with a mountain-pass type procedure to prove the existence of positive solution for $\hbar$ sufficiently small

Weak solutions of quasilinear elliptic eystems via the cohomological index

In this paper we study a class of quasilinear elliptic systems of the type \cases -\divg(a_1(x,\nabla u_1,\nabla u_2))=f_1(x,u_1,u_2) & \text{in } \Omega,\\ -\divg(a_2(x,\nabla u_1,\nabla u_2))=f_2(x,u_1,u_2) & \text{in } \Omega,\\ u_1 = u_2 = 0 & \text{on } \partial \Omega, \endcases with $\Omega$ bounded domain in $\R^N$. We assume that $A\colon \Omega \times {\mathbb{R}}^N\times{\mathbb{R}}^N\rightarrow{\mathbb{R}}$, $F\colon \Omega \times {\mathbb{R}} \times {\mathbb{R}} \rightarrow {\mathbb{R}}$ exist such that $a=(a_1,a_2)=\nabla A$ satisfies the so called Leray-Lions conditions and $f_1={\partial F}/{\partial u_1}$, $f_2={\partial F}/{\partial u_2}$ are Carathéodory functions with {\it subcritical growth}. The approach relies on variational methods and, in particular, on a cohomological local splitting which allows one to prove the existence of a nontrivial solution

On a planar Hartree–Fock type system

This work deals with the existence of solutions for a class of Hartree–Fock type system in the two dimensional Euclidean space. Our approach is variational and based on a minimization technique in the Nehari manifold. The main steps in the prove are some trick estimates from the sign-changing logarithm potential in an appropriate subspace of H1(R2) introduced by Stubbe [23] (see also Cingolani-Weth [8]).Instituto de Ciências Exatas (IE)Departamento de Matemática (IE MAT