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Register a new userExistence of positive solutions for nonlinear systems
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2007
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This paper deals with the existence of positive solutions for the nonlinear system q(t)phi(p(t)u_{i}(t)))+f^{i}(t,textbf{u})=0,quad 0<t<1,quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)hbox{rm :} [0,1]to (0,oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)phi(p(t)u_{i}(t)))+lambda h_{i}(t)g^{i} (textbf{u})=0, quad 0<t<1,quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones.
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It is well known that a single nonlinear fractional Schrodinger equation with a potential $V(x)$ and a small parameter $varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schrodinger systems with a small parameter $varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}left(xright) $. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of fractional Laplacian system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored.
In this paper, we study a class of Schr{o}dinger-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.
We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian and with a reaction which is parametric and exhibits the combined effects of a singular term and of a superdiffusive one. We prove an existence and nonexistence result for positive solutions depending on the value of the parameter $lambda in overset{circ}{mathbb{R}}_+=(0,+infty)$.
In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: begin{equation}label{eqA0.1} onumber begin{cases} -Delta u+lambda_1u=mu_1u^3+alpha_1|u|^{p-2}u+beta v^2uquad&hbox{in}~R^4, -Delta v+lambda_2v=mu_2v^3+alpha_2|v|^{p-2}v+beta u^2vquad&hbox{in}~R^4, end{cases} end{equation} under the mass constraint $$int_{R^4}u^2=a_1^2quadtext{and}quadint_{R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $mu_1,mu_2,beta>0$; $alpha_1,alpha_2in R$, $p!in! (2,4)$ and $lambda_1,lambda_2!in!R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $alpha_i<0(i=1,2)$. Then turning to the case of $alpha_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)to (0,0)$ and $a_1sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr{o}dinger systems with Sobolev critical exponent. When $3leq p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.
We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a W^{1,1}_0 solution which is distributional or entropic, according to the growth assumptions on a lower order term in divergence form.
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