We consider a nonlinear elliptic equation driven by the sum of a p-Laplacian, where 1<q\leq 2\leq p<\infty -superlinear Carathéodory reaction term which doesn't satisfy the usual ...Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the problem has at least three nontrivial solutions; among them one is positive and one is negative.
In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the ...following Kirchhoff-type fractional Laplace equation:
$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$where ( − Δ)s is the fractional Laplace operator, s ∈ (0, 1), N > 2s, Ω is an open bounded subset of ℝN with smooth boundary ∂Ω, $M:{\open R}_0^ + \to {\open R}^ + $ is a continuous function satisfying certain assumptions, and f(x, u) is superlinear at infinity. By computing the critical groups at zero and at infinity, we obtain the existence of non-trivial solutions for the above problem via Morse theory. To the best of our knowledge, our results are new in the study of Kirchhoff–type Laplacian problems.
In this paper we consider the existence of solutions for the quasilinear Schrödinger equation −Δu−kΔ(1+u2)1∕2u2(1+u2)1∕2+V(x)u=g(u)in H1(RN)∩Lloc∞(RN), where N≥3, V is a continuous potential allowed ...to be indefinite, g is a subcritical growth function, and k is a real parameter. By using local linking arguments and computing the critical groups of the energy functional, we obtain the existence of nontrivial solution for the equation.
We consider a nonlinear Robin problem driven by the p-Laplacian plus an indefinite potential. The conditions on the source term are minimal. We prove two multiplicity theorems with sign information ...for all the solutions. In the semilinear case (p=2), we show that we can have multiple nodal solutions. We apply our results to a special class of logistic equations with equidiffusive reaction.
é. We allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue, and we prove several multiplicity results. Our approach uses critical point theory, Morse theory and the ...reduction method (the Lyapunov-Schmidt method).>
We consider a nonautonomous (
p
,
q
)-equation with unbalanced growth and a reaction which exhibits the combined effects of a parametric “concave" (
(
p
-
1
)
-sublinear) term and of a
(
p
-
1
)
...-linear perturbation, which asymptotically stays above the principal eigenvalue
λ
^
1
a
>
0
of the Dirichlet
-
Δ
p
a
operator. Using variational tools, truncation and comparison techniques and critical groups, we show that for all small values of the parameter the problem has at least three nontrivial bounded solutions with sign information (positive, negative, nodal), which are ordered.
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