Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques

Abstract

This paper discusses the existence and multiplicity of solutions for a class of p(x)-Kirchhoff type problems with Dirichlet boundary data of the following form − a + b Ω 1 p(x) |∇u| p(x) dx div |∇u| p(x)−2 ∇u = f (x, u) , in Ω u = 0 on ∂Ω , where Ω is a smooth open subset of R N and p ∈ C(Ω) with N < p − = inf x∈Ω p(x) ≤ p + = sup x∈Ω p(x) < +∞, a, b are positive constants and f : Ω × R → R is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

On a m(x)$$ m(x) $$‐polyharmonic Kirchhoff problem without any growth near 0 and Ambrosetti–Rabinowitz conditions

“…According to [37, 38], we notice that $$$ {W}_0&amp;amp;#x0005E;{r,{m}&amp;amp;#x0005E;{&amp;amp;#x0002B;}}\left(\Omega \right)\subset {W}_0&amp;amp;#x0005E;{r,m(x)}\left(\Omega \right) $$$. Consider $$$ \left\{{e}_1,{e}_2,\dots \right\} $$$, a Schauder basis of the space $$$ {W}_0&amp;amp;#x0005E;{r,{m}&amp;amp;#x0005E;{&amp;amp;#x0002B;}}\left(\Omega \right) $$$ which means that each $$$ z\in {W}_0&amp;amp;#x0005E;{r,{m}&amp;amp;#x0005E;{&amp;amp;#x0002B;}}\left(\Omega \right) $$$ has a unique representation $$$ z&amp;amp;#x0003D;\sum \limits_{i&amp;amp;#x0003D;1}&amp;amp;#x0005E;{\infty }{a}_i{e}_i $$$, where $$$ {a}_i $$$ are real numbers.…”

Multiple weak solutions for pi(x)‐Kirchhoff‐type quasilinear elliptic systems

“…(Ω, m i )(see earlier studies [36,37]). With the same argument as proof of Lemma 2.2, consider {e 1 , e 2 , ...} as a Schauder basis of the space D…”

“…Proof Notice that $$$ {D}_0&amp;amp;amp;#x0005E;{1,{p}_i&amp;amp;amp;#x0005E;{&amp;amp;amp;#x0002B;}}\left(\Omega, {m}_i\right)\subset {D}_0&amp;amp;amp;#x0005E;{1,{p}_i(x)}\left(\Omega, {m}_i\right) $$$(see earlier studies [36, 37]). With the same argument as proof of Lemma 2.2, consider $$$ \left\{{e}_1,{e}_2,\dots \right\} $$$ as a Schauder basis of the space $$$ {D}_0&amp;amp;amp;#x0005E;{1,{p}_i&amp;amp;amp;#x0005E;{&amp;amp;amp;#x0002B;}}\left(\Omega, {m}_i\right) $$$.…”

Existence and multiplicity of solutions for $m(x)-$polyharmonic elliptic Kirchhoff type equations without Ambrosetti-Rabinowitz conditions

“…According to [26,40,54,64], we notice that W r,m + where a i are real numbers. We consider for each j ∈ N * that E j , the subspace of W r,m + 0 (Ω) generated by j vectors {e 1 , e 2 , .., e j }.…”